From d39bf69d694e3731e3c2b6eb5316afe87cb16c7c Mon Sep 17 00:00:00 2001 From: Olga Hrydziuszko Date: Tue, 24 Sep 2024 21:27:15 +0200 Subject: [PATCH] Initiate survival session --- session-survival/.Rprofile | 1 + session-survival/.gitignore | 2 + session-survival/_quarto.yml | 28 + session-survival/data/bmtcrr.rda | Bin 0 -> 1505 bytes session-survival/data/data-bmt.rds | Bin 0 -> 233 bytes session-survival/data/data-bmtcrr.rds | Bin 0 -> 1493 bytes session-survival/data/data-opioid.rds | Bin 0 -> 10592 bytes session-survival/data/data_cancer_brain.rds | Bin 0 -> 1233 bytes session-survival/data/data_cancer_brain.tsv | 89 + session-survival/data/opioid_rmph.rData | Bin 0 -> 10621 bytes session-survival/docs/index.html | 621 +++++ session-survival/docs/intro.html | 1344 +++++++++++ .../figure-html/fig-censoring-1.png | Bin 0 -> 63166 bytes .../figure-html/fig-left-interval-1.png | Bin 0 -> 61749 bytes .../figure-html/fig-tooth-filling-1-1.png | Bin 0 -> 57463 bytes 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12 + .../docs/site_libs/bootstrap/bootstrap.min.js | 7 + .../docs/site_libs/clipboard/clipboard.min.js | 7 + .../docs/site_libs/quarto-html/anchor.min.js | 9 + .../docs/site_libs/quarto-html/popper.min.js | 6 + .../quarto-syntax-highlighting.css | 203 ++ .../docs/site_libs/quarto-html/quarto.js | 899 +++++++ .../docs/site_libs/quarto-html/tippy.css | 1 + .../site_libs/quarto-html/tippy.umd.min.js | 2 + .../docs/site_libs/quarto-nav/headroom.min.js | 7 + .../docs/site_libs/quarto-nav/quarto-nav.js | 289 +++ .../quarto-search/autocomplete.umd.js | 3 + .../docs/site_libs/quarto-search/fuse.min.js | 9 + .../site_libs/quarto-search/quarto-search.js | 1286 ++++++++++ session-survival/index.qmd | 19 + session-survival/intro.qmd | 424 ++++ session-survival/lab.qmd | 250 ++ session-survival/notes.qmd | 49 + session-survival/references.bib | 137 ++ session-survival/references.qmd | 4 + session-survival/regression.qmd | 270 +++ session-survival/renv.lock | 1552 ++++++++++++ session-survival/session-survival.Rproj | 13 + session-survival/styles.css | 17 + 55 files changed, 12703 insertions(+) create mode 100644 session-survival/.Rprofile create mode 100644 session-survival/.gitignore create mode 100644 session-survival/_quarto.yml create mode 100644 session-survival/data/bmtcrr.rda create mode 100644 session-survival/data/data-bmt.rds create mode 100644 session-survival/data/data-bmtcrr.rds create mode 100644 session-survival/data/data-opioid.rds create mode 100644 session-survival/data/data_cancer_brain.rds create mode 100644 session-survival/data/data_cancer_brain.tsv create mode 100644 session-survival/data/opioid_rmph.rData create mode 100644 session-survival/docs/index.html create mode 100644 session-survival/docs/intro.html create mode 100644 session-survival/docs/intro_files/figure-html/fig-censoring-1.png create mode 100644 session-survival/docs/intro_files/figure-html/fig-left-interval-1.png create mode 100644 session-survival/docs/intro_files/figure-html/fig-tooth-filling-1-1.png create mode 100644 session-survival/docs/intro_files/figure-html/fig-tooth-filling-2-1.png create mode 100644 session-survival/docs/intro_files/figure-html/fig-tooth-filling-20-part-1.png create mode 100644 session-survival/docs/intro_files/figure-html/fig-tooth-filling-2b-1.png create mode 100644 session-survival/docs/intro_files/figure-html/fig-tooth-filling-cumhaz-1.png create mode 100644 session-survival/docs/lab.html create mode 100644 session-survival/docs/lab_files/figure-html/KM-logrank-Cox-1.png create mode 100644 session-survival/docs/lab_files/figure-html/KM-logrank-Cox-2.png create mode 100644 session-survival/docs/lab_files/figure-html/KM-logrank-Cox-3.png create mode 100644 session-survival/docs/lab_files/figure-html/cmpr-bmt-1.png create mode 100644 session-survival/docs/lab_files/figure-html/cmpr-bmt-2.png create mode 100644 session-survival/docs/references.html create mode 100644 session-survival/docs/regression.html create mode 100644 session-survival/docs/regression_files/figure-html/forest-plot-1.png create mode 100644 session-survival/docs/search.json create mode 100644 session-survival/docs/site_libs/bootstrap/bootstrap-icons.css create mode 100644 session-survival/docs/site_libs/bootstrap/bootstrap-icons.woff create mode 100644 session-survival/docs/site_libs/bootstrap/bootstrap.min.css create mode 100644 session-survival/docs/site_libs/bootstrap/bootstrap.min.js create mode 100644 session-survival/docs/site_libs/clipboard/clipboard.min.js create mode 100644 session-survival/docs/site_libs/quarto-html/anchor.min.js create mode 100644 session-survival/docs/site_libs/quarto-html/popper.min.js create mode 100644 session-survival/docs/site_libs/quarto-html/quarto-syntax-highlighting.css create mode 100644 session-survival/docs/site_libs/quarto-html/quarto.js create mode 100644 session-survival/docs/site_libs/quarto-html/tippy.css create mode 100644 session-survival/docs/site_libs/quarto-html/tippy.umd.min.js create mode 100644 session-survival/docs/site_libs/quarto-nav/headroom.min.js create mode 100644 session-survival/docs/site_libs/quarto-nav/quarto-nav.js create mode 100644 session-survival/docs/site_libs/quarto-search/autocomplete.umd.js create mode 100644 session-survival/docs/site_libs/quarto-search/fuse.min.js create mode 100644 session-survival/docs/site_libs/quarto-search/quarto-search.js create mode 100644 session-survival/index.qmd create mode 100644 session-survival/intro.qmd create mode 100644 session-survival/lab.qmd create mode 100644 session-survival/notes.qmd create mode 100644 session-survival/references.bib create mode 100644 session-survival/references.qmd create mode 100644 session-survival/regression.qmd create mode 100644 session-survival/renv.lock create mode 100644 session-survival/session-survival.Rproj create mode 100644 session-survival/styles.css diff --git a/session-survival/.Rprofile b/session-survival/.Rprofile new file mode 100644 index 00000000..81b960f5 --- /dev/null +++ b/session-survival/.Rprofile @@ -0,0 +1 @@ +source("renv/activate.R") diff --git a/session-survival/.gitignore b/session-survival/.gitignore new file mode 100644 index 00000000..f4d91c9e --- /dev/null +++ b/session-survival/.gitignore @@ -0,0 +1,2 @@ +/.quarto/ +renv/ diff --git a/session-survival/_quarto.yml b/session-survival/_quarto.yml new file mode 100644 index 00000000..bce17f9f --- /dev/null +++ b/session-survival/_quarto.yml @@ -0,0 +1,28 @@ +project: + type: book + output-dir: docs + +book: + title: "Survival Analysis" + author: "Olga Dethlefsen" + # date: "8/31/2022" + chapters: + - index.qmd + - intro.qmd + - regression.qmd + - lab.qmd + - references.qmd + +bibliography: references.bib + +format: + html: + theme: spacebar + code-fold: true + mermaid: + theme: forest + + + + + diff --git a/session-survival/data/bmtcrr.rda b/session-survival/data/bmtcrr.rda new file mode 100644 index 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Survival Analysis

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Author
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Olga Dethlefsen

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Preface

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Aims

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  • to introduce survival analysis
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Learning outcomes

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  • TBD
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Do you see a mistake or a typo? We would be grateful if you let us know via edu.ml-biostats@nbis.se

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This repository contains teaching and learning materials prepared for and used during “Introduction to biostatistics and Machine Learning” and “Biostatistics and Machine Learning II” courses, organized by NBIS, National Bioinformatics Infrastructure Sweden. The courses are open for PhD students, postdoctoral researcher and other employees within Swedish universities. The materials are geared towards life scientists wanting to be able to understand and use the basic statistical and machine learning methods. More about the course https://nbisweden.github.io/workshop-mlbiostatistics/

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1  Introduction

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1.1 Why survival analysis?

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  • The term survival analysis refers to statistical methods developed to study time-to-event data, originally applied to time until death in 17th-century mortality studies, which is where the term “survival” originates.

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  • Over the decades its scope has broadened to include a wide range of applications in both medical studies and beyond, in fields such as marketing, engineering or criminology and many more.

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  • The common factor in survival analysis is the study of censored data that arise when studying a unique kind of outcome variable: time until an event occurs.

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  • This event, can be death, as previously mentioned, but also disease onset or relapse in medical studies. In other fields, we can study time until next purchase (marketing), time until machine failure (engineering), time until a released prisoner commits another crime (criminology), and many more.

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  • In fact, survival analysis is applicable whenever censored data occurs, and these cases are not limited to studying time-based events. We can use survival analysis methods to study the number of treatment sessions until a patient recovers or the number of attempts until a student passes an exam.

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  • Censoring occurs when the event of interest has not yet been observed by the end of the study. This makes survival analysis especially useful for incomplete data sets, where we can still make use of the available information to estimate the time to event.

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  • With survival analysis, we can compare groups (e.g., treatment vs. control) and determine which factors increase or decrease the hazard (risk) of the event occurring over time, even when adjusting for other variables.

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  • We can incorporate time-dependent variables, allowing the analysis to account for changing risk factors, such as fluctuating health conditions or environmental factors, and include multiple states and/or events. For example, a patient may progress through different stages of a disease (multiple states), or a patient dying from different causes may prevent the occurrence of the primary event being studied (multiple events, competing risks).

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1.2 Survival and censoring times

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Let’s imagine a mortality study of terminally ill patients receiving a certain medical treatment. We follow a group of patients over 24 months follow-up period and record the time until death or until the end of the study. For each patient, we suppose there is:

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  • a true survival time \(T\) and
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The survival time represents the time at which the event of interest occurs, here death. The censoring time represents the time at which the patient is lost to follow-up (e.g. withdraws from the study) or the study ends.

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We observe the random variable \[Y = min (T, C)\]

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If the event occurs before censoring (i.e. \(T < C\)) then we observe the true survival time \(T\). Otherwise, if censoring occurs before the event (i.e. \(T > C\)), we observe the censoring time \(C\).

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We also observe a status indicator

+

\[\begin{equation} + \sigma = + \left\{ + \begin{array}{cc} + 1 & \mathrm{if\ } T \le C \\ + 0 & \mathrm{if\ } T>C \\ + \end{array} + \right. +\end{equation}\]

+

A figure below illustrates the data for 10 patients in this ficitious study.

+
+
+Code +
# Example data for 10 patients
+data <- data.frame(
+  patient = factor(1:10),     # Patient IDs
+  time = c(23, 24, 9, 7, 14, 16, 19, 20, 24, 24),  # Time to event or censoring
+  status = c(1, 0, 1, 0, 1, 1, 0, 1, 0, 1)       # Event status (1 = event, 0 = censored)
+)
+
+# Plot the data with horizontal lines
+ggplot(data, aes(x = time, y = patient)) +
+  geom_segment(aes(x = 0, xend = time, y = patient, yend = patient), color = mycols[2]) +  #
+  geom_point(aes(shape = factor(status)), size = 3) +  
+  geom_vline(xintercept = 24, linetype = "dashed") +
+  scale_shape_manual(values = c(1, 16),               
+                     labels = c("Censored", "Event")) +
+  labs(x = "Time (months)", y = "Patient", shape = "Status") +  
+  theme_minimal() +
+  theme(legend.position = "top")
+
+
+
+
+
+ +
+
+Figure 1.1: Example of right-censored data in a mortatily study lasting 24 months. The horizontal lines represent the time to event or censoring for each patient. The points indicate whether the event occurred (event) or not (censored). +
+
+
+
+
+

We we observe 10 \((Y, \sigma)\) pairs, which we denote as \((y_1, \sigma_1, \dots, (y_n, \sigma_n))\), where \(n = 10\). From Figure 1.1, we can see that:

+
    +
  • for patient #1 we get \((y_1= t_1, \space \sigma_1 = 1)\) since we observe event (death) as ca. 23 months, before the end of the follow-up at 24 months.
    • +
  • for patient #2 we get \((y_2= c_1, \space \sigma_1 = 0)\) since the patient is still alive at the end of the study at 24 moths, meaning that the patient has survied at least 24 months (censored)
    • +
  • for patient #3 we get \((y_3= t_3, \space \sigma_3 = 1)\) since we observe event (death) at 9 months
    • +
  • for patients #4, we get \((y_4= c_4, \space \sigma_4 = 0)\), anther censored data point. The study is not over yet, the patient has not diet but is lost to follow-up, e.g. due to withdrawing from the study
    • +
  • analogously, for patients #5, #6, #8 and #10 we observe survival times, and for patients #7 and #9 we observe censoring times.
    • +
+
+
+

1.3 Independent censoring

+

In order to analyze survival data, we need to make some assumptions about the censoring process. One of the key assumptions in survival analysis is independent censoring. This assumption states that the reason for censoring is unrelated to the likelihood of the event occurring. In other words, censored individuals should have the same probability of experiencing the event as those who remain in the study.

+

For example, in a cancer study, if a number of patients leave the study for reasons unrelated to their health status, their data can be treated as censored without biasing the results, since censoring does not provide any information about the outcome. However, if a number of patients drop out early because they are very sick, this would violate the assumption of the independent censoring.

+
+
+

1.4 Left and interval censoring

+

In addition to right-censored data, we may encounter left and/or interval-censored data. Left censoring occurs when the event happened before the observation period started, whereas interval censoring occurs when the event is known to have happened between two specific time points within the study.

+

Imagine an age study in which left censoring occurs when a patient was diagnosed with a disease before the study began, but the exact age of diagnosis is unknown. For example, if a patient is known to have been diagnosed before age 25, but the exact age is not recorded, they are left-censored. Interval censoring occurs when a patient’s diagnosis happens between two observed ages, but the exact age of diagnosis is unknown. For instance, if a patient was healthy at age 45 and diagnosed by age 55, but the precise age of diagnosis isn’t known, their data is interval-censored.

+
+
+Code +
# Create a data frame for 8 patients
+# Left-censored patients were diagnosed before the study, and interval-censored patients were diagnosed between visits
+data <- data.frame(
+  patient_id = 1:8,
+  start_age = c(NA, 30, 40, 35, NA, 25, 45, 50),   # Start of observation or known age for interval censoring (NA for left-censored)
+  end_age = c(25, 35, 45, 50, 20, 40, 55, 60),     # Age of diagnosis or end of observation
+  type = c("Left Censored", "Fully Observed", "Fully Observed", "Fully Observed", 
+                     "Left Censored", "Interval Censored", "Interval Censored", "Fully Observed")
+)
+
+# Plot the data
+ggplot(data, aes(x = end_age, y = factor(patient_id), color = type)) +
+  # Add points for observed ages (diagnosis age or censoring)
+  geom_point(size = 4) +
+  
+  # For interval-censored patients, add a horizontal line to show the interval
+  geom_segment(data = subset(data, type == "Interval Censored"), aes(x = start_age, xend = end_age, y = factor(patient_id), yend = factor(patient_id)),
+               color = mycols[2], linetype = "dashed") +
+  
+  # For left-censored patients, add an arrow showing the uncertainty before diagnosis
+  geom_segment(data = subset(data, type == "Left Censored"), aes(x = 0, xend = end_age, y = factor(patient_id), yend = factor(patient_id)),
+               color = mycols[1], linetype = "dotted", arrow = arrow(length = unit(0.2, "cm"))) +
+  
+  # Customize the plot labels and appearance
+  labs(x = "Age", y = "Patient ID") +
+  scale_color_manual(values = c("Left Censored" = mycols[1], "Fully Observed" = "black", "Interval Censored" = mycols[2])) +
+  theme_minimal() + 
+  theme(legend.position = "top", legend.title = element_blank())
+
+
+
+
+
+ +
+
+Figure 1.2: Example of left and interval censoring in an age study. Left-censored patients were diagnosed before the study, and interval-censored patients were diagnosed between visits. +
+
+
+
+
+

In this session, we will focus on right censored data.

+
+
+

1.5 Kaplan-Meier estimator

+

The survival curve, or survival function, is a decreasing function that quantifies the probability of surviving past time \(t\) and is defined as \[S(t) = Pr(T > t) \tag{1.1}\]

+

The Kaplan-Meier (KM) estimator is a non-parametric statistic used to estimate the survival function from lifetime data and is given by: \[\hat{S}(t) = \prod_{j=1}^{t} \left( 1 - \frac{d_j}{n_j} \right) \tag{1.2}\]

+

where:

+
    +
  • \(d_j\): number of failures at time \(t_j\)
    • +
  • \(n_j\): number of patients at risk just before time \(t_ij\)
    • +
  • and the product is taken over all time intervals in which a death occurred, up to and including \(t\)
    • +
+

Kapalan-Meier estimator is also known as product-limit estimator.

+
+

1.5.1 Tooth filling example

+
+

1.5.1.1 Complete follow-up data

+

Suppose we have 10 patients who receive a tooth filling, and we want to track how long these fillings last before they fail (fall out or need replacement). The failure times (in months) for each patient are recorded:

+
+
+Code +
data_tooth <- data.frame(
+  patientID = 1:10,
+  time = c(10, 8, 11,  5, 4,  3,  7,  6, 12,  8), 
+  surv = 1
+)
+
+str(data_tooth)
+
+
+
'data.frame':   10 obs. of  3 variables:
+ $ patientID: int  1 2 3 4 5 6 7 8 9 10
+ $ time     : num  10 8 11 5 4 3 7 6 12 8
+ $ surv     : num  1 1 1 1 1 1 1 1 1 1
+
+
+

Note that the surv column is a binary indicator of survival status (1 = event, 0 = censored). In this example, we assume all patients have experienced the event (failure) and have complete follow-up data.

+

To calculate the Kaplan-Meier Curve:

+
    +
  1. We sort the data by time to failure.2.
    2. +
  2. For each time, we calculate the survival probabilities, i.e. the probability the filling lasting beyond that time.
    3. +
  3. We apply the Kaplan-Meier formula (Equation 1.2) to calculate the probability of surviving up to time \(t\), as the product of all individual survival probabilities at each time up to \(t\):
    4. +
+

In our example:

+ +++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Time
(months)		Patients at Risk
\(n_i\)		Failures
\(d_i\)		Survival Probability
at \(t_i\)		Cumulative Survival
\(S(t)\)
31011 - \(\frac{1}{10}\) = 0.90.9
4911 - \(\frac{1}{9}\) = 0.8880.9 \(\times\) 0.888 = 0.8
5811 - \(\frac{1}{8}\) = 0.8750.8 \(\times\) 0.875 = 0.7
6711 - \(\frac{1}{7}\) = 0.8570.7 \(\times\) 0.857 = 0.6
7611 - \(\frac{1}{6}\) = 0.8330.6 \(\times\) 0.833 = 0.5
8521 - \(\frac{2}{5}\) = 0.60.5 \(\times\) 0.6 = 0.3
10311 - \(\frac{1}{3}\) = 0.6670.3 \(\times\) 0.667 = 0.2
11211 - \(\frac{1}{2}\) = 0.50.2 \(\times\) 0.5 = 0.1
12111 - \(\frac{1}{1}\) = 00.1 \(\times\) 0 = 0
+

We can see that:

+
    +
  • At 3 months: 10 patients are at risk, and 1 failure occurs, so the survival probability for this period is 0.9.
    • +
  • For 4 months: 9 patients are at risk (since one patient failed at 3 months), and the survival probability is now cumulative (0.9 from the first interval multiplied by 0.888 from this interval).
    • +
  • At time 8 months: two failures occur and the survival probability drops more sharply.
    • +
  • At time 12 months: the survival probability reaches 0, indicating that all patients experienced the event (tooth filling failure) by the end of the study.
    • +
+

We can visualize the Kaplan-Meier curve for this data:

+
+
+Code +
fit.surv <- survfit(Surv(time, surv) ~ 1, data = data_tooth)
+plot(fit.surv, conf.int=FALSE,  xlab = "Months", ylab = "Estiamted Probabiliy of Survival")
+
+
+
+
+
+ +
+
+Figure 1.3: Kaplan-Meier survival curve for tooth filling data. The curve shows the estimated probability of survival (filling intact) over time. +
+
+
+
+
+
+
+

1.5.1.2 Including censored data

+

In real life, not all patients may have complete follow-up data. Some patients might drop out of the study (e.g., move away or lose contact). This is called censoring, and these patients are still considered “alive” (their fillings intact) at the time they are censored, but we do not know what happens to them afterward. The filling can also be intact at the end of the study, but we do not know how long it will last.

+

Let’s update the data with censoring for two patients:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
PatientTime to Failure (months)Censored
13No
24No
35Yes
46No
57No
68No
78No
810Yes
911No
1012No
+

Now, we can adjust the KM calculations to account for censored data. For censored patients, we do not count them in the number of events (failures), but they are still included in the “at-risk” population until they are censored.

+ ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Time
(months)		Patients at Risk
\(n_i\)		Failures
\(d_i\)		CSurvival Probability
at \(t_i\)		Cumulative Survival
\(S(t)\)
31010\(1 - \frac{1}{10} = 0.9\)0.9
4910\(1 - \frac{1}{9} = 0.888\)0.9 \(\times\) 0.888 = 0.8
58010.8
6710\(1 - \frac{1}{7} = 0.857\)0.8 \(\times\) 0.857 = 0.686
7610\(1 - \frac{1}{6} = 0.833\)0.686 \(\times\) 0.833 = 0.571
8520\(1 - \frac{2}{5} = 0.0.6\)0.571 \(\times\) 0.6 = 0.343
103010.343
11210\(1 - \frac{1}{2} = 0.5\)0.343 \(\times\) 0.5 = 0.171
12110\(1 - \frac{1}{1} = 0\)0.171 \(\times\) 0 = 0
+

Censoring affects the calculations:

+
    +
  • At 5 months, one patient is censored, so we do not record a failure, and the cumulative survival rate remains unchanged.
    • +
  • At 10 months, one more patient is censored. Again, there is no effect on the failure probability, but the number of at-risk patients decreases.
    • +
  • In other words, if a patient is censored, they contribute to the number of at-risk patients until their censoring time, but they do not contribute to the event (failure) counts.
    • +
+

We can see how the censoring data affect the KM curve:

+
+
+Code +
data_tooth[1, "surv"] <- 0
+data_tooth[4, "surv"] <- 0
+fit.surv <- survfit(Surv(time, surv) ~ 1, data = data_tooth)
+plot(fit.surv, mark.time=TRUE, conf.int=FALSE,  xlab = "Months", ylab = "Estiamted Probabiliy of Survival")
+
+
+
+
+
+ +
+
+Figure 1.4: Kaplan-Meier survival curve for tooth filling data. The curve shows the estimated probability of survival (filling intact) over time. +
+
+
+
+
+
+
+
+

1.5.2 Interpretations

+

The Kaplan-Meier curve shows the cumulative survival probabilities. A steeper slope indicates a higher event rate (death rate) and therefore a worse survival prognosis. A flatter slope indicates a lower event rate and therefore a better survival prognosis. The curve may have plateaus or flat areas, indicating periods of relatively stable survival.

+

At specific time points, you can estimate the survival probability by locating the time point on the horizontal axis and dropping a vertical line to the curve. Then, read the corresponding survival probability from the vertical axis.

+
    +
  • So, what is the probability that a tooth filling will last 8 months? We can estimate this by locating 8 months on the x-axis and dropping a vertical line to the curve. The estimated survival probability at 10 months is approximately 0.343, meaning there is a 34.3.7% chance that a tooth filling will last 10 months.
    • +
+
+
+Code +
plot(fit.surv, conf.int=FALSE,  xlab = "Months", ylab = "Estiamted Probabiliy of Survival")
+abline(v = 10, h = 0.343, col = "red", lty = 2)
+
+
+
+
+
+ +
+
+Figure 1.5: Kaplan-Meier survival curve for tooth filling data. Red lines indicate the estimated survival probability (tooth being intact) at 10 months. +
+
+
+
+
+

If the interest is the average length of survival, or life expectancy, following the start of treatment, this may be crudely estimated from the survival curve as the time corresponding to a cumulative probability of survival at 0.5. Alternatively, it can be calculated as: \[\textrm{Life expectancy} = 0.5 + \sum(\textrm{length of interval} \times \textrm{cumulative chance of survival})\]

+

If there are multiple curves representing different groups, you can compare their shapes and patterns. If the curves are parallel, it suggests that the groups have similar survival experiences. If the curves diverge or cross, it indicates differences in survival between the groups.

+
+
+Code +
# Create data frame for survival data
+data_tooth_ext <- data.frame(
+  time = c(3, 5, 6, 7, 9, 10, 12, 15, 18, 20, 
+           4, 6, 7, 9, 11, 13, 14, 16, 18, 22),
+  surv = c(1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 
+               1, 1, 0, 1, 1, 1, 1, 0, 1, 1),
+  gender = c(rep("M", 10), rep("F", 10))
+)
+
+# Create a survival object and
+surv_object <- Surv(time = data_tooth_ext$time, event = data_tooth_ext$surv)
+
+# Fit Kaplan-Meier curves for gender
+fit <- survfit(surv_object ~ gender, data = data_tooth_ext)
+
+# Plot Kaplan-Meier survival curves
+ggsurvplot(fit, 
+           data = data_tooth_ext,
+           pval = FALSE,             # Show p-value for difference in survival
+           conf.int = FALSE,         # Add confidence interval
+           risk.table = TRUE,       # Show risk table at the bottom
+           ggtheme = theme_minimal(), # Use a minimal theme
+           xlab = "Time (Months)",
+           ylab = "Survival Probability", 
+           palette = mycols[1:2])
+
+
+
+
+
+ +
+
+Figure 1.6: Kaplan-Meier survival curves for a new extended study of tooth filling data, including 20 participants (10 men and 10 women). +
+
+
+
+
+
+
+
+

1.6 Comparisons of hazards

+

Often, in survival analysis we would like to compare the survival patterns of different groups, e.g. is there a group difference between survival times (tooth filling intact) between men and women in the tooth filling study?

+

The differences between the survival curves are not constant. For example, both groups start at 1 and the survival probability deceases, with lower survival probability for men at 10 months, but then again higher survival probability at 15 months.

+

We solve the problem of allowing for differences in survival time by comparing hazards in the two groups over the duration of the follow-up. We also assume that the ratio of the hazards in the two groups remains constant over time, even if the underlying hazards change. We assume, that at all times \(t\): \[\frac{h_1(t)}{h_0(t)} = constant\]

+

where:

+
    +
  • \(h_1(t)\) and \(h_0(t)\) are the hazards in the exposed and unexposed groups at time \(t\) (or in one group and the other).
    • +
+

This assumption is called the proportional hazards assumption. It is difficult to estimate the hazard directly from the data, since this would give a series of “spikes” when an even occurs, interspersed with zeros when there is no disease event. Instead we use the cumulative hazard function, \(H(t)\). This is the total hazard experienced up to time \(t\), and is estimated by the sum of the risks at each time \(t\) at which an event occurs \[H(t) = \sum_{i=1}^{t}\frac{d_i}{n_i}\]

+

If the ratio of the hazards in both groups is constant over time, if follows that the ratio of the cumulative hazard functions must also equal this constant \[\frac{H_1(t)}{H_0(t)} = \frac{h_1(t)}{h_0(t)} = constant\] and applying the rules of logarithms: \[log(H_1(t) - log(H_0(t)) = log(constant)\]

+

Therefore, if the proportional hazards assumptions is met, then graphs of the log of the cumulative hazard functions should be parallel.

+
+
+Code +
# Plot Kaplan-Meier survival curves
+ggsurvplot(fit, 
+           data = data_tooth_ext,
+           fun = "cumhaz",           # Add cumulative hazard function
+           pval = FALSE,             # Show p-value for difference in survival
+           conf.int = FALSE,         # Add confidence interval
+           risk.table = FALSE,       # Show risk table at the bottom
+           ggtheme = theme_minimal(), # Use a minimal theme
+           xlab = "Time (Months)",
+           ylab = "Cumulative hazard (log)", 
+           palette = mycols[1:2])
+
+
+
+
+
+ +
+
+Figure 1.7: Cumulative hazard functions for the two groups in the extended tooth filling study. +
+
+
+
+
+

Note that it can be shown mathematically that the cumulative hazard is related to the survival function by \(H(t) = -log(S(t))\).

+
+
+

1.7 Mantel-Cox (log-rank test)

+

Assuming the proportional hazards assumption holds, we can test for differences in survival between groups using the Mantel-Cox test, also known as the log-rank test. It tests the null hypothesis of no difference in survival between two or more independent groups. The test compares the entire survival experience between groups and can be thought of as a test of whether the survival curves are identical (overlapping).

+

The log-rank test statistic is constructed by comparing the observed number of events (e.g. deaths, failures) in each group to the expected number of events, assuming that the survival experiences of the groups are the same. The test statistic is based on a chi-square distribution, which tests whether the difference between the observed and expected numbers of events is larger than what we would expect by chance.

+

Let’s start with two groups, A and B. If at time \(t_j\) there were \(d_j\) deaths and there were \(n'_{jA}\) and \(n'_{jB}\) subjects alive just before \(t_j\) in groups A and B respectively, then the data can be arranged in \(2\times2\) table:

+ ++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
DiedSurvivedTotal
Group A\(d_{jA}\)\(r_{jA} = n'_{_jA} - d_{jA}\)\(n'_{jA}\)
Group B\(d_{jB}\)\(r_{jB} = n_{_jB} - d_{jB}\)\(n'_{jB}\)
Total\(d_j = d_{jA} + d_{jB}\)\(r_j = r_{jA} + r_{jB}\)\(n'_j = n'_{jA} + n'_{jB}\)
+

where:

+
    +
  • Group A and Group B represent the two groups being compared (e.g., two different treatments, genders, etc.).
    • +
  • Died (\(d_{jA}\), \(d_{jB}\)): The number of individuals who experienced the event (e.g., death, tooth filling falling out) in each group.
    • +
  • Survived (\(r_{jA}\), \(r_{jB}\)): The number of individuals who did not experience the event (survivors, with tooth filling intact) in each group. This is calculated as the total number in the group minus the number of deaths.
    • +
  • Total (\(n_A\), \(n_B\)): The total number of individuals in each group at risk just before the time of an event
    • +
+

The log-rank test compares the observed number of events to the expected number of events if the survival curves were the same for all groups. The expected number of events in group A at each event time is denoted by: \[E(d_{jA}) = \frac{n'_{jA}d_i}{n'_j}\] and analogously for group B \[E(d_{jB}) = \frac{n'_{jB}d_j}{n'_j}\]

+

The difference between observed and expected number of events is evidence against the null hypothesis. The Mantel-Cox chi-squared (log-rank) its simple for is the combination of these differences over all the times at which deaths (events) occurred: \[\chi^2_{MC} = \frac{(O_A - E_A)^2}{E_A} + \frac{(O_B - E_B)^2}{E_B}\]

+

with 1 \(df\).

+

The test can be easily expanded to more than two groups, by covering more groups in the summation and having \(k-1\) degrees of freedom, where \(k\) is the number of groups.

+

In more advanced version, the log-rank test incorporates variance into the test statistics. This ensures more accurate and reliable comparisons of survival distributions, particularly when groups differ in size, have varying numbers of individuals at risk, or experience uneven censoring: \[\chi^2_{MC} = \frac{(O_A - E_A)^2}{V_A} + \frac{(O_B - E_B)^2}{V_B}\] where \(V\) stands for total variance across all times at which event occurs, with variance at these points calculated as \[V(d_{jA}) = \frac{d_j(n'_j - d_j)n'_{jA}n'_{jB}}{n'^2_j(n'_j -1)}\] for group A (and analogously for other groups in the study).

+

In our extended tooth filling study, we can compare the survival experiences between men and women:

+
+
+Code +
# Perform the log-rank test
+log_rank_test <- survdiff(surv_object ~ gender, data = data_tooth_ext)
+
+# Display the result of the log-rank test
+print(log_rank_test)
+
+
+
Call:
+survdiff(formula = surv_object ~ gender, data = data_tooth_ext)
+
+          N Observed Expected (O-E)^2/E (O-E)^2/V
+gender=F 10        8     9.05     0.121     0.302
+gender=M 10        8     6.95     0.158     0.302
+
+ Chisq= 0.3  on 1 degrees of freedom, p= 0.6
+
+
+

In this example, the chi-square values is 0.3, with \(1\) degrees of freedom. The critical value for \(\chi^2\) at \(\alpha = 0.05\) is approximately \(3.84\) and since 0.3 < \(3.84\) we would conclude that we cannot reject the null hypothesis of no difference in survival between the groups (indicated also by the obtained p-value 0.58).

+ + +
+ +
+ + +
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zn2?kyUPzL%>+S2q87&${hs|Vw5tn#X9~)F`8$pbI-+zuY2<_xRc&6*E=J#iI)V1Jo z{a@|@#uJHRQd1r*num+dIz~o6preP@h0pB$#c^d@F!_Zk3QDOgjD8!%M-d(9l4F_f zE;u+R;s;hO&Mv3Mk1~W-%Q?uSj-CI(jyNxroV&_8hi{Fg_sA%$+ir85)6PT~vE`0~_A z7lsp$EzL(V8D9_m@$~=B4mCse+o^=VDI1$XcvhJ~iDbKeH3vpr4lqmZ&AP&G%na|u zoI%z=M@sv*J{UmVM6hN0Pn&!9|IIiT@y(-7p*GEz%bbmi4bt0WCh($ literal 0 HcmV?d00001 diff --git a/session-survival/docs/lab.html b/session-survival/docs/lab.html new file mode 100644 index 00000000..b2b3a7e8 --- /dev/null +++ b/session-survival/docs/lab.html @@ -0,0 +1,1269 @@ + + + + + + + + + +Survival Analysis - 3  R examples + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
+ + +
+ + + +
+ +
+
+

3  R examples

+
+ + + +
+ + + + +
+ + + +
+ + +
+

3.1 KM, log-rank, Cox proportional hazards model

+
+
rm(list=ls())
+
+# load libraries
+library(tidyverse)
+library(skimr)
+library(survival)
+
+# load data
+data_brain <- readRDS("data/data_cancer_brain.rds")
+
+# preview data
+skim(data_brain)
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Data summary
Namedata_brain
Number of rows88
Number of columns8
_______________________
Column type frequency:
factor4
numeric4
________________________
Group variablesNone
+

Variable type: factor

+ ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
skim_variablen_missingcomplete_rateorderedn_uniquetop_counts
sex01.00FALSE2Fem: 45, Mal: 43
diagnosis10.99FALSE4Men: 42, HG : 22, Oth: 14, LG : 9
loc01.00FALSE2Sup: 69, Inf: 19
stereo01.00FALSE2SRT: 65, SRS: 23
+

Variable type: numeric

+ +++++++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
skim_variablen_missingcomplete_ratemeansdp0p25p50p75p100hist
ki0181.0210.5140.0080.0080.0090.0100.00▁▁▃▇▇
gtv018.668.660.012.506.5112.134.64▇▃▁▁▁
status010.400.490.000.000.001.01.00▇▁▁▁▅
time0127.4620.120.0710.3924.0341.682.56▇▅▅▂▁
+
+
# status = 0, indicates a censored observation 
+# status = 1, indicates an uncensored observation
+
+# Kaplan-Meier survival curve
+fit.surv <- survfit(Surv(time, status) ~ 1, data = data_brain)
+plot(fit.surv, xlab = "Months", ylab = "Estiamted Probabiliy of Survival")
+
+
+
+

+
+
+
+

+# KM stratify by sex
+fit.surv <- survfit(Surv(time, status) ~ sex, data = data_brain)
+plot(fit.surv, mark.time = TRUE, col = c(2, 4), xlab = "Months", ylab = "Estiamted Probabiliy of Survival")
+
+
+
+

+
+
+
+

+# Log-rank test to compare survival of men and women
+logrank_test <- survdiff(Surv(time, status) ~ sex, data = data_brain)
+print(logrank_test)
+## Call:
+## survdiff(formula = Surv(time, status) ~ sex, data = data_brain)
+## 
+##             N Observed Expected (O-E)^2/E (O-E)^2/V
+## sex=Female 45       15     18.5     0.676      1.44
+## sex=Male   43       20     16.5     0.761      1.44
+## 
+##  Chisq= 1.4  on 1 degrees of freedom, p= 0.2
+
+# Fit Cox proportional hazards model
+fit.cox <- coxph(Surv(time, status) ~ sex, data = data_brain)
+summary(fit.cox)
+## Call:
+## coxph(formula = Surv(time, status) ~ sex, data = data_brain)
+## 
+##   n= 88, number of events= 35 
+## 
+##           coef exp(coef) se(coef)     z Pr(>|z|)
+## sexMale 0.4077    1.5033   0.3420 1.192    0.233
+## 
+##         exp(coef) exp(-coef) lower .95 upper .95
+## sexMale     1.503     0.6652     0.769     2.939
+## 
+## Concordance= 0.565  (se = 0.045 )
+## Likelihood ratio test= 1.44  on 1 df,   p=0.2
+## Wald test            = 1.42  on 1 df,   p=0.2
+## Score (logrank) test = 1.44  on 1 df,   p=0.2
+
+# Test the proportional hazards assumption in the two gender groups
+ph_test <- cox.zph(fit.cox)
+print(ph_test)
+##        chisq df    p
+## sex    0.588  1 0.44
+## GLOBAL 0.588  1 0.44
+
+# Fit Cox model with multiple predictors
+fit.cox_multi <- coxph(Surv(time, status) ~ sex + diagnosis + loc + ki + gtv + stereo, data = data_brain)
+summary(fit.cox_multi)
+## Call:
+## coxph(formula = Surv(time, status) ~ sex + diagnosis + loc + 
+##     ki + gtv + stereo, data = data_brain)
+## 
+##   n= 87, number of events= 35 
+##    (1 observation deleted due to missingness)
+## 
+##                        coef exp(coef) se(coef)      z Pr(>|z|)    
+## sexMale             0.18375   1.20171  0.36036  0.510  0.61012    
+## diagnosisLG glioma  0.91502   2.49683  0.63816  1.434  0.15161    
+## diagnosisHG glioma  2.15457   8.62414  0.45052  4.782 1.73e-06 ***
+## diagnosisOther      0.88570   2.42467  0.65787  1.346  0.17821    
+## locSupratentorial   0.44119   1.55456  0.70367  0.627  0.53066    
+## ki                 -0.05496   0.94653  0.01831 -3.001  0.00269 ** 
+## gtv                 0.03429   1.03489  0.02233  1.536  0.12466    
+## stereoSRT           0.17778   1.19456  0.60158  0.296  0.76760    
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+## 
+##                    exp(coef) exp(-coef) lower .95 upper .95
+## sexMale               1.2017     0.8321    0.5930    2.4352
+## diagnosisLG glioma    2.4968     0.4005    0.7148    8.7215
+## diagnosisHG glioma    8.6241     0.1160    3.5664   20.8546
+## diagnosisOther        2.4247     0.4124    0.6678    8.8031
+## locSupratentorial     1.5546     0.6433    0.3914    6.1741
+## ki                    0.9465     1.0565    0.9132    0.9811
+## gtv                   1.0349     0.9663    0.9906    1.0812
+## stereoSRT             1.1946     0.8371    0.3674    3.8839
+## 
+## Concordance= 0.794  (se = 0.04 )
+## Likelihood ratio test= 41.37  on 8 df,   p=2e-06
+## Wald test            = 38.7  on 8 df,   p=6e-06
+## Score (logrank) test = 46.59  on 8 df,   p=2e-07
+
+# Plot survival curves for each diagnosis category, adjusting for other predictors
+# we set the values of the other predictors to the mean for quantitative predictors, 
+# and to the modal value for factors
+
+modeldata <- data.frame(
+  diagnosis = levels(data_brain$diagnosis), 
+  sex = rep("Female", 4), 
+  loc = rep("Supratentorial", 4), 
+  ki = rep(mean(data_brain$ki), 4),
+  gtv = rep(mean(data_brain$gtv), 4),
+  stereo = rep("SRT", 4)
+)
+
+survplots <- survfit(fit.cox_multi, newdata = modeldata)
+plot(survplots, mark.time = FALSE, xlab = "Months", ylab = "Estiamted Probabiliy of Survival", col = 2:5)
+legend("bottomleft", legend = levels(data_brain$diagnosis), col = 2:5, lty = 1)
+
+
+
+

+
+
+
+
+
+
+

3.2 Time-varying predictors

+

adpoted from https://www.bookdown.org/rwnahhas/RMPH/

+

A dataset with time-varying predictors will have multiple rows per individual, with different rows having different values for the time-varying predictors, reflecting how they change over time. Additionally, rather than having a single event time variable, each row will have two time variables indicating the beginning and end of the time interval represented by that row of data.

+
+
data_opioid <- readRDS("data/data-opioid.rds")
+
+data_opioid %>% 
+  filter(RANDID == 10) %>%
+  select(RANDID, wave, START, STOP, heroin, age_at_init, sex, dep_lifetime)
+##   RANDID wave START STOP heroin age_at_init  sex dep_lifetime
+## 1     10    1  3.78 4.26      0          19 Male            0
+## 2     10    2  4.26 4.78      0          19 Male            1
+## 3     10    3  4.78 5.29      0          19 Male            1
+## 4     10    4  5.29 5.84      0          19 Male            1
+## 5     10    5  5.84 6.27      0          19 Male            1
+## 6     10    6  6.27 6.79      1          19 Male            1
+
+

Here, we see a preview of the data_opioid dataset containing longitudinal information for 362 individuals, who at baseline had used non-prescribed pharmaceutical opioids (NPPO, “pain pills”), but were not dependent on NPPOs and had never used heroin.

+
    +
  • Each row contains the time variables START and STOP which define the time interval (years from initiation of NPPO use) associated with that row.
    • +
  • Time-invariant variables in the dataset are constant over all rows for the same individual, while time-varying variables can change between rows.
    • +
  • Each (START, STOP] interval defines a period of time during which no variables changed.
    • +
  • Two time-varying variables in the dataset are heroin use (heroin) (the event indicator variable) and lifetime opioid dependence (dep_lifetime)
    +
    • +
  • For instance, for this particular individual, male, started using NPPOs at age 19 (time invariant variables). He did not meet the criteria for lifetime opioid dependence at baseline (wave = 0), but did at the next interview (wave = 1) and he first reported using heroin at his 6th interview.
    • +
+

To model the heroin usage using both time-invariant and time-varying predictors, we can use the coxph function:

+
+
cox.timevar <- coxph(Surv(START, STOP, heroin) ~
+                     age_at_init + sex + dep_lifetime,
+                   data = data_opioid)
+
+summary(cox.timevar)
+## Call:
+## coxph(formula = Surv(START, STOP, heroin) ~ age_at_init + sex + 
+##     dep_lifetime, data = data_opioid)
+## 
+##   n= 1853, number of events= 27 
+## 
+##                 coef exp(coef) se(coef)      z Pr(>|z|)   
+## age_at_init  -0.3932    0.6749   0.1350 -2.911  0.00360 **
+## sexMale       0.1515    1.1635   0.3925  0.386  0.69956   
+## dep_lifetime  1.0580    2.8807   0.3991  2.651  0.00802 **
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+## 
+##              exp(coef) exp(-coef) lower .95 upper .95
+## age_at_init     0.6749     1.4817    0.5180    0.8794
+## sexMale         1.1635     0.8594    0.5391    2.5111
+## dep_lifetime    2.8807     0.3471    1.3176    6.2981
+## 
+## Concordance= 0.716  (se = 0.042 )
+## Likelihood ratio test= 15.24  on 3 df,   p=0.002
+## Wald test            = 14.53  on 3 df,   p=0.002
+## Score (logrank) test = 15.45  on 3 df,   p=0.001
+
+

Whether a predictor is time-varying or time-invariant, its HR can be interpreted as a comparison of the hazard between groups of individuals with different values of that predictor. Thus, in this example, we could conclude that after adjusting for age at NPPO initiation and sex, those with lifetime opioid dependence have 2.88 times the hazard of using heroin as those who do not (AHR = 2.88; 95% CI = 1.32, 6.30; p = 0.008). The HR for a time-varying predictor, however, can also be interpreted as the effect of within-individual change on the hazard. If an individual without opioid dependence transitions to dependence, their hazard of transitioning to heroin is multiplied by 2.88.

+
+
+

3.3 Competing risks

+
+

3.3.1 Comparing CIFs

+

Let’s demonstrate how to analyze competing risks using the cmprsk package in R. This package includes functions for estimating the cumulative incidence function (CIF) in the presence of competing risks.

+

Bone marrow transplant (BMT) data from the cmprsk package includes information for 35 leukemia cancer patients who underwent bone marrow transplantation. The data includes the following variables:

+
    +
  • dis: disease; 0 = ALL; 1 = AML (ALL, Acute lymphoblastic leukemia; AML, Acute myeloid leukemia)
    • +
  • ftime: follow-up time
    • +
  • status: 0 = censored (survival); 1 = Transplant-related mortality; 2 = relapse
    • +
+
+
library(cmprsk)
+library(survminer)
+
+# preview data
+data_bmt <- readRDS("data/data-bmt.rds")
+str(data_bmt)
+## 'data.frame':    35 obs. of  3 variables:
+##  $ dis   : int  0 0 0 0 0 1 0 0 1 1 ...
+##  $ ftime : int  13 1 72 7 8 67 9 5 70 4 ...
+##  $ status: int  2 1 0 2 2 0 2 2 0 0 ...
+
+# label levels
+data_bmt <- data_bmt %>%
+  mutate(dis = factor(dis, levels = c(0,1), labels = c("ALL", "AML")), 
+         status = factor(status, levels = c(0,1,2), labels = c("Censored","Mortality","Relapse")))
+
+str(data_bmt)
+## 'data.frame':    35 obs. of  3 variables:
+##  $ dis   : Factor w/ 2 levels "ALL","AML": 1 1 1 1 1 2 1 1 2 2 ...
+##  $ ftime : int  13 1 72 7 8 67 9 5 70 4 ...
+##  $ status: Factor w/ 3 levels "Censored","Mortality",..: 3 2 1 3 3 1 3 3 1 1 ...
+
+# Estimate the cumulative incidence functions for each event type
+cif <- cuminc(ftime = data_bmt$ftime, # Failure time variable
+              fstatus = data_bmt$status) # Codes for different causes of failure
+
+# Print the CIF object
+print(cif)
+## Estimates and Variances:
+## $est
+##                    20        40        60
+## 1 Censored  0.1142857 0.2000000 0.2000000
+## 1 Mortality 0.2571429 0.2571429 0.2571429
+## 1 Relapse   0.3714286 0.4285714 0.4285714
+## 
+## $var
+##                      20          40          60
+## 1 Censored  0.002998057 0.004910750 0.004910750
+## 1 Mortality 0.005636800 0.005636800 0.005636800
+## 1 Relapse   0.006878377 0.007269181 0.007269181
+
+# Plot the CIFs for all even types
+plot(cif, lty = 1, col = c("green", "red", "blue"), xlab = "Time", ylab = "Cumulative Incidence",
+     main = "Cumulative Incidence Functions for Competing Risks", curvlab = c("Censored", "Mortality", "Relapse"))
+
+
+
+

+
+
+
+

+# Estimate the cumulative incidence functions fo each even type per group
+cif <- cuminc(ftime = data_bmt$ftime, 
+              fstatus = data_bmt$status, 
+              group = data_bmt$dis) # Estimates will calculated within groups
+
+# Plot the CIFs for both event types using survminer package
+ggcompetingrisks(cif, multiple_panels = FALSE, legend = "right")
+
+
+
+

+
+
+
+

+# Show Gray's test for equality of CIFs
+print(cif$Tests)
+##                stat          pv df
+## Censored  5.9785107 0.014481226  1
+## Mortality 0.9133497 0.339227192  1
+## Relapse   9.4874094 0.002068867  1
+
+

Gray’s test for equality of CIFs across groups (ALL vs. AML) is split across different types of events, here (Censored, Mortality, and Relapse).

+
    +
  • For Censored, the p-value is small and suggest that there isa statistically significant difference in the CIF for censored events between the groups.
    • +
  • For Mortality, the p-value is greater than 0.05, suggesting that there is no statistically significant difference in the CIF for mortality between the groups.
    • +
  • For Relapse, the p-value is less than 0.05, indicating a statistically significant difference in the CIF for relapse between the groups.
    • +
  • The above suggest that the groups are behaving differently in terms of the time to relapse and censored events, but not for mortality.
    • +
+
+
+

3.3.2 Competing Risks Regression

+

adopted from https://www.nature.com/articles/bmt2009359

+

Suppose that the BMT study was extended to include more participants and additional covariates and now includes:

+
    +
  • 177 observations
    • +
  • Sex: gender of the individual
    • +
  • D: disease; 0 = ALL; 1 = AML
    • +
  • Phase: phase at transplant (Relapse, CR1, CR2, CR3)
    • +
  • Age: age at the beginning of follow-up
    • +
  • Status: 0 = censored; 1 = Transplant-related mortality; 2 = relapse
    • +
  • Source: source of stem cells (BM+PB, PB)
    • +
  • ftime: failure time
    • +
+

We are interested in modeling time to relapse in the presence of transplant-related death (competing event). We want to sutdy the effect on relapse of sex, disease type, phase at transplant, source of stem cells and age.

+
+
# load additional libraries
+library(fastDummies)
+
+# load data
+data_bmtcrr <- readRDS("data/data-bmtcrr.rds")
+
+# preview data
+str(data_bmtcrr)
+## 'data.frame':    177 obs. of  7 variables:
+##  $ Sex   : Factor w/ 2 levels "F","M": 2 1 2 1 1 2 2 1 2 1 ...
+##  $ D     : Factor w/ 2 levels "ALL","AML": 1 2 1 1 1 1 1 1 1 1 ...
+##  $ Phase : Factor w/ 4 levels "CR1","CR2","CR3",..: 4 2 3 2 2 4 1 1 1 4 ...
+##  $ Age   : int  48 23 7 26 36 17 7 17 26 8 ...
+##  $ Status: int  2 1 0 2 2 2 0 2 0 1 ...
+##  $ Source: Factor w/ 2 levels "BM+PB","PB": 1 1 1 1 1 1 1 1 1 1 ...
+##  $ ftime : num  0.67 9.5 131.77 24.03 1.47 ...
+
+# prepare matrix of covariates
+# factor variables need to be coded as dummy variables
+x_sex <- dummy_cols(data_bmtcrr$Sex) %>% dplyr::select(.data_F)
+x_phase <- dummy_cols(data_bmtcrr$Phase) %>% dplyr::select(.data_CR1, .data_CR2, .data_CR3)
+x_d <- dummy_cols(data_bmtcrr$D) %>% dplyr::select(.data_AML)
+x_source <- dummy_cols(data_bmtcrr$Source) %>% dplyr::select(.data_PB)
+x <- data.frame(age = data_bmtcrr$Age, sex = x_sex, x_d, x_phase, x_source)
+colnames(x) <- c("age", "sex_F", "D_AML", "P:CR1", "P:CR2", "P:CR3", "source:PB")
+
+# The first regression model for relapse can be produced by typing
+mod1 <- crr(ftime = data_bmtcrr$ftime, 
+            fstatus = data_bmtcrr$Status, 
+            cov1 = x)
+
+summary(mod1)
+## Competing Risks Regression
+## 
+## Call:
+## crr(ftime = data_bmtcrr$ftime, fstatus = data_bmtcrr$Status, 
+##     cov1 = x)
+## 
+##              coef exp(coef) se(coef)      z p-value
+## age       -0.0185     0.982   0.0119 -1.554  0.1200
+## sex_F     -0.0352     0.965   0.2900 -0.122  0.9000
+## D_AML     -0.4723     0.624   0.3054 -1.547  0.1200
+## P:CR1     -1.1018     0.332   0.3764 -2.927  0.0034
+## P:CR2     -1.0200     0.361   0.3558 -2.867  0.0041
+## P:CR3     -0.7314     0.481   0.5766 -1.268  0.2000
+## source:PB  0.9211     2.512   0.5530  1.666  0.0960
+## 
+##           exp(coef) exp(-coef)  2.5% 97.5%
+## age           0.982      1.019 0.959 1.005
+## sex_F         0.965      1.036 0.547 1.704
+## D_AML         0.624      1.604 0.343 1.134
+## P:CR1         0.332      3.009 0.159 0.695
+## P:CR2         0.361      2.773 0.180 0.724
+## P:CR3         0.481      2.078 0.155 1.490
+## source:PB     2.512      0.398 0.850 7.426
+## 
+## Num. cases = 177
+## Pseudo Log-likelihood = -267 
+## Pseudo likelihood ratio test = 24.4  on 7 df,
+
+

The first part of the output shows for each term in the design matrix the estimated coefficient \(\hat{\beta}_j\), the relative risk \(\mathrm{exp}(\hat{\beta}_j)\), the standard error, the z-value and the corresponding P-value for assessing significance.

+

Here, Sex is not significant, followed by Age and D (disease type), whereas Source is only marginally significant. Phase is a factor with relapse as baseline, so each P-value provides a test for the difference of each level with respect to the baseline.

+

An overall P-value for Phase can be obtained through the Wald test via aod R package:

+
+
library(aod)
+wald.test(mod1$var, mod1$coef, Terms = 4:6)
+## Wald test:
+## ----------
+## 
+## Chi-squared test:
+## X2 = 14.0, df = 3, P(> X2) = 0.0029
+
+

The first argument to the function wald.test() is the estimated covariance matrix for the coefficients, followed by the vector of coefficients estimates, and the position of coefficients for which we want to assess significance. In our case, the P-value indicates that Phase is statistically significant.

+

The second part of the output for competing risks regression shows the relative risk for each term, and a 95% confidence interval. The relative risk or subdistribution hazard ratio for a categorical covariate is the ratio of subdistribution hazards for the actual group with respect to the baseline, with all other covariates being equal. If the covariate is continuous then the relative risk refers to the effect of a one unit increase in the covariate, with all other covariates being equal. In our data, exp(−0.0352)=0.965 is the relative risk of a woman with respect to a man, and exp(−0.0185)=0.982 is the relative risk for a 1 year increase in age.

+

The last part of the output shows the pseudo log-likelihood at maximum and the pseudo likelihood ratio test, that is, the difference in the objective function at the global null and at the final estimates. As this objective function is not a true likelihood, this test statistic is not asymptotically distributed as a χ2. As a consequence, model comparison based on likelihood ratio approach cannot be performed directly, but significance must be evaluated through simulations. However, a model selection criterion can be easily adopted as described in the following section.

+

For an example on model selection, model diagnostics or adding time-varying covariates in presence of competing events, please refer to the original tutorial article Scrucca, Santucci, and Aversa (2010) https://www.nature.com/articles/bmt2009359.

+ + + +
+
+ +
+ + +
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+
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+ + + +
+ +
+
+

References

+
+ + + +
+ + + + +
+ + + +
+ + +
+
+Fine, Jason P., and Robert J. Gray. 1999. “A Proportional Hazards +Model for the Subdistribution of a Competing Risk.” Journal +of the American Statistical Association 94. https://doi.org/10.1080/01621459.1999.10474144. +
+
+Gray, Robert J. 1998. “A Class of k-Sample Tests for Comparing the +Cumulative Incidence of a Competing Risk.” The Annals of +Statistics 16. https://doi.org/10.1214/aos/1176350951. +
+
+Scrucca, L, A Santucci, and F Aversa. 2010. “Regression Modeling +of Competing Risk Using r: An in Depth Guide for Clinicians.” +Bone Marrow Transplantation 45: 1388–95. https://doi.org/10.1038/bmt.2009.359. +
+
+ + + +
+ + +
+ + + + + \ No newline at end of file diff --git a/session-survival/docs/regression.html b/session-survival/docs/regression.html new file mode 100644 index 00000000..6fa116f2 --- /dev/null +++ b/session-survival/docs/regression.html @@ -0,0 +1,983 @@ + + + + + + + + + +Survival Analysis - 2  Regression with survival response + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+ +
+ +
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+ + + +
+ +
+
+

2  Regression with survival response

+
+ + + +
+ + + + +
+ + + +
+ + +
+

2.1 Cox proportional hazards model

+

If survival time is of interest, such as in mortality studies, a common approach is to postulate a distribution for survival time and estimate the parameters of this distribution from the data. For instance, the exponential distribution can be used if the death rate is independent of time, the Weibull distribution allows for increasing or decreasing hazard rates, and the Gompertz distribution models hazard rates that exponentially increase or decrease over time.

+

More commonly, it is the relationship between survival time and one or more predictor variables (covariates) that is of interest. The Cox proportional hazards model is a widely used semi-parametric model to study this relationship. It is used to estimate the hazard ratio for individuals based on their covariates, without needing to specify the baseline hazard function.

+

Estimation of \(\boldsymbol{\beta}\) and inferences are developed by considering the information supplied at each time that a death (event) occurred.

+

Consider:

+
    +
  • a death occurring at time \(t_j\), and
    • +
  • suppose that there were \(n_j\) subjects alive just before \(t_j\),
    • +
  • that the values of \(\boldsymbol{x}\) for these subjects are \(\boldsymbol{x_1}, \boldsymbol{x_2}, \cdots, \boldsymbol{x_{n'j}}\)
    • +
  • and that the subject that dies is denoted, by the subscript 1
    • +
+

Then:

+
    +
  • The set of \(n'_j\) subjects at risk is referred to as the risk set.
    • +
  • The risk of death at time \(t_j\) for each subject in the risk set is given by: \[h(t) = h_0(t)\textrm{exp}(\boldsymbol{\beta}^T\boldsymbol{x})\]
    • +
+

where:

+
    +
  • \(\boldsymbol{\beta}^T\boldsymbol{x}\) is the matrix representation of the regression function, \(\beta_1x_1 + \beta_2x_2 + \cdots + \beta_px_p\)
    • +
  • and \(h_0(t)\) is baseline hazard function
    • +
+

The risk of death at time \(t_j\) in the risk set does not supply absolute measures of risk, but does supply the relative risks for each subject, since, although \(h_0(t)\) is unknown, it is the same for each subject. Thus the probability that the death observed at \(t_j\) was of the subject who did die at that time is: \[p_j = \frac{\textrm{exp}(\boldsymbol{\beta}^T\boldsymbol{x_1})}{\sum \textrm{exp}(\beta^T\boldsymbol{x_i})}\] where summation is over all remembers of the risk set.

+

Similar terms are derived for each time that a death occurred and are combined to form a likelihood (product of these probabilities over all event times \(t_j\)): \[PL(\boldsymbol{\beta}) = \prod_{j=1}^{k} \frac{\exp(\boldsymbol{\beta}^T \boldsymbol{x_1})}{\sum_{i \in \text{risk set at } t_j} \exp(\boldsymbol{\beta}^T \boldsymbol{x_i})}\] Technically this is called partial likelihood, since the component terms are derived conditionally on the times that deaths occurred and the composition of the risk set at these times. This partial likelihood is used to estimate the regression coefficients \(\boldsymbol{\beta}\) in the Cox model. Maximizing this partial likelihood gives the best estimates for \(\boldsymbol{\beta}\), quantifying the effect of the covariates on the hazard.

+

To estimate \(\beta\), we maximize the partial likelihood function with respect to \(\beta\). No closed-form solution is available, and so iterative algorithms are used.

+
+
+

2.2 Example with brain cancer data

+

The BrainCancer data set from the ISLR2 package contains survival times for patients with primary brain tumors undergoing treatment with radiation therapy. Variables included in the study are:

+
    +
  • sex: male or female
    • +
  • diagnosis: meningioma, LG glioma, HG glioma, or other
    • +
  • loc: tumor location, infratentorial or supratentorial
    • +
  • gtv: gross tumor volume \(cm^3\)
    • +
  • ki: Karnofsky index
    • +
  • stereo: stereotatcic radiosurgery (SRS) or fractionated stereotactic (SRT) radiotherapy
    • +
+
+
+Code +
# preview data
+data_brain <- BrainCancer
+
+data_brain %>%
+  str()
+
+
+
'data.frame':   88 obs. of  8 variables:
+ $ sex      : Factor w/ 2 levels "Female","Male": 1 2 1 1 2 1 2 2 1 2 ...
+ $ diagnosis: Factor w/ 4 levels "Meningioma","LG glioma",..: 1 3 1 2 3 1 1 2 1 3 ...
+ $ loc      : Factor w/ 2 levels "Infratentorial",..: 1 2 1 2 2 2 2 2 2 2 ...
+ $ ki       : int  90 90 70 80 90 80 80 80 70 100 ...
+ $ gtv      : num  6.11 19.35 7.95 7.61 5.06 ...
+ $ stereo   : Factor w/ 2 levels "SRS","SRT": 1 2 1 2 2 1 2 2 2 2 ...
+ $ status   : int  0 1 0 1 1 0 0 0 0 0 ...
+ $ time     : num  57.64 8.98 26.46 47.8 6.3 ...
+
+
+Code +
# Cox model with multiple predictors
+fit.cox_multi <- coxph(Surv(time, status) ~ sex + diagnosis + ki, data = data_brain)
+summary(fit.cox_multi)
+
+
+
Call:
+coxph(formula = Surv(time, status) ~ sex + diagnosis + ki, data = data_brain)
+
+  n= 87, number of events= 35 
+   (1 observation deleted due to missingness)
+
+                       coef exp(coef) se(coef)      z Pr(>|z|)    
+sexMale             0.11149   1.11795  0.35670  0.313 0.754608    
+diagnosisLG glioma  1.01441   2.75774  0.62159  1.632 0.102687    
+diagnosisHG glioma  2.22962   9.29629  0.44520  5.008  5.5e-07 ***
+diagnosisOther      0.82773   2.28813  0.57751  1.433 0.151774    
+ki                 -0.06624   0.93591  0.01753 -3.778 0.000158 ***
+---
+Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
+                   exp(coef) exp(-coef) lower .95 upper .95
+sexMale               1.1179     0.8945    0.5556    2.2493
+diagnosisLG glioma    2.7577     0.3626    0.8156    9.3251
+diagnosisHG glioma    9.2963     0.1076    3.8847   22.2467
+diagnosisOther        2.2881     0.4370    0.7377    7.0967
+ki                    0.9359     1.0685    0.9043    0.9686
+
+Concordance= 0.782  (se = 0.038 )
+Likelihood ratio test= 36.59  on 5 df,   p=7e-07
+Wald test            = 34.11  on 5 df,   p=2e-06
+Score (logrank) test = 40.79  on 5 df,   p=1e-07
+
+
+
+

2.2.1 Coefficient interpration

+

Above, we fitted Cox model using sex and diagnosis as predictors. The results, indicate, for instance that:

+
    +
  • The estimated hazard ratio (HR) for a male patient is \(e^{0.11} = 1.11\). This means that men have 1.11 times the hazard of dying than women while keeping other variables constant (sometimes referred to as adjusted hazard ration, AHR).
    • +
  • However the associated Z value is small, resulting in large p-value of 0.75, which indicates that this difference is not statistically significant.
    • +
  • We can also see that one-unit increase in the Karnofsky index is corresponds to a multiplier of \(e^{-0.06} = 0.94\) in the chance of dying. The higher the Karnofsky index score, the lower the chance of dying at any given point in time, with this effect being highly significant \(p\)-value of 0.000158.
    • +
  • Note that if you want to find HR associated with more than 1-unit dffererence, e.g. for patients with 10 units higher Karnofsky index, we would calculate HR as \(e^{-0.06 \times 10} = 0.54\). This means that patients with 10 units higher Karnofsky index have 0.54 times the hazard of dying compared to those with lower Karnofsky index.
    • +
+

Sometimes, in addition to reporting the numeric results of a Cox regression, forest plots are used to visualize HRs and their 95% confidence intervals.

+
+
+Code +
# make forest plot with ggforest()
+ggforest(fit.cox_multi, data = data_brain)
+
+
+
+
+

+
+
+
+
+
+
+

2.2.2 Global statistical signficance

+

The output of the Cox regression contains also three methods for an overall test of whether the predictors in the model have a significant effect on the hazard (or risk) of the event occurring. Essentially, these check whether the model as a whole provides a better fit than a null model (a model without any covariates).

+

Likelihood Ratio Test (LRT) compares the log-likelihoods of two models:

+
    +
  • The null model, which includes only the baseline hazard (i.e., no covariates).
    • +
  • The full model, which includes the covariates of interest.
    • +
+

The test statistic is computed as: \[ +2 \times (\text{log-likelihood of the full model} - \text{log-likelihood of the null model}) +\] This statistic follows a chi-squared distribution, and the associated p-value indicates whether the full model (with covariates) is significantly better at explaining the data than the null model.

+

Wald Test

+

The Wald test checks whether the estimated regression coefficients are significantly different from zero. For each predictor, the test examines whether its coefficient, \(\beta_i\), is significantly different from 0, implying that the predictor has a significant effect on the hazard. The Wald test can also be used as a global test of significance.

+

Score Test (Log-Rank Test)

+

The Score test (also called the log-rank test in this context) evaluates the contribution of each predictor to the model’s fit, considering the expected number of events. Like the likelihood ratio test, the Score test also compares the null and full models and uses a chi-squared distribution to compute significance.

+
+
+

2.2.3 Concordance

+

In the context of a Cox proportional hazards model, concordance (also known as the C-index or concordance index) is a measure of how well the model predicts the order of events. Specifically, it assesses the ability of the Cox model to correctly predict which of two individuals will experience an event first, based on their risk scores.

+

The concordance index (C-index) ranges between 0.5 and 1:

+
    +
  • A C-index of 0.5 indicates that the model’s predictions are no better than random chance. This means the model cannot distinguish between individuals who experience the event earlier and those who experience it later.
    • +
  • A C-index of 1 means perfect prediction, indicating that the model always correctly predicts the order of events.
    • +
+

The C-index is similar to the area under the ROC curve (AUC) used in binary classification models, but it is adapted to survival analysis where the goal is to rank individuals by their risk of experiencing the event.

+

In our example, a concordance of 0.782 means that the Cox model can correctly predict the ordering of survival times 78.2% of the time. In other words, if you randomly select two individuals and compare their predicted risk scores, the model will correctly predict which individual experiences the event earlier in 78.2% of cases. The standard error is 0.038 which provides a measure of uncertainty around the concordance estimate. A smaller standard error indicates more confidence in the estimate.

+
+
+

2.2.4 Predictions

+

In Cox regression, we can estimate:

+
    +
  • the survival probability at a specific time, \(S(t|X=x)\)
    • +
  • and the hazard ratio for an individual relative to a reference individual, $h(t|X=x) / \(S(t|X=x_{ref})\), where \(X_{ref}\) is a reference individual with known covariate values.
    • +
+

For predicted survival, the choice of time matters. A predicted HR will not depend on time due to proportional hazards assumption.

+
+

2.2.4.1 Survival

+

Let’s predict the probability of survival at 40 days, for a new patient (man, with LG glioma, Karnofsky index of 80).

+
+
# new patient data
+new_pat <- data.frame(time = 40, 
+                      sex = "Male", 
+                      diagnosis = "LG glioma",
+                      ki = 80, 
+                      status = 0)
+
+predict(fit.cox_multi, newdata = new_pat, type = "survival", se.fit = T)
+
+
$fit
+[1] 0.4824421
+
+$se.fit
+[1] 0.1834607
+
+
+

The probability of surviving through 40 days is 0.48. To get confidence interval we could use:

+
+
unlist(summary(survfit(fit.cox_multi,
+                       newdata = new_pat,
+                       se.fit =T, conf.int = 0.95),
+        times=40)[c("surv", "std.err", "lower", "upper")])
+
+
     surv   std.err     lower     upper 
+0.4824421 0.1834607 0.2289573 1.0000000
+
+
+

So the probability of surviving through 40 days is 0.48 with 95% CI [0.23, 1].

+
+
+

2.2.4.2 HR

+

To predict the adjusted hazard ratio for a new patient, we use again the predict() function, this type with type = "risk". We also need to specify the reference individual. Some options include:

+
    +
  • reference = "zero": the reference individual has continuous predictor values set to 0 and categorical predictors each at their reference level. This is not a good choice if a value of 0 is not plausible for continuous variables in the model, e.g. age.
    • +
  • reference = "sample": the reference individual has continuous predictors each their respective sample mean and categorical predictors at their reference level.
    • +
  • reference = "strata": this leads to the same results as in reference = "sample" unless the model includes astrata()`, in which case continuous variables will be set their within-stratum means.
    • +
+
+
# predict HR for our new patient
+predict(fit.cox_multi, 
+        newdata = new_pat, 
+        type = "risk", 
+        reference = "strata", 
+        se.fit = T)
+
+
$fit
+       1 
+3.276632 
+
+$se.fit
+       1 
+1.201369
+
+
+

Our new patient 3.27 times the hazard of dying compared to a reference individual with level of each categorical predictor and the mean of each continuous predictor, which in our case are (Female for gender, Meningioma for diagnosis, and mean Karnofsky index of 81.02).

+
+
+
+
+

2.3 Time-dependent covariates

+

In all above example the predictors did not vary over time. Each individual had only one value for each predictor. Since survival data is the result of follow-up over time, it is possible to have predictors that do vary over time. For example, in a study of time to heart attack, researchers could record various other conditions’ occurrence over time, such as hypertension or angina. In a study of juvenile recidivism, researchers could record how education or employment status change over time. Cox regression is able to handle such time-varying predictors (also known as “time dependent covariates”).

+

For time-varying covariates, the hazard function is extended to account for changes in the covariates over time: \[h(t | \boldsymbol{x}(t)) = h_0(t) \cdot \exp\left(\boldsymbol{\beta}^T \boldsymbol{x}(t)\right)\]

+

where:

+
    +
  • \(h(t | \boldsymbol{x}(t))\) is the hazard at time \(t\), conditional on the covariate values at that time.
    • +
  • \(\boldsymbol{x}(t)\) represents the covariate vector at time \(t\). In this case, one or more covariates can change over time.
    • +
  • \(\boldsymbol{\beta}\) is the vector of regression coefficients.
    • +
  • \(h_0(t)\) is the baseline hazard function, which remains time-dependent but does not depend on the covariates.
    • +
+

In practice, time-varying covariates are handled by breaking the follow-up time into intervals where the covariate values are assumed to remain constant, and then updating the covariate values at each interval. Suppose a covariate (e.g., blood pressure, treatment status, or exposure) varies over time. For an individual, we divide their follow-up time into periods where the covariate value is constant, and then treat each period as a separate “observation” in the Cox model. The Cox model then uses these updated covariate values when estimating the hazard at each time point.

+

The partial likelihood function in the Cox model is modified to incorporate the updated covariate values \(\boldsymbol{x}(t)\). For each event time \(t_j\), the likelihood is based on the covariate values that are valid at that specific time.

+

The partial likelihood becomes:

+

\[PL(\boldsymbol{\beta}) = \prod_{j=1}^{k} \frac{\exp\left(\boldsymbol{\beta}^T \boldsymbol{x}_1(t_j)\right)}{\sum_{i \in \text{risk set at } t_j} \exp\left(\boldsymbol{\beta}^T \boldsymbol{x}_i(t_j)\right)}\]

+

where:

+
    +
  • \(\boldsymbol{x}_1(t_j)\) is the covariate vector for the subject who experienced the event at time \(t_j\),
    • +
  • and \(\boldsymbol{x}_i(t_j)\) is the covariate vector for each subject in the risk set at time \(t_j\).
    • +
+

A dataset with time-varying predictors will have multiple rows per individual, with different rows having different values for the time-varying predictors, reflecting how they change over time. Additionally, rather than having a single event time variable, each row will have two time variables indicating the beginning and end of the time interval represented by that row of data.

+

For instance, in the study of heroin usage, we can see a male participant who started using non-prescribed pharmaceutical opioids (NPPOs) at age 19 (time independent covariates). He did not meet the criteria for lifetime opioid dependence at baseline (wave 1), but did at the next interview (wave2), and he first reported using heroin at his 6th interview (both time varying variables).

+
+
+Code +
data_opioid <- readRDS("data/data-opioid.rds")
+
+data_opioid %>% 
+  filter(RANDID == 10) %>%
+  select(RANDID, wave, START, STOP, heroin, age_at_init, sex, dep_lifetime)
+
+
+
  RANDID wave START STOP heroin age_at_init  sex dep_lifetime
+1     10    1  3.78 4.26      0          19 Male            0
+2     10    2  4.26 4.78      0          19 Male            1
+3     10    3  4.78 5.29      0          19 Male            1
+4     10    4  5.29 5.84      0          19 Male            1
+5     10    5  5.84 6.27      0          19 Male            1
+6     10    6  6.27 6.79      1          19 Male            1
+
+
+
+
+

2.4 Multiple events: competing risks

+

In survival analysis, competing risks occur when an individual is at risk of more than one mutually exclusive event, and the occurrence of one event precludes the occurrence of the others. Traditional survival models, such as the Cox proportional hazards model, typically focus on a single type of event (e.g., death or failure), but in many real-world scenarios, individuals may face different types of events that compete with each other.

+

A competing risk refers to events that prevent the event of primary interest from happening. For example: - In a study of heart disease, death due to cancer is a competing risk if the primary event of interest is death due to heart disease. - In a study of cancer treatment, death from any cause is a competing risk if the event of interest is relapse.

+

In these situations, standard survival analysis methods may overestimate the probability of the primary event because they do not account for the possibility of competing events.

+

The Cumulative Incidence Function (CIF) is commonly used in competing risks analysis. It represents the probability of experiencing a specific event (e.g., death from heart disease) by a certain time, accounting for the presence of competing risks (e.g., death from cancer).

+

CIF is a product of two estimates.

+
    +
  1. The estimate of hazard at ordered failure time \(t_j\) for the event of interest: \[\hat{h_c(t_j)}=\frac{d_{cj}}{n_j}\] where:
    2. +
+
    +
  • \(m_{cj}\) denotes the number of events for risk \(c\) at time \(t_j\)
    • +
  • and \(n_j\) is the number of subjects at risk at time \(t_j\)
    • +
+
    +
  1. The estimate of overall probability of surviving previous time: \[\hat{S}(t_{j-1})\] We consider the overall survival as a subject must have survived all other competing events in order to fail from event type \(c\) at time \(t_j\).
    2. +
+

The estimated incidence probability of failing from even type \(c\) at time \(t_j\) is then: \[\hat{I_c}(t_j) = \hat{S}(t_{j-1}) \times \hat{h_c(t_j)}\] The probability of failing form even type \(c\) at time \(t_j\) is a product of surviving the previous time periods and the cause specific hazard at time \(t_j\).

+

The CIF for event type \(c\) at time \(t_j\) is then the cumulative sum up to time \(t_j\), i.e. from \(f' = 1\) to \(f'=f\), of these incidence probabilities over all event type \(c\) failure times, which is expressed: \[CIF_c(t_j) = \sum_{f'=1}^{f}\hat{I_c}(t_j) = \sum_{f'=1}^{f}\hat{S}(t_{f'-1})\times\hat{h_c}(t_{f'})\]

+

In 1999 Gray (1998) proposed a non-parametric test to compare two or more CIFs. The test is analogous to the log-rank test comparing KM curves, using a modified Chi-squared test statistic. This test does not require the independent censoring assumption.

+

In Fine and Gray Fine and Gray (1999) proposed a proportional hazards model aims at modeling the CIF with covariates, by treating the CIF curve as a subdistribution function. The subdistribution function is analogous to the Cox proportional hazard model, except that it models a hazard function (as known as subdistribution hazard) derived from a CIF.

+ + + +
+ +
+ + +
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We would be grateful if you let us know via edu.ml-biostats@nbis.se\nThis repository contains teaching and learning materials prepared for and used during “Introduction to biostatistics and Machine Learning” and “Biostatistics and Machine Learning II” courses, organized by NBIS, National Bioinformatics Infrastructure Sweden. The courses are open for PhD students, postdoctoral researcher and other employees within Swedish universities. The materials are geared towards life scientists wanting to be able to understand and use the basic statistical and machine learning methods. More about the course https://nbisweden.github.io/workshop-mlbiostatistics/", + "crumbs": [ + "Preface" + ] + }, + { + "objectID": "intro.html", + "href": "intro.html", + "title": "1  Introduction", + "section": "", + "text": "1.1 Why survival analysis?", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#why-survival-analysis", + "href": "intro.html#why-survival-analysis", + "title": "1  Introduction", + "section": "", + "text": "The term survival analysis refers to statistical methods developed to study time-to-event data, originally applied to time until death in 17th-century mortality studies, which is where the term “survival” originates.\nOver the decades its scope has broadened to include a wide range of applications in both medical studies and beyond, in fields such as marketing, engineering or criminology and many more.\nThe common factor in survival analysis is the study of censored data that arise when studying a unique kind of outcome variable: time until an event occurs.\nThis event, can be death, as previously mentioned, but also disease onset or relapse in medical studies. In other fields, we can study time until next purchase (marketing), time until machine failure (engineering), time until a released prisoner commits another crime (criminology), and many more.\nIn fact, survival analysis is applicable whenever censored data occurs, and these cases are not limited to studying time-based events. We can use survival analysis methods to study the number of treatment sessions until a patient recovers or the number of attempts until a student passes an exam.\nCensoring occurs when the event of interest has not yet been observed by the end of the study. This makes survival analysis especially useful for incomplete data sets, where we can still make use of the available information to estimate the time to event.\nWith survival analysis, we can compare groups (e.g., treatment vs. control) and determine which factors increase or decrease the hazard (risk) of the event occurring over time, even when adjusting for other variables.\nWe can incorporate time-dependent variables, allowing the analysis to account for changing risk factors, such as fluctuating health conditions or environmental factors, and include multiple states and/or events. For example, a patient may progress through different stages of a disease (multiple states), or a patient dying from different causes may prevent the occurrence of the primary event being studied (multiple events, competing risks).", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#survival-and-censoring-times", + "href": "intro.html#survival-and-censoring-times", + "title": "1  Introduction", + "section": "1.2 Survival and censoring times", + "text": "1.2 Survival and censoring times\nLet’s imagine a mortality study of terminally ill patients receiving a certain medical treatment. We follow a group of patients over 24 months follow-up period and record the time until death or until the end of the study. For each patient, we suppose there is:\n\na true survival time \\(T\\) and\na censoring time \\(C\\).\n\nThe survival time represents the time at which the event of interest occurs, here death. The censoring time represents the time at which the patient is lost to follow-up (e.g. withdraws from the study) or the study ends.\nWe observe the random variable \\[Y = min (T, C)\\]\nIf the event occurs before censoring (i.e. \\(T < C\\)) then we observe the true survival time \\(T\\). Otherwise, if censoring occurs before the event (i.e. \\(T > C\\)), we observe the censoring time \\(C\\).\nWe also observe a status indicator\n\\[\\begin{equation}\n \\sigma =\n \\left\\{\n \\begin{array}{cc}\n 1 & \\mathrm{if\\ } T \\le C \\\\\n 0 & \\mathrm{if\\ } T>C \\\\\n \\end{array}\n \\right.\n\\end{equation}\\]\nA figure below illustrates the data for 10 patients in this ficitious study.\n\n\nCode\n# Example data for 10 patients\ndata <- data.frame(\n patient = factor(1:10), # Patient IDs\n time = c(23, 24, 9, 7, 14, 16, 19, 20, 24, 24), # Time to event or censoring\n status = c(1, 0, 1, 0, 1, 1, 0, 1, 0, 1) # Event status (1 = event, 0 = censored)\n)\n\n# Plot the data with horizontal lines\nggplot(data, aes(x = time, y = patient)) +\n geom_segment(aes(x = 0, xend = time, y = patient, yend = patient), color = mycols[2]) + #\n geom_point(aes(shape = factor(status)), size = 3) + \n geom_vline(xintercept = 24, linetype = \"dashed\") +\n scale_shape_manual(values = c(1, 16), \n labels = c(\"Censored\", \"Event\")) +\n labs(x = \"Time (months)\", y = \"Patient\", shape = \"Status\") + \n theme_minimal() +\n theme(legend.position = \"top\")\n\n\n\n\n\n\n\n\nFigure 1.1: Example of right-censored data in a mortatily study lasting 24 months. The horizontal lines represent the time to event or censoring for each patient. The points indicate whether the event occurred (event) or not (censored).\n\n\n\n\n\nWe we observe 10 \\((Y, \\sigma)\\) pairs, which we denote as \\((y_1, \\sigma_1, \\dots, (y_n, \\sigma_n))\\), where \\(n = 10\\). From Figure 1.1, we can see that:\n\nfor patient #1 we get \\((y_1= t_1, \\space \\sigma_1 = 1)\\) since we observe event (death) as ca. 23 months, before the end of the follow-up at 24 months.\nfor patient #2 we get \\((y_2= c_1, \\space \\sigma_1 = 0)\\) since the patient is still alive at the end of the study at 24 moths, meaning that the patient has survied at least 24 months (censored)\nfor patient #3 we get \\((y_3= t_3, \\space \\sigma_3 = 1)\\) since we observe event (death) at 9 months\nfor patients #4, we get \\((y_4= c_4, \\space \\sigma_4 = 0)\\), anther censored data point. The study is not over yet, the patient has not diet but is lost to follow-up, e.g. due to withdrawing from the study\nanalogously, for patients #5, #6, #8 and #10 we observe survival times, and for patients #7 and #9 we observe censoring times.", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#independent-censoring", + "href": "intro.html#independent-censoring", + "title": "1  Introduction", + "section": "1.3 Independent censoring", + "text": "1.3 Independent censoring\nIn order to analyze survival data, we need to make some assumptions about the censoring process. One of the key assumptions in survival analysis is independent censoring. This assumption states that the reason for censoring is unrelated to the likelihood of the event occurring. In other words, censored individuals should have the same probability of experiencing the event as those who remain in the study.\nFor example, in a cancer study, if a number of patients leave the study for reasons unrelated to their health status, their data can be treated as censored without biasing the results, since censoring does not provide any information about the outcome. However, if a number of patients drop out early because they are very sick, this would violate the assumption of the independent censoring.", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#left-and-interval-censoring", + "href": "intro.html#left-and-interval-censoring", + "title": "1  Introduction", + "section": "1.4 Left and interval censoring", + "text": "1.4 Left and interval censoring\nIn addition to right-censored data, we may encounter left and/or interval-censored data. Left censoring occurs when the event happened before the observation period started, whereas interval censoring occurs when the event is known to have happened between two specific time points within the study.\nImagine an age study in which left censoring occurs when a patient was diagnosed with a disease before the study began, but the exact age of diagnosis is unknown. For example, if a patient is known to have been diagnosed before age 25, but the exact age is not recorded, they are left-censored. Interval censoring occurs when a patient’s diagnosis happens between two observed ages, but the exact age of diagnosis is unknown. For instance, if a patient was healthy at age 45 and diagnosed by age 55, but the precise age of diagnosis isn’t known, their data is interval-censored.\n\n\nCode\n# Create a data frame for 8 patients\n# Left-censored patients were diagnosed before the study, and interval-censored patients were diagnosed between visits\ndata <- data.frame(\n patient_id = 1:8,\n start_age = c(NA, 30, 40, 35, NA, 25, 45, 50), # Start of observation or known age for interval censoring (NA for left-censored)\n end_age = c(25, 35, 45, 50, 20, 40, 55, 60), # Age of diagnosis or end of observation\n type = c(\"Left Censored\", \"Fully Observed\", \"Fully Observed\", \"Fully Observed\", \n \"Left Censored\", \"Interval Censored\", \"Interval Censored\", \"Fully Observed\")\n)\n\n# Plot the data\nggplot(data, aes(x = end_age, y = factor(patient_id), color = type)) +\n # Add points for observed ages (diagnosis age or censoring)\n geom_point(size = 4) +\n \n # For interval-censored patients, add a horizontal line to show the interval\n geom_segment(data = subset(data, type == \"Interval Censored\"), aes(x = start_age, xend = end_age, y = factor(patient_id), yend = factor(patient_id)),\n color = mycols[2], linetype = \"dashed\") +\n \n # For left-censored patients, add an arrow showing the uncertainty before diagnosis\n geom_segment(data = subset(data, type == \"Left Censored\"), aes(x = 0, xend = end_age, y = factor(patient_id), yend = factor(patient_id)),\n color = mycols[1], linetype = \"dotted\", arrow = arrow(length = unit(0.2, \"cm\"))) +\n \n # Customize the plot labels and appearance\n labs(x = \"Age\", y = \"Patient ID\") +\n scale_color_manual(values = c(\"Left Censored\" = mycols[1], \"Fully Observed\" = \"black\", \"Interval Censored\" = mycols[2])) +\n theme_minimal() + \n theme(legend.position = \"top\", legend.title = element_blank())\n\n\n\n\n\n\n\n\nFigure 1.2: Example of left and interval censoring in an age study. Left-censored patients were diagnosed before the study, and interval-censored patients were diagnosed between visits.\n\n\n\n\n\nIn this session, we will focus on right censored data.", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#kaplan-meier-estimator", + "href": "intro.html#kaplan-meier-estimator", + "title": "1  Introduction", + "section": "1.5 Kaplan-Meier estimator", + "text": "1.5 Kaplan-Meier estimator\nThe survival curve, or survival function, is a decreasing function that quantifies the probability of surviving past time \\(t\\) and is defined as \\[S(t) = Pr(T > t) \\tag{1.1}\\]\nThe Kaplan-Meier (KM) estimator is a non-parametric statistic used to estimate the survival function from lifetime data and is given by: \\[\\hat{S}(t) = \\prod_{j=1}^{t} \\left( 1 - \\frac{d_j}{n_j} \\right) \\tag{1.2}\\]\nwhere:\n\n\\(d_j\\): number of failures at time \\(t_j\\)\n\\(n_j\\): number of patients at risk just before time \\(t_ij\\)\nand the product is taken over all time intervals in which a death occurred, up to and including \\(t\\)\n\nKapalan-Meier estimator is also known as product-limit estimator.\n\n1.5.1 Tooth filling example\n\n1.5.1.1 Complete follow-up data\nSuppose we have 10 patients who receive a tooth filling, and we want to track how long these fillings last before they fail (fall out or need replacement). The failure times (in months) for each patient are recorded:\n\n\nCode\ndata_tooth <- data.frame(\n patientID = 1:10,\n time = c(10, 8, 11, 5, 4, 3, 7, 6, 12, 8), \n surv = 1\n)\n\nstr(data_tooth)\n\n\n'data.frame': 10 obs. of 3 variables:\n $ patientID: int 1 2 3 4 5 6 7 8 9 10\n $ time : num 10 8 11 5 4 3 7 6 12 8\n $ surv : num 1 1 1 1 1 1 1 1 1 1\n\n\nNote that the surv column is a binary indicator of survival status (1 = event, 0 = censored). In this example, we assume all patients have experienced the event (failure) and have complete follow-up data.\nTo calculate the Kaplan-Meier Curve:\n\nWe sort the data by time to failure.2.\nFor each time, we calculate the survival probabilities, i.e. the probability the filling lasting beyond that time.\nWe apply the Kaplan-Meier formula (Equation 1.2) to calculate the probability of surviving up to time \\(t\\), as the product of all individual survival probabilities at each time up to \\(t\\):\n\nIn our example:\n\n\n\n\n\n\n\n\n\n\nTime (months)\nPatients at Risk \\(n_i\\)\nFailures \\(d_i\\)\nSurvival Probability at \\(t_i\\)\nCumulative Survival \\(S(t)\\)\n\n\n\n\n3\n10\n1\n1 - \\(\\frac{1}{10}\\) = 0.9\n0.9\n\n\n4\n9\n1\n1 - \\(\\frac{1}{9}\\) = 0.888\n0.9 \\(\\times\\) 0.888 = 0.8\n\n\n5\n8\n1\n1 - \\(\\frac{1}{8}\\) = 0.875\n0.8 \\(\\times\\) 0.875 = 0.7\n\n\n6\n7\n1\n1 - \\(\\frac{1}{7}\\) = 0.857\n0.7 \\(\\times\\) 0.857 = 0.6\n\n\n7\n6\n1\n1 - \\(\\frac{1}{6}\\) = 0.833\n0.6 \\(\\times\\) 0.833 = 0.5\n\n\n8\n5\n2\n1 - \\(\\frac{2}{5}\\) = 0.6\n0.5 \\(\\times\\) 0.6 = 0.3\n\n\n10\n3\n1\n1 - \\(\\frac{1}{3}\\) = 0.667\n0.3 \\(\\times\\) 0.667 = 0.2\n\n\n11\n2\n1\n1 - \\(\\frac{1}{2}\\) = 0.5\n0.2 \\(\\times\\) 0.5 = 0.1\n\n\n12\n1\n1\n1 - \\(\\frac{1}{1}\\) = 0\n0.1 \\(\\times\\) 0 = 0\n\n\n\nWe can see that:\n\nAt 3 months: 10 patients are at risk, and 1 failure occurs, so the survival probability for this period is 0.9.\nFor 4 months: 9 patients are at risk (since one patient failed at 3 months), and the survival probability is now cumulative (0.9 from the first interval multiplied by 0.888 from this interval).\nAt time 8 months: two failures occur and the survival probability drops more sharply.\nAt time 12 months: the survival probability reaches 0, indicating that all patients experienced the event (tooth filling failure) by the end of the study.\n\nWe can visualize the Kaplan-Meier curve for this data:\n\n\nCode\nfit.surv <- survfit(Surv(time, surv) ~ 1, data = data_tooth)\nplot(fit.surv, conf.int=FALSE, xlab = \"Months\", ylab = \"Estiamted Probabiliy of Survival\")\n\n\n\n\n\n\n\n\nFigure 1.3: Kaplan-Meier survival curve for tooth filling data. The curve shows the estimated probability of survival (filling intact) over time.\n\n\n\n\n\n\n\n1.5.1.2 Including censored data\nIn real life, not all patients may have complete follow-up data. Some patients might drop out of the study (e.g., move away or lose contact). This is called censoring, and these patients are still considered “alive” (their fillings intact) at the time they are censored, but we do not know what happens to them afterward. The filling can also be intact at the end of the study, but we do not know how long it will last.\nLet’s update the data with censoring for two patients:\n\n\n\nPatient\nTime to Failure (months)\nCensored\n\n\n\n\n1\n3\nNo\n\n\n2\n4\nNo\n\n\n3\n5\nYes\n\n\n4\n6\nNo\n\n\n5\n7\nNo\n\n\n6\n8\nNo\n\n\n7\n8\nNo\n\n\n8\n10\nYes\n\n\n9\n11\nNo\n\n\n10\n12\nNo\n\n\n\nNow, we can adjust the KM calculations to account for censored data. For censored patients, we do not count them in the number of events (failures), but they are still included in the “at-risk” population until they are censored.\n\n\n\n\n\n\n\n\n\n\n\nTime(months)\nPatients at Risk\\(n_i\\)\nFailures\\(d_i\\)\nC\nSurvival Probability at \\(t_i\\)\nCumulative Survival\\(S(t)\\)\n\n\n\n\n3\n10\n1\n0\n\\(1 - \\frac{1}{10} = 0.9\\)\n0.9\n\n\n4\n9\n1\n0\n\\(1 - \\frac{1}{9} = 0.888\\)\n0.9 \\(\\times\\) 0.888 = 0.8\n\n\n5\n8\n0\n1\n\n0.8\n\n\n6\n7\n1\n0\n\\(1 - \\frac{1}{7} = 0.857\\)\n0.8 \\(\\times\\) 0.857 = 0.686\n\n\n7\n6\n1\n0\n\\(1 - \\frac{1}{6} = 0.833\\)\n0.686 \\(\\times\\) 0.833 = 0.571\n\n\n8\n5\n2\n0\n\\(1 - \\frac{2}{5} = 0.0.6\\)\n0.571 \\(\\times\\) 0.6 = 0.343\n\n\n10\n3\n0\n1\n\n0.343\n\n\n11\n2\n1\n0\n\\(1 - \\frac{1}{2} = 0.5\\)\n0.343 \\(\\times\\) 0.5 = 0.171\n\n\n12\n1\n1\n0\n\\(1 - \\frac{1}{1} = 0\\)\n0.171 \\(\\times\\) 0 = 0\n\n\n\nCensoring affects the calculations:\n\nAt 5 months, one patient is censored, so we do not record a failure, and the cumulative survival rate remains unchanged.\nAt 10 months, one more patient is censored. Again, there is no effect on the failure probability, but the number of at-risk patients decreases.\nIn other words, if a patient is censored, they contribute to the number of at-risk patients until their censoring time, but they do not contribute to the event (failure) counts.\n\nWe can see how the censoring data affect the KM curve:\n\n\nCode\ndata_tooth[1, \"surv\"] <- 0\ndata_tooth[4, \"surv\"] <- 0\nfit.surv <- survfit(Surv(time, surv) ~ 1, data = data_tooth)\nplot(fit.surv, mark.time=TRUE, conf.int=FALSE, xlab = \"Months\", ylab = \"Estiamted Probabiliy of Survival\")\n\n\n\n\n\n\n\n\nFigure 1.4: Kaplan-Meier survival curve for tooth filling data. The curve shows the estimated probability of survival (filling intact) over time.\n\n\n\n\n\n\n\n\n1.5.2 Interpretations\nThe Kaplan-Meier curve shows the cumulative survival probabilities. A steeper slope indicates a higher event rate (death rate) and therefore a worse survival prognosis. A flatter slope indicates a lower event rate and therefore a better survival prognosis. The curve may have plateaus or flat areas, indicating periods of relatively stable survival.\nAt specific time points, you can estimate the survival probability by locating the time point on the horizontal axis and dropping a vertical line to the curve. Then, read the corresponding survival probability from the vertical axis.\n\nSo, what is the probability that a tooth filling will last 8 months? We can estimate this by locating 8 months on the x-axis and dropping a vertical line to the curve. The estimated survival probability at 10 months is approximately 0.343, meaning there is a 34.3.7% chance that a tooth filling will last 10 months.\n\n\n\nCode\nplot(fit.surv, conf.int=FALSE, xlab = \"Months\", ylab = \"Estiamted Probabiliy of Survival\")\nabline(v = 10, h = 0.343, col = \"red\", lty = 2)\n\n\n\n\n\n\n\n\nFigure 1.5: Kaplan-Meier survival curve for tooth filling data. Red lines indicate the estimated survival probability (tooth being intact) at 10 months.\n\n\n\n\n\nIf the interest is the average length of survival, or life expectancy, following the start of treatment, this may be crudely estimated from the survival curve as the time corresponding to a cumulative probability of survival at 0.5. Alternatively, it can be calculated as: \\[\\textrm{Life expectancy} = 0.5 + \\sum(\\textrm{length of interval} \\times \\textrm{cumulative chance of survival})\\]\nIf there are multiple curves representing different groups, you can compare their shapes and patterns. If the curves are parallel, it suggests that the groups have similar survival experiences. If the curves diverge or cross, it indicates differences in survival between the groups.\n\n\nCode\n# Create data frame for survival data\ndata_tooth_ext <- data.frame(\n time = c(3, 5, 6, 7, 9, 10, 12, 15, 18, 20, \n 4, 6, 7, 9, 11, 13, 14, 16, 18, 22),\n surv = c(1, 1, 1, 1, 1, 0, 1, 1, 0, 1, \n 1, 1, 0, 1, 1, 1, 1, 0, 1, 1),\n gender = c(rep(\"M\", 10), rep(\"F\", 10))\n)\n\n# Create a survival object and\nsurv_object <- Surv(time = data_tooth_ext$time, event = data_tooth_ext$surv)\n\n# Fit Kaplan-Meier curves for gender\nfit <- survfit(surv_object ~ gender, data = data_tooth_ext)\n\n# Plot Kaplan-Meier survival curves\nggsurvplot(fit, \n data = data_tooth_ext,\n pval = FALSE, # Show p-value for difference in survival\n conf.int = FALSE, # Add confidence interval\n risk.table = TRUE, # Show risk table at the bottom\n ggtheme = theme_minimal(), # Use a minimal theme\n xlab = \"Time (Months)\",\n ylab = \"Survival Probability\", \n palette = mycols[1:2])\n\n\n\n\n\n\n\n\nFigure 1.6: Kaplan-Meier survival curves for a new extended study of tooth filling data, including 20 participants (10 men and 10 women).", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#comparisons-of-hazards", + "href": "intro.html#comparisons-of-hazards", + "title": "1  Introduction", + "section": "1.6 Comparisons of hazards", + "text": "1.6 Comparisons of hazards\nOften, in survival analysis we would like to compare the survival patterns of different groups, e.g. is there a group difference between survival times (tooth filling intact) between men and women in the tooth filling study?\nThe differences between the survival curves are not constant. For example, both groups start at 1 and the survival probability deceases, with lower survival probability for men at 10 months, but then again higher survival probability at 15 months.\nWe solve the problem of allowing for differences in survival time by comparing hazards in the two groups over the duration of the follow-up. We also assume that the ratio of the hazards in the two groups remains constant over time, even if the underlying hazards change. We assume, that at all times \\(t\\): \\[\\frac{h_1(t)}{h_0(t)} = constant\\]\nwhere:\n\n\\(h_1(t)\\) and \\(h_0(t)\\) are the hazards in the exposed and unexposed groups at time \\(t\\) (or in one group and the other).\n\nThis assumption is called the proportional hazards assumption. It is difficult to estimate the hazard directly from the data, since this would give a series of “spikes” when an even occurs, interspersed with zeros when there is no disease event. Instead we use the cumulative hazard function, \\(H(t)\\). This is the total hazard experienced up to time \\(t\\), and is estimated by the sum of the risks at each time \\(t\\) at which an event occurs \\[H(t) = \\sum_{i=1}^{t}\\frac{d_i}{n_i}\\]\nIf the ratio of the hazards in both groups is constant over time, if follows that the ratio of the cumulative hazard functions must also equal this constant \\[\\frac{H_1(t)}{H_0(t)} = \\frac{h_1(t)}{h_0(t)} = constant\\] and applying the rules of logarithms: \\[log(H_1(t) - log(H_0(t)) = log(constant)\\]\nTherefore, if the proportional hazards assumptions is met, then graphs of the log of the cumulative hazard functions should be parallel.\n\n\nCode\n# Plot Kaplan-Meier survival curves\nggsurvplot(fit, \n data = data_tooth_ext,\n fun = \"cumhaz\", # Add cumulative hazard function\n pval = FALSE, # Show p-value for difference in survival\n conf.int = FALSE, # Add confidence interval\n risk.table = FALSE, # Show risk table at the bottom\n ggtheme = theme_minimal(), # Use a minimal theme\n xlab = \"Time (Months)\",\n ylab = \"Cumulative hazard (log)\", \n palette = mycols[1:2])\n\n\n\n\n\n\n\n\nFigure 1.7: Cumulative hazard functions for the two groups in the extended tooth filling study.\n\n\n\n\n\nNote that it can be shown mathematically that the cumulative hazard is related to the survival function by \\(H(t) = -log(S(t))\\).", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "intro.html#mantel-cox-log-rank-test", + "href": "intro.html#mantel-cox-log-rank-test", + "title": "1  Introduction", + "section": "1.7 Mantel-Cox (log-rank test)", + "text": "1.7 Mantel-Cox (log-rank test)\nAssuming the proportional hazards assumption holds, we can test for differences in survival between groups using the Mantel-Cox test, also known as the log-rank test. It tests the null hypothesis of no difference in survival between two or more independent groups. The test compares the entire survival experience between groups and can be thought of as a test of whether the survival curves are identical (overlapping).\nThe log-rank test statistic is constructed by comparing the observed number of events (e.g. deaths, failures) in each group to the expected number of events, assuming that the survival experiences of the groups are the same. The test statistic is based on a chi-square distribution, which tests whether the difference between the observed and expected numbers of events is larger than what we would expect by chance.\nLet’s start with two groups, A and B. If at time \\(t_j\\) there were \\(d_j\\) deaths and there were \\(n'_{jA}\\) and \\(n'_{jB}\\) subjects alive just before \\(t_j\\) in groups A and B respectively, then the data can be arranged in \\(2\\times2\\) table:\n\n\n\n\n\n\n\n\n\n\nDied\nSurvived\nTotal\n\n\n\n\nGroup A\n\\(d_{jA}\\)\n\\(r_{jA} = n'_{_jA} - d_{jA}\\)\n\\(n'_{jA}\\)\n\n\nGroup B\n\\(d_{jB}\\)\n\\(r_{jB} = n_{_jB} - d_{jB}\\)\n\\(n'_{jB}\\)\n\n\nTotal\n\\(d_j = d_{jA} + d_{jB}\\)\n\\(r_j = r_{jA} + r_{jB}\\)\n\\(n'_j = n'_{jA} + n'_{jB}\\)\n\n\n\nwhere:\n\nGroup A and Group B represent the two groups being compared (e.g., two different treatments, genders, etc.).\nDied (\\(d_{jA}\\), \\(d_{jB}\\)): The number of individuals who experienced the event (e.g., death, tooth filling falling out) in each group.\nSurvived (\\(r_{jA}\\), \\(r_{jB}\\)): The number of individuals who did not experience the event (survivors, with tooth filling intact) in each group. This is calculated as the total number in the group minus the number of deaths.\nTotal (\\(n_A\\), \\(n_B\\)): The total number of individuals in each group at risk just before the time of an event\n\nThe log-rank test compares the observed number of events to the expected number of events if the survival curves were the same for all groups. The expected number of events in group A at each event time is denoted by: \\[E(d_{jA}) = \\frac{n'_{jA}d_i}{n'_j}\\] and analogously for group B \\[E(d_{jB}) = \\frac{n'_{jB}d_j}{n'_j}\\]\nThe difference between observed and expected number of events is evidence against the null hypothesis. The Mantel-Cox chi-squared (log-rank) its simple for is the combination of these differences over all the times at which deaths (events) occurred: \\[\\chi^2_{MC} = \\frac{(O_A - E_A)^2}{E_A} + \\frac{(O_B - E_B)^2}{E_B}\\]\nwith 1 \\(df\\).\nThe test can be easily expanded to more than two groups, by covering more groups in the summation and having \\(k-1\\) degrees of freedom, where \\(k\\) is the number of groups.\nIn more advanced version, the log-rank test incorporates variance into the test statistics. This ensures more accurate and reliable comparisons of survival distributions, particularly when groups differ in size, have varying numbers of individuals at risk, or experience uneven censoring: \\[\\chi^2_{MC} = \\frac{(O_A - E_A)^2}{V_A} + \\frac{(O_B - E_B)^2}{V_B}\\] where \\(V\\) stands for total variance across all times at which event occurs, with variance at these points calculated as \\[V(d_{jA}) = \\frac{d_j(n'_j - d_j)n'_{jA}n'_{jB}}{n'^2_j(n'_j -1)}\\] for group A (and analogously for other groups in the study).\nIn our extended tooth filling study, we can compare the survival experiences between men and women:\n\n\nCode\n# Perform the log-rank test\nlog_rank_test <- survdiff(surv_object ~ gender, data = data_tooth_ext)\n\n# Display the result of the log-rank test\nprint(log_rank_test)\n\n\nCall:\nsurvdiff(formula = surv_object ~ gender, data = data_tooth_ext)\n\n N Observed Expected (O-E)^2/E (O-E)^2/V\ngender=F 10 8 9.05 0.121 0.302\ngender=M 10 8 6.95 0.158 0.302\n\n Chisq= 0.3 on 1 degrees of freedom, p= 0.6 \n\n\nIn this example, the chi-square values is 0.3, with \\(1\\) degrees of freedom. The critical value for \\(\\chi^2\\) at \\(\\alpha = 0.05\\) is approximately \\(3.84\\) and since 0.3 < \\(3.84\\) we would conclude that we cannot reject the null hypothesis of no difference in survival between the groups (indicated also by the obtained p-value 0.58).", + "crumbs": [ + "1  Introduction" + ] + }, + { + "objectID": "regression.html", + "href": "regression.html", + "title": "2  Regression with survival response", + "section": "", + "text": "2.1 Cox proportional hazards model\nIf survival time is of interest, such as in mortality studies, a common approach is to postulate a distribution for survival time and estimate the parameters of this distribution from the data. For instance, the exponential distribution can be used if the death rate is independent of time, the Weibull distribution allows for increasing or decreasing hazard rates, and the Gompertz distribution models hazard rates that exponentially increase or decrease over time.\nMore commonly, it is the relationship between survival time and one or more predictor variables (covariates) that is of interest. The Cox proportional hazards model is a widely used semi-parametric model to study this relationship. It is used to estimate the hazard ratio for individuals based on their covariates, without needing to specify the baseline hazard function.\nEstimation of \\(\\boldsymbol{\\beta}\\) and inferences are developed by considering the information supplied at each time that a death (event) occurred.\nConsider:\nThen:\nwhere:\nThe risk of death at time \\(t_j\\) in the risk set does not supply absolute measures of risk, but does supply the relative risks for each subject, since, although \\(h_0(t)\\) is unknown, it is the same for each subject. Thus the probability that the death observed at \\(t_j\\) was of the subject who did die at that time is: \\[p_j = \\frac{\\textrm{exp}(\\boldsymbol{\\beta}^T\\boldsymbol{x_1})}{\\sum \\textrm{exp}(\\beta^T\\boldsymbol{x_i})}\\] where summation is over all remembers of the risk set.\nSimilar terms are derived for each time that a death occurred and are combined to form a likelihood (product of these probabilities over all event times \\(t_j\\)): \\[PL(\\boldsymbol{\\beta}) = \\prod_{j=1}^{k} \\frac{\\exp(\\boldsymbol{\\beta}^T \\boldsymbol{x_1})}{\\sum_{i \\in \\text{risk set at } t_j} \\exp(\\boldsymbol{\\beta}^T \\boldsymbol{x_i})}\\] Technically this is called partial likelihood, since the component terms are derived conditionally on the times that deaths occurred and the composition of the risk set at these times. This partial likelihood is used to estimate the regression coefficients \\(\\boldsymbol{\\beta}\\) in the Cox model. Maximizing this partial likelihood gives the best estimates for \\(\\boldsymbol{\\beta}\\), quantifying the effect of the covariates on the hazard.\nTo estimate \\(\\beta\\), we maximize the partial likelihood function with respect to \\(\\beta\\). No closed-form solution is available, and so iterative algorithms are used.", + "crumbs": [ + "2  Regression with survival response" + ] + }, + { + "objectID": "regression.html#cox-proportional-hazards-model", + "href": "regression.html#cox-proportional-hazards-model", + "title": "2  Regression with survival response", + "section": "", + "text": "a death occurring at time \\(t_j\\), and\nsuppose that there were \\(n_j\\) subjects alive just before \\(t_j\\),\nthat the values of \\(\\boldsymbol{x}\\) for these subjects are \\(\\boldsymbol{x_1}, \\boldsymbol{x_2}, \\cdots, \\boldsymbol{x_{n'j}}\\)\nand that the subject that dies is denoted, by the subscript 1\n\n\n\nThe set of \\(n'_j\\) subjects at risk is referred to as the risk set.\nThe risk of death at time \\(t_j\\) for each subject in the risk set is given by: \\[h(t) = h_0(t)\\textrm{exp}(\\boldsymbol{\\beta}^T\\boldsymbol{x})\\]\n\n\n\n\\(\\boldsymbol{\\beta}^T\\boldsymbol{x}\\) is the matrix representation of the regression function, \\(\\beta_1x_1 + \\beta_2x_2 + \\cdots + \\beta_px_p\\)\nand \\(h_0(t)\\) is baseline hazard function", + "crumbs": [ + "2  Regression with survival response" + ] + }, + { + "objectID": "regression.html#example-with-brain-cancer-data", + "href": "regression.html#example-with-brain-cancer-data", + "title": "2  Regression with survival response", + "section": "2.2 Example with brain cancer data", + "text": "2.2 Example with brain cancer data\nThe BrainCancer data set from the ISLR2 package contains survival times for patients with primary brain tumors undergoing treatment with radiation therapy. Variables included in the study are:\n\nsex: male or female\ndiagnosis: meningioma, LG glioma, HG glioma, or other\nloc: tumor location, infratentorial or supratentorial\ngtv: gross tumor volume \\(cm^3\\)\nki: Karnofsky index\nstereo: stereotatcic radiosurgery (SRS) or fractionated stereotactic (SRT) radiotherapy\n\n\n\nCode\n# preview data\ndata_brain <- BrainCancer\n\ndata_brain %>%\n str()\n\n\n'data.frame': 88 obs. of 8 variables:\n $ sex : Factor w/ 2 levels \"Female\",\"Male\": 1 2 1 1 2 1 2 2 1 2 ...\n $ diagnosis: Factor w/ 4 levels \"Meningioma\",\"LG glioma\",..: 1 3 1 2 3 1 1 2 1 3 ...\n $ loc : Factor w/ 2 levels \"Infratentorial\",..: 1 2 1 2 2 2 2 2 2 2 ...\n $ ki : int 90 90 70 80 90 80 80 80 70 100 ...\n $ gtv : num 6.11 19.35 7.95 7.61 5.06 ...\n $ stereo : Factor w/ 2 levels \"SRS\",\"SRT\": 1 2 1 2 2 1 2 2 2 2 ...\n $ status : int 0 1 0 1 1 0 0 0 0 0 ...\n $ time : num 57.64 8.98 26.46 47.8 6.3 ...\n\n\nCode\n# Cox model with multiple predictors\nfit.cox_multi <- coxph(Surv(time, status) ~ sex + diagnosis + ki, data = data_brain)\nsummary(fit.cox_multi)\n\n\nCall:\ncoxph(formula = Surv(time, status) ~ sex + diagnosis + ki, data = data_brain)\n\n n= 87, number of events= 35 \n (1 observation deleted due to missingness)\n\n coef exp(coef) se(coef) z Pr(>|z|) \nsexMale 0.11149 1.11795 0.35670 0.313 0.754608 \ndiagnosisLG glioma 1.01441 2.75774 0.62159 1.632 0.102687 \ndiagnosisHG glioma 2.22962 9.29629 0.44520 5.008 5.5e-07 ***\ndiagnosisOther 0.82773 2.28813 0.57751 1.433 0.151774 \nki -0.06624 0.93591 0.01753 -3.778 0.000158 ***\n---\nSignif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n\n exp(coef) exp(-coef) lower .95 upper .95\nsexMale 1.1179 0.8945 0.5556 2.2493\ndiagnosisLG glioma 2.7577 0.3626 0.8156 9.3251\ndiagnosisHG glioma 9.2963 0.1076 3.8847 22.2467\ndiagnosisOther 2.2881 0.4370 0.7377 7.0967\nki 0.9359 1.0685 0.9043 0.9686\n\nConcordance= 0.782 (se = 0.038 )\nLikelihood ratio test= 36.59 on 5 df, p=7e-07\nWald test = 34.11 on 5 df, p=2e-06\nScore (logrank) test = 40.79 on 5 df, p=1e-07\n\n\n\n2.2.1 Coefficient interpration\nAbove, we fitted Cox model using sex and diagnosis as predictors. The results, indicate, for instance that:\n\nThe estimated hazard ratio (HR) for a male patient is \\(e^{0.11} = 1.11\\). This means that men have 1.11 times the hazard of dying than women while keeping other variables constant (sometimes referred to as adjusted hazard ration, AHR).\nHowever the associated Z value is small, resulting in large p-value of 0.75, which indicates that this difference is not statistically significant.\nWe can also see that one-unit increase in the Karnofsky index is corresponds to a multiplier of \\(e^{-0.06} = 0.94\\) in the chance of dying. The higher the Karnofsky index score, the lower the chance of dying at any given point in time, with this effect being highly significant \\(p\\)-value of 0.000158.\nNote that if you want to find HR associated with more than 1-unit dffererence, e.g. for patients with 10 units higher Karnofsky index, we would calculate HR as \\(e^{-0.06 \\times 10} = 0.54\\). This means that patients with 10 units higher Karnofsky index have 0.54 times the hazard of dying compared to those with lower Karnofsky index.\n\nSometimes, in addition to reporting the numeric results of a Cox regression, forest plots are used to visualize HRs and their 95% confidence intervals.\n\n\nCode\n# make forest plot with ggforest()\nggforest(fit.cox_multi, data = data_brain)\n\n\n\n\n\n\n\n\n\n\n\n2.2.2 Global statistical signficance\nThe output of the Cox regression contains also three methods for an overall test of whether the predictors in the model have a significant effect on the hazard (or risk) of the event occurring. Essentially, these check whether the model as a whole provides a better fit than a null model (a model without any covariates).\nLikelihood Ratio Test (LRT) compares the log-likelihoods of two models:\n\nThe null model, which includes only the baseline hazard (i.e., no covariates).\nThe full model, which includes the covariates of interest.\n\nThe test statistic is computed as: \\[\n2 \\times (\\text{log-likelihood of the full model} - \\text{log-likelihood of the null model})\n\\] This statistic follows a chi-squared distribution, and the associated p-value indicates whether the full model (with covariates) is significantly better at explaining the data than the null model.\nWald Test\nThe Wald test checks whether the estimated regression coefficients are significantly different from zero. For each predictor, the test examines whether its coefficient, \\(\\beta_i\\), is significantly different from 0, implying that the predictor has a significant effect on the hazard. The Wald test can also be used as a global test of significance.\nScore Test (Log-Rank Test)\nThe Score test (also called the log-rank test in this context) evaluates the contribution of each predictor to the model’s fit, considering the expected number of events. Like the likelihood ratio test, the Score test also compares the null and full models and uses a chi-squared distribution to compute significance.\n\n\n2.2.3 Concordance\nIn the context of a Cox proportional hazards model, concordance (also known as the C-index or concordance index) is a measure of how well the model predicts the order of events. Specifically, it assesses the ability of the Cox model to correctly predict which of two individuals will experience an event first, based on their risk scores.\nThe concordance index (C-index) ranges between 0.5 and 1:\n\nA C-index of 0.5 indicates that the model’s predictions are no better than random chance. This means the model cannot distinguish between individuals who experience the event earlier and those who experience it later.\nA C-index of 1 means perfect prediction, indicating that the model always correctly predicts the order of events.\n\nThe C-index is similar to the area under the ROC curve (AUC) used in binary classification models, but it is adapted to survival analysis where the goal is to rank individuals by their risk of experiencing the event.\nIn our example, a concordance of 0.782 means that the Cox model can correctly predict the ordering of survival times 78.2% of the time. In other words, if you randomly select two individuals and compare their predicted risk scores, the model will correctly predict which individual experiences the event earlier in 78.2% of cases. The standard error is 0.038 which provides a measure of uncertainty around the concordance estimate. A smaller standard error indicates more confidence in the estimate.\n\n\n2.2.4 Predictions\nIn Cox regression, we can estimate:\n\nthe survival probability at a specific time, \\(S(t|X=x)\\)\nand the hazard ratio for an individual relative to a reference individual, $h(t|X=x) / \\(S(t|X=x_{ref})\\), where \\(X_{ref}\\) is a reference individual with known covariate values.\n\nFor predicted survival, the choice of time matters. A predicted HR will not depend on time due to proportional hazards assumption.\n\n2.2.4.1 Survival\nLet’s predict the probability of survival at 40 days, for a new patient (man, with LG glioma, Karnofsky index of 80).\n\n# new patient data\nnew_pat <- data.frame(time = 40, \n sex = \"Male\", \n diagnosis = \"LG glioma\",\n ki = 80, \n status = 0)\n\npredict(fit.cox_multi, newdata = new_pat, type = \"survival\", se.fit = T)\n\n$fit\n[1] 0.4824421\n\n$se.fit\n[1] 0.1834607\n\n\nThe probability of surviving through 40 days is 0.48. To get confidence interval we could use:\n\nunlist(summary(survfit(fit.cox_multi,\n newdata = new_pat,\n se.fit =T, conf.int = 0.95),\n times=40)[c(\"surv\", \"std.err\", \"lower\", \"upper\")])\n\n surv std.err lower upper \n0.4824421 0.1834607 0.2289573 1.0000000 \n\n\nSo the probability of surviving through 40 days is 0.48 with 95% CI [0.23, 1].\n\n\n2.2.4.2 HR\nTo predict the adjusted hazard ratio for a new patient, we use again the predict() function, this type with type = \"risk\". We also need to specify the reference individual. Some options include:\n\nreference = \"zero\": the reference individual has continuous predictor values set to 0 and categorical predictors each at their reference level. This is not a good choice if a value of 0 is not plausible for continuous variables in the model, e.g. age.\nreference = \"sample\": the reference individual has continuous predictors each their respective sample mean and categorical predictors at their reference level.\nreference = \"strata\": this leads to the same results as in reference = \"sample\" unless the model includes astrata()`, in which case continuous variables will be set their within-stratum means.\n\n\n# predict HR for our new patient\npredict(fit.cox_multi, \n newdata = new_pat, \n type = \"risk\", \n reference = \"strata\", \n se.fit = T)\n\n$fit\n 1 \n3.276632 \n\n$se.fit\n 1 \n1.201369 \n\n\nOur new patient 3.27 times the hazard of dying compared to a reference individual with level of each categorical predictor and the mean of each continuous predictor, which in our case are (Female for gender, Meningioma for diagnosis, and mean Karnofsky index of 81.02).", + "crumbs": [ + "2  Regression with survival response" + ] + }, + { + "objectID": "regression.html#time-dependent-covariates", + "href": "regression.html#time-dependent-covariates", + "title": "2  Regression with survival response", + "section": "2.3 Time-dependent covariates", + "text": "2.3 Time-dependent covariates\nIn all above example the predictors did not vary over time. Each individual had only one value for each predictor. Since survival data is the result of follow-up over time, it is possible to have predictors that do vary over time. For example, in a study of time to heart attack, researchers could record various other conditions’ occurrence over time, such as hypertension or angina. In a study of juvenile recidivism, researchers could record how education or employment status change over time. Cox regression is able to handle such time-varying predictors (also known as “time dependent covariates”).\nFor time-varying covariates, the hazard function is extended to account for changes in the covariates over time: \\[h(t | \\boldsymbol{x}(t)) = h_0(t) \\cdot \\exp\\left(\\boldsymbol{\\beta}^T \\boldsymbol{x}(t)\\right)\\]\nwhere:\n\n\\(h(t | \\boldsymbol{x}(t))\\) is the hazard at time \\(t\\), conditional on the covariate values at that time.\n\\(\\boldsymbol{x}(t)\\) represents the covariate vector at time \\(t\\). In this case, one or more covariates can change over time.\n\\(\\boldsymbol{\\beta}\\) is the vector of regression coefficients.\n\\(h_0(t)\\) is the baseline hazard function, which remains time-dependent but does not depend on the covariates.\n\nIn practice, time-varying covariates are handled by breaking the follow-up time into intervals where the covariate values are assumed to remain constant, and then updating the covariate values at each interval. Suppose a covariate (e.g., blood pressure, treatment status, or exposure) varies over time. For an individual, we divide their follow-up time into periods where the covariate value is constant, and then treat each period as a separate “observation” in the Cox model. The Cox model then uses these updated covariate values when estimating the hazard at each time point.\nThe partial likelihood function in the Cox model is modified to incorporate the updated covariate values \\(\\boldsymbol{x}(t)\\). For each event time \\(t_j\\), the likelihood is based on the covariate values that are valid at that specific time.\nThe partial likelihood becomes:\n\\[PL(\\boldsymbol{\\beta}) = \\prod_{j=1}^{k} \\frac{\\exp\\left(\\boldsymbol{\\beta}^T \\boldsymbol{x}_1(t_j)\\right)}{\\sum_{i \\in \\text{risk set at } t_j} \\exp\\left(\\boldsymbol{\\beta}^T \\boldsymbol{x}_i(t_j)\\right)}\\]\nwhere:\n\n\\(\\boldsymbol{x}_1(t_j)\\) is the covariate vector for the subject who experienced the event at time \\(t_j\\),\nand \\(\\boldsymbol{x}_i(t_j)\\) is the covariate vector for each subject in the risk set at time \\(t_j\\).\n\nA dataset with time-varying predictors will have multiple rows per individual, with different rows having different values for the time-varying predictors, reflecting how they change over time. Additionally, rather than having a single event time variable, each row will have two time variables indicating the beginning and end of the time interval represented by that row of data.\nFor instance, in the study of heroin usage, we can see a male participant who started using non-prescribed pharmaceutical opioids (NPPOs) at age 19 (time independent covariates). He did not meet the criteria for lifetime opioid dependence at baseline (wave 1), but did at the next interview (wave2), and he first reported using heroin at his 6th interview (both time varying variables).\n\n\nCode\ndata_opioid <- readRDS(\"data/data-opioid.rds\")\n\ndata_opioid %>% \n filter(RANDID == 10) %>%\n select(RANDID, wave, START, STOP, heroin, age_at_init, sex, dep_lifetime)\n\n\n RANDID wave START STOP heroin age_at_init sex dep_lifetime\n1 10 1 3.78 4.26 0 19 Male 0\n2 10 2 4.26 4.78 0 19 Male 1\n3 10 3 4.78 5.29 0 19 Male 1\n4 10 4 5.29 5.84 0 19 Male 1\n5 10 5 5.84 6.27 0 19 Male 1\n6 10 6 6.27 6.79 1 19 Male 1", + "crumbs": [ + "2  Regression with survival response" + ] + }, + { + "objectID": "regression.html#multiple-events-competing-risks", + "href": "regression.html#multiple-events-competing-risks", + "title": "2  Regression with survival response", + "section": "2.4 Multiple events: competing risks", + "text": "2.4 Multiple events: competing risks\nIn survival analysis, competing risks occur when an individual is at risk of more than one mutually exclusive event, and the occurrence of one event precludes the occurrence of the others. Traditional survival models, such as the Cox proportional hazards model, typically focus on a single type of event (e.g., death or failure), but in many real-world scenarios, individuals may face different types of events that compete with each other.\nA competing risk refers to events that prevent the event of primary interest from happening. For example: - In a study of heart disease, death due to cancer is a competing risk if the primary event of interest is death due to heart disease. - In a study of cancer treatment, death from any cause is a competing risk if the event of interest is relapse.\nIn these situations, standard survival analysis methods may overestimate the probability of the primary event because they do not account for the possibility of competing events.\nThe Cumulative Incidence Function (CIF) is commonly used in competing risks analysis. It represents the probability of experiencing a specific event (e.g., death from heart disease) by a certain time, accounting for the presence of competing risks (e.g., death from cancer).\nCIF is a product of two estimates.\n\nThe estimate of hazard at ordered failure time \\(t_j\\) for the event of interest: \\[\\hat{h_c(t_j)}=\\frac{d_{cj}}{n_j}\\] where:\n\n\n\\(m_{cj}\\) denotes the number of events for risk \\(c\\) at time \\(t_j\\)\nand \\(n_j\\) is the number of subjects at risk at time \\(t_j\\)\n\n\nThe estimate of overall probability of surviving previous time: \\[\\hat{S}(t_{j-1})\\] We consider the overall survival as a subject must have survived all other competing events in order to fail from event type \\(c\\) at time \\(t_j\\).\n\nThe estimated incidence probability of failing from even type \\(c\\) at time \\(t_j\\) is then: \\[\\hat{I_c}(t_j) = \\hat{S}(t_{j-1}) \\times \\hat{h_c(t_j)}\\] The probability of failing form even type \\(c\\) at time \\(t_j\\) is a product of surviving the previous time periods and the cause specific hazard at time \\(t_j\\).\nThe CIF for event type \\(c\\) at time \\(t_j\\) is then the cumulative sum up to time \\(t_j\\), i.e. from \\(f' = 1\\) to \\(f'=f\\), of these incidence probabilities over all event type \\(c\\) failure times, which is expressed: \\[CIF_c(t_j) = \\sum_{f'=1}^{f}\\hat{I_c}(t_j) = \\sum_{f'=1}^{f}\\hat{S}(t_{f'-1})\\times\\hat{h_c}(t_{f'})\\]\nIn 1999 Gray (1998) proposed a non-parametric test to compare two or more CIFs. The test is analogous to the log-rank test comparing KM curves, using a modified Chi-squared test statistic. This test does not require the independent censoring assumption.\nIn Fine and Gray Fine and Gray (1999) proposed a proportional hazards model aims at modeling the CIF with covariates, by treating the CIF curve as a subdistribution function. The subdistribution function is analogous to the Cox proportional hazard model, except that it models a hazard function (as known as subdistribution hazard) derived from a CIF.\n\n\n\n\nFine, Jason P., and Robert J. Gray. 1999. “A Proportional Hazards Model for the Subdistribution of a Competing Risk.” Journal of the American Statistical Association 94. https://doi.org/10.1080/01621459.1999.10474144.\n\n\nGray, Robert J. 1998. “A Class of k-Sample Tests for Comparing the Cumulative Incidence of a Competing Risk.” The Annals of Statistics 16. https://doi.org/10.1214/aos/1176350951.", + "crumbs": [ + "2  Regression with survival response" + ] + }, + { + "objectID": "lab.html", + "href": "lab.html", + "title": "3  R examples", + "section": "", + "text": "3.1 KM, log-rank, Cox proportional hazards model\nrm(list=ls())\n\n# load libraries\nlibrary(tidyverse)\nlibrary(skimr)\nlibrary(survival)\n\n# load data\ndata_brain <- readRDS(\"data/data_cancer_brain.rds\")\n\n# preview data\nskim(data_brain)\n\n\nData summary\n\n\nName\ndata_brain\n\n\nNumber of rows\n88\n\n\nNumber of columns\n8\n\n\n_______________________\n\n\n\nColumn type frequency:\n\n\n\nfactor\n4\n\n\nnumeric\n4\n\n\n________________________\n\n\n\nGroup variables\nNone\n\n\n\nVariable type: factor\n\n\n\n\n\n\n\n\n\n\n\nskim_variable\nn_missing\ncomplete_rate\nordered\nn_unique\ntop_counts\n\n\n\n\nsex\n0\n1.00\nFALSE\n2\nFem: 45, Mal: 43\n\n\ndiagnosis\n1\n0.99\nFALSE\n4\nMen: 42, HG : 22, Oth: 14, LG : 9\n\n\nloc\n0\n1.00\nFALSE\n2\nSup: 69, Inf: 19\n\n\nstereo\n0\n1.00\nFALSE\n2\nSRT: 65, SRS: 23\n\n\n\nVariable type: numeric\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nskim_variable\nn_missing\ncomplete_rate\nmean\nsd\np0\np25\np50\np75\np100\nhist\n\n\n\n\nki\n0\n1\n81.02\n10.51\n40.00\n80.00\n80.00\n90.0\n100.00\n▁▁▃▇▇\n\n\ngtv\n0\n1\n8.66\n8.66\n0.01\n2.50\n6.51\n12.1\n34.64\n▇▃▁▁▁\n\n\nstatus\n0\n1\n0.40\n0.49\n0.00\n0.00\n0.00\n1.0\n1.00\n▇▁▁▁▅\n\n\ntime\n0\n1\n27.46\n20.12\n0.07\n10.39\n24.03\n41.6\n82.56\n▇▅▅▂▁\n\n\n\n\n# status = 0, indicates a censored observation \n# status = 1, indicates an uncensored observation\n\n# Kaplan-Meier survival curve\nfit.surv <- survfit(Surv(time, status) ~ 1, data = data_brain)\nplot(fit.surv, xlab = \"Months\", ylab = \"Estiamted Probabiliy of Survival\")\n\n\n\n\n\n\n\n\n# KM stratify by sex\nfit.surv <- survfit(Surv(time, status) ~ sex, data = data_brain)\nplot(fit.surv, mark.time = TRUE, col = c(2, 4), xlab = \"Months\", ylab = \"Estiamted Probabiliy of Survival\")\n\n\n\n\n\n\n\n\n# Log-rank test to compare survival of men and women\nlogrank_test <- survdiff(Surv(time, status) ~ sex, data = data_brain)\nprint(logrank_test)\n## Call:\n## survdiff(formula = Surv(time, status) ~ sex, data = data_brain)\n## \n## N Observed Expected (O-E)^2/E (O-E)^2/V\n## sex=Female 45 15 18.5 0.676 1.44\n## sex=Male 43 20 16.5 0.761 1.44\n## \n## Chisq= 1.4 on 1 degrees of freedom, p= 0.2\n\n# Fit Cox proportional hazards model\nfit.cox <- coxph(Surv(time, status) ~ sex, data = data_brain)\nsummary(fit.cox)\n## Call:\n## coxph(formula = Surv(time, status) ~ sex, data = data_brain)\n## \n## n= 88, number of events= 35 \n## \n## coef exp(coef) se(coef) z Pr(>|z|)\n## sexMale 0.4077 1.5033 0.3420 1.192 0.233\n## \n## exp(coef) exp(-coef) lower .95 upper .95\n## sexMale 1.503 0.6652 0.769 2.939\n## \n## Concordance= 0.565 (se = 0.045 )\n## Likelihood ratio test= 1.44 on 1 df, p=0.2\n## Wald test = 1.42 on 1 df, p=0.2\n## Score (logrank) test = 1.44 on 1 df, p=0.2\n\n# Test the proportional hazards assumption in the two gender groups\nph_test <- cox.zph(fit.cox)\nprint(ph_test)\n## chisq df p\n## sex 0.588 1 0.44\n## GLOBAL 0.588 1 0.44\n\n# Fit Cox model with multiple predictors\nfit.cox_multi <- coxph(Surv(time, status) ~ sex + diagnosis + loc + ki + gtv + stereo, data = data_brain)\nsummary(fit.cox_multi)\n## Call:\n## coxph(formula = Surv(time, status) ~ sex + diagnosis + loc + \n## ki + gtv + stereo, data = data_brain)\n## \n## n= 87, number of events= 35 \n## (1 observation deleted due to missingness)\n## \n## coef exp(coef) se(coef) z Pr(>|z|) \n## sexMale 0.18375 1.20171 0.36036 0.510 0.61012 \n## diagnosisLG glioma 0.91502 2.49683 0.63816 1.434 0.15161 \n## diagnosisHG glioma 2.15457 8.62414 0.45052 4.782 1.73e-06 ***\n## diagnosisOther 0.88570 2.42467 0.65787 1.346 0.17821 \n## locSupratentorial 0.44119 1.55456 0.70367 0.627 0.53066 \n## ki -0.05496 0.94653 0.01831 -3.001 0.00269 ** \n## gtv 0.03429 1.03489 0.02233 1.536 0.12466 \n## stereoSRT 0.17778 1.19456 0.60158 0.296 0.76760 \n## ---\n## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n## \n## exp(coef) exp(-coef) lower .95 upper .95\n## sexMale 1.2017 0.8321 0.5930 2.4352\n## diagnosisLG glioma 2.4968 0.4005 0.7148 8.7215\n## diagnosisHG glioma 8.6241 0.1160 3.5664 20.8546\n## diagnosisOther 2.4247 0.4124 0.6678 8.8031\n## locSupratentorial 1.5546 0.6433 0.3914 6.1741\n## ki 0.9465 1.0565 0.9132 0.9811\n## gtv 1.0349 0.9663 0.9906 1.0812\n## stereoSRT 1.1946 0.8371 0.3674 3.8839\n## \n## Concordance= 0.794 (se = 0.04 )\n## Likelihood ratio test= 41.37 on 8 df, p=2e-06\n## Wald test = 38.7 on 8 df, p=6e-06\n## Score (logrank) test = 46.59 on 8 df, p=2e-07\n\n# Plot survival curves for each diagnosis category, adjusting for other predictors\n# we set the values of the other predictors to the mean for quantitative predictors, \n# and to the modal value for factors\n\nmodeldata <- data.frame(\n diagnosis = levels(data_brain$diagnosis), \n sex = rep(\"Female\", 4), \n loc = rep(\"Supratentorial\", 4), \n ki = rep(mean(data_brain$ki), 4),\n gtv = rep(mean(data_brain$gtv), 4),\n stereo = rep(\"SRT\", 4)\n)\n\nsurvplots <- survfit(fit.cox_multi, newdata = modeldata)\nplot(survplots, mark.time = FALSE, xlab = \"Months\", ylab = \"Estiamted Probabiliy of Survival\", col = 2:5)\nlegend(\"bottomleft\", legend = levels(data_brain$diagnosis), col = 2:5, lty = 1)", + "crumbs": [ + "3  R examples" + ] + }, + { + "objectID": "lab.html#time-varying-predictors", + "href": "lab.html#time-varying-predictors", + "title": "3  R examples", + "section": "3.2 Time-varying predictors", + "text": "3.2 Time-varying predictors\nadpoted from https://www.bookdown.org/rwnahhas/RMPH/\nA dataset with time-varying predictors will have multiple rows per individual, with different rows having different values for the time-varying predictors, reflecting how they change over time. Additionally, rather than having a single event time variable, each row will have two time variables indicating the beginning and end of the time interval represented by that row of data.\n\ndata_opioid <- readRDS(\"data/data-opioid.rds\")\n\ndata_opioid %>% \n filter(RANDID == 10) %>%\n select(RANDID, wave, START, STOP, heroin, age_at_init, sex, dep_lifetime)\n## RANDID wave START STOP heroin age_at_init sex dep_lifetime\n## 1 10 1 3.78 4.26 0 19 Male 0\n## 2 10 2 4.26 4.78 0 19 Male 1\n## 3 10 3 4.78 5.29 0 19 Male 1\n## 4 10 4 5.29 5.84 0 19 Male 1\n## 5 10 5 5.84 6.27 0 19 Male 1\n## 6 10 6 6.27 6.79 1 19 Male 1\n\nHere, we see a preview of the data_opioid dataset containing longitudinal information for 362 individuals, who at baseline had used non-prescribed pharmaceutical opioids (NPPO, “pain pills”), but were not dependent on NPPOs and had never used heroin.\n\nEach row contains the time variables START and STOP which define the time interval (years from initiation of NPPO use) associated with that row.\nTime-invariant variables in the dataset are constant over all rows for the same individual, while time-varying variables can change between rows.\nEach (START, STOP] interval defines a period of time during which no variables changed.\nTwo time-varying variables in the dataset are heroin use (heroin) (the event indicator variable) and lifetime opioid dependence (dep_lifetime)\n\nFor instance, for this particular individual, male, started using NPPOs at age 19 (time invariant variables). He did not meet the criteria for lifetime opioid dependence at baseline (wave = 0), but did at the next interview (wave = 1) and he first reported using heroin at his 6th interview.\n\nTo model the heroin usage using both time-invariant and time-varying predictors, we can use the coxph function:\n\ncox.timevar <- coxph(Surv(START, STOP, heroin) ~\n age_at_init + sex + dep_lifetime,\n data = data_opioid)\n\nsummary(cox.timevar)\n## Call:\n## coxph(formula = Surv(START, STOP, heroin) ~ age_at_init + sex + \n## dep_lifetime, data = data_opioid)\n## \n## n= 1853, number of events= 27 \n## \n## coef exp(coef) se(coef) z Pr(>|z|) \n## age_at_init -0.3932 0.6749 0.1350 -2.911 0.00360 **\n## sexMale 0.1515 1.1635 0.3925 0.386 0.69956 \n## dep_lifetime 1.0580 2.8807 0.3991 2.651 0.00802 **\n## ---\n## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n## \n## exp(coef) exp(-coef) lower .95 upper .95\n## age_at_init 0.6749 1.4817 0.5180 0.8794\n## sexMale 1.1635 0.8594 0.5391 2.5111\n## dep_lifetime 2.8807 0.3471 1.3176 6.2981\n## \n## Concordance= 0.716 (se = 0.042 )\n## Likelihood ratio test= 15.24 on 3 df, p=0.002\n## Wald test = 14.53 on 3 df, p=0.002\n## Score (logrank) test = 15.45 on 3 df, p=0.001\n\nWhether a predictor is time-varying or time-invariant, its HR can be interpreted as a comparison of the hazard between groups of individuals with different values of that predictor. Thus, in this example, we could conclude that after adjusting for age at NPPO initiation and sex, those with lifetime opioid dependence have 2.88 times the hazard of using heroin as those who do not (AHR = 2.88; 95% CI = 1.32, 6.30; p = 0.008). The HR for a time-varying predictor, however, can also be interpreted as the effect of within-individual change on the hazard. If an individual without opioid dependence transitions to dependence, their hazard of transitioning to heroin is multiplied by 2.88.", + "crumbs": [ + "3  R examples" + ] + }, + { + "objectID": "lab.html#competing-risks", + "href": "lab.html#competing-risks", + "title": "3  R examples", + "section": "3.3 Competing risks", + "text": "3.3 Competing risks\n\n3.3.1 Comparing CIFs\nLet’s demonstrate how to analyze competing risks using the cmprsk package in R. This package includes functions for estimating the cumulative incidence function (CIF) in the presence of competing risks.\nBone marrow transplant (BMT) data from the cmprsk package includes information for 35 leukemia cancer patients who underwent bone marrow transplantation. The data includes the following variables:\n\ndis: disease; 0 = ALL; 1 = AML (ALL, Acute lymphoblastic leukemia; AML, Acute myeloid leukemia)\nftime: follow-up time\nstatus: 0 = censored (survival); 1 = Transplant-related mortality; 2 = relapse\n\n\nlibrary(cmprsk)\nlibrary(survminer)\n\n# preview data\ndata_bmt <- readRDS(\"data/data-bmt.rds\")\nstr(data_bmt)\n## 'data.frame': 35 obs. of 3 variables:\n## $ dis : int 0 0 0 0 0 1 0 0 1 1 ...\n## $ ftime : int 13 1 72 7 8 67 9 5 70 4 ...\n## $ status: int 2 1 0 2 2 0 2 2 0 0 ...\n\n# label levels\ndata_bmt <- data_bmt %>%\n mutate(dis = factor(dis, levels = c(0,1), labels = c(\"ALL\", \"AML\")), \n status = factor(status, levels = c(0,1,2), labels = c(\"Censored\",\"Mortality\",\"Relapse\")))\n\nstr(data_bmt)\n## 'data.frame': 35 obs. of 3 variables:\n## $ dis : Factor w/ 2 levels \"ALL\",\"AML\": 1 1 1 1 1 2 1 1 2 2 ...\n## $ ftime : int 13 1 72 7 8 67 9 5 70 4 ...\n## $ status: Factor w/ 3 levels \"Censored\",\"Mortality\",..: 3 2 1 3 3 1 3 3 1 1 ...\n\n# Estimate the cumulative incidence functions for each event type\ncif <- cuminc(ftime = data_bmt$ftime, # Failure time variable\n fstatus = data_bmt$status) # Codes for different causes of failure\n\n# Print the CIF object\nprint(cif)\n## Estimates and Variances:\n## $est\n## 20 40 60\n## 1 Censored 0.1142857 0.2000000 0.2000000\n## 1 Mortality 0.2571429 0.2571429 0.2571429\n## 1 Relapse 0.3714286 0.4285714 0.4285714\n## \n## $var\n## 20 40 60\n## 1 Censored 0.002998057 0.004910750 0.004910750\n## 1 Mortality 0.005636800 0.005636800 0.005636800\n## 1 Relapse 0.006878377 0.007269181 0.007269181\n\n# Plot the CIFs for all even types\nplot(cif, lty = 1, col = c(\"green\", \"red\", \"blue\"), xlab = \"Time\", ylab = \"Cumulative Incidence\",\n main = \"Cumulative Incidence Functions for Competing Risks\", curvlab = c(\"Censored\", \"Mortality\", \"Relapse\"))\n\n\n\n\n\n\n\n\n# Estimate the cumulative incidence functions fo each even type per group\ncif <- cuminc(ftime = data_bmt$ftime, \n fstatus = data_bmt$status, \n group = data_bmt$dis) # Estimates will calculated within groups\n\n# Plot the CIFs for both event types using survminer package\nggcompetingrisks(cif, multiple_panels = FALSE, legend = \"right\")\n\n\n\n\n\n\n\n\n# Show Gray's test for equality of CIFs\nprint(cif$Tests)\n## stat pv df\n## Censored 5.9785107 0.014481226 1\n## Mortality 0.9133497 0.339227192 1\n## Relapse 9.4874094 0.002068867 1\n\nGray’s test for equality of CIFs across groups (ALL vs. AML) is split across different types of events, here (Censored, Mortality, and Relapse).\n\nFor Censored, the p-value is small and suggest that there isa statistically significant difference in the CIF for censored events between the groups.\nFor Mortality, the p-value is greater than 0.05, suggesting that there is no statistically significant difference in the CIF for mortality between the groups.\nFor Relapse, the p-value is less than 0.05, indicating a statistically significant difference in the CIF for relapse between the groups.\nThe above suggest that the groups are behaving differently in terms of the time to relapse and censored events, but not for mortality.\n\n\n\n3.3.2 Competing Risks Regression\nadopted from https://www.nature.com/articles/bmt2009359\nSuppose that the BMT study was extended to include more participants and additional covariates and now includes:\n\n177 observations\nSex: gender of the individual\nD: disease; 0 = ALL; 1 = AML\nPhase: phase at transplant (Relapse, CR1, CR2, CR3)\nAge: age at the beginning of follow-up\nStatus: 0 = censored; 1 = Transplant-related mortality; 2 = relapse\nSource: source of stem cells (BM+PB, PB)\nftime: failure time\n\nWe are interested in modeling time to relapse in the presence of transplant-related death (competing event). We want to sutdy the effect on relapse of sex, disease type, phase at transplant, source of stem cells and age.\n\n# load additional libraries\nlibrary(fastDummies)\n\n# load data\ndata_bmtcrr <- readRDS(\"data/data-bmtcrr.rds\")\n\n# preview data\nstr(data_bmtcrr)\n## 'data.frame': 177 obs. of 7 variables:\n## $ Sex : Factor w/ 2 levels \"F\",\"M\": 2 1 2 1 1 2 2 1 2 1 ...\n## $ D : Factor w/ 2 levels \"ALL\",\"AML\": 1 2 1 1 1 1 1 1 1 1 ...\n## $ Phase : Factor w/ 4 levels \"CR1\",\"CR2\",\"CR3\",..: 4 2 3 2 2 4 1 1 1 4 ...\n## $ Age : int 48 23 7 26 36 17 7 17 26 8 ...\n## $ Status: int 2 1 0 2 2 2 0 2 0 1 ...\n## $ Source: Factor w/ 2 levels \"BM+PB\",\"PB\": 1 1 1 1 1 1 1 1 1 1 ...\n## $ ftime : num 0.67 9.5 131.77 24.03 1.47 ...\n\n# prepare matrix of covariates\n# factor variables need to be coded as dummy variables\nx_sex <- dummy_cols(data_bmtcrr$Sex) %>% dplyr::select(.data_F)\nx_phase <- dummy_cols(data_bmtcrr$Phase) %>% dplyr::select(.data_CR1, .data_CR2, .data_CR3)\nx_d <- dummy_cols(data_bmtcrr$D) %>% dplyr::select(.data_AML)\nx_source <- dummy_cols(data_bmtcrr$Source) %>% dplyr::select(.data_PB)\nx <- data.frame(age = data_bmtcrr$Age, sex = x_sex, x_d, x_phase, x_source)\ncolnames(x) <- c(\"age\", \"sex_F\", \"D_AML\", \"P:CR1\", \"P:CR2\", \"P:CR3\", \"source:PB\")\n\n# The first regression model for relapse can be produced by typing\nmod1 <- crr(ftime = data_bmtcrr$ftime, \n fstatus = data_bmtcrr$Status, \n cov1 = x)\n\nsummary(mod1)\n## Competing Risks Regression\n## \n## Call:\n## crr(ftime = data_bmtcrr$ftime, fstatus = data_bmtcrr$Status, \n## cov1 = x)\n## \n## coef exp(coef) se(coef) z p-value\n## age -0.0185 0.982 0.0119 -1.554 0.1200\n## sex_F -0.0352 0.965 0.2900 -0.122 0.9000\n## D_AML -0.4723 0.624 0.3054 -1.547 0.1200\n## P:CR1 -1.1018 0.332 0.3764 -2.927 0.0034\n## P:CR2 -1.0200 0.361 0.3558 -2.867 0.0041\n## P:CR3 -0.7314 0.481 0.5766 -1.268 0.2000\n## source:PB 0.9211 2.512 0.5530 1.666 0.0960\n## \n## exp(coef) exp(-coef) 2.5% 97.5%\n## age 0.982 1.019 0.959 1.005\n## sex_F 0.965 1.036 0.547 1.704\n## D_AML 0.624 1.604 0.343 1.134\n## P:CR1 0.332 3.009 0.159 0.695\n## P:CR2 0.361 2.773 0.180 0.724\n## P:CR3 0.481 2.078 0.155 1.490\n## source:PB 2.512 0.398 0.850 7.426\n## \n## Num. cases = 177\n## Pseudo Log-likelihood = -267 \n## Pseudo likelihood ratio test = 24.4 on 7 df,\n\nThe first part of the output shows for each term in the design matrix the estimated coefficient \\(\\hat{\\beta}_j\\), the relative risk \\(\\mathrm{exp}(\\hat{\\beta}_j)\\), the standard error, the z-value and the corresponding P-value for assessing significance.\nHere, Sex is not significant, followed by Age and D (disease type), whereas Source is only marginally significant. Phase is a factor with relapse as baseline, so each P-value provides a test for the difference of each level with respect to the baseline.\nAn overall P-value for Phase can be obtained through the Wald test via aod R package:\n\nlibrary(aod)\nwald.test(mod1$var, mod1$coef, Terms = 4:6)\n## Wald test:\n## ----------\n## \n## Chi-squared test:\n## X2 = 14.0, df = 3, P(> X2) = 0.0029\n\nThe first argument to the function wald.test() is the estimated covariance matrix for the coefficients, followed by the vector of coefficients estimates, and the position of coefficients for which we want to assess significance. In our case, the P-value indicates that Phase is statistically significant.\nThe second part of the output for competing risks regression shows the relative risk for each term, and a 95% confidence interval. The relative risk or subdistribution hazard ratio for a categorical covariate is the ratio of subdistribution hazards for the actual group with respect to the baseline, with all other covariates being equal. If the covariate is continuous then the relative risk refers to the effect of a one unit increase in the covariate, with all other covariates being equal. In our data, exp(−0.0352)=0.965 is the relative risk of a woman with respect to a man, and exp(−0.0185)=0.982 is the relative risk for a 1 year increase in age.\nThe last part of the output shows the pseudo log-likelihood at maximum and the pseudo likelihood ratio test, that is, the difference in the objective function at the global null and at the final estimates. As this objective function is not a true likelihood, this test statistic is not asymptotically distributed as a χ2. As a consequence, model comparison based on likelihood ratio approach cannot be performed directly, but significance must be evaluated through simulations. However, a model selection criterion can be easily adopted as described in the following section.\nFor an example on model selection, model diagnostics or adding time-varying covariates in presence of competing events, please refer to the original tutorial article Scrucca, Santucci, and Aversa (2010) https://www.nature.com/articles/bmt2009359.\n\n\n\n\nScrucca, L, A Santucci, and F Aversa. 2010. “Regression Modeling of Competing Risk Using r: An in Depth Guide for Clinicians.” Bone Marrow Transplantation 45: 1388–95. https://doi.org/10.1038/bmt.2009.359.", + "crumbs": [ + "3  R examples" + ] + }, + { + "objectID": "references.html", + "href": "references.html", + "title": "References", + "section": "", + "text": "Fine, Jason P., and Robert J. Gray. 1999. “A Proportional Hazards\nModel for the Subdistribution of a Competing Risk.” Journal\nof the American Statistical Association 94. https://doi.org/10.1080/01621459.1999.10474144.\n\n\nGray, Robert J. 1998. “A Class of k-Sample Tests for Comparing the\nCumulative Incidence of a Competing Risk.” The Annals of\nStatistics 16. https://doi.org/10.1214/aos/1176350951.\n\n\nScrucca, L, A Santucci, and F Aversa. 2010. “Regression Modeling\nof Competing Risk Using r: An in Depth Guide for Clinicians.”\nBone Marrow Transplantation 45: 1388–95. https://doi.org/10.1038/bmt.2009.359.", + "crumbs": [ + "References" + ] + } +] \ No newline at end of file diff --git a/session-survival/docs/site_libs/bootstrap/bootstrap-icons.css b/session-survival/docs/site_libs/bootstrap/bootstrap-icons.css new file mode 100644 index 00000000..285e4448 --- /dev/null +++ b/session-survival/docs/site_libs/bootstrap/bootstrap-icons.css @@ -0,0 +1,2078 @@ +/*! + * Bootstrap Icons v1.11.1 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.offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand-xxl .offcanvas .offcanvas-header{display:none}.navbar-expand-xxl .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}}.navbar-expand{flex-wrap:nowrap;-webkit-flex-wrap:nowrap;justify-content:flex-start;-webkit-justify-content:flex-start}.navbar-expand .navbar-nav{flex-direction:row;-webkit-flex-direction:row}.navbar-expand .navbar-nav .dropdown-menu{position:absolute}.navbar-expand .navbar-nav .nav-link{padding-right:var(--bs-navbar-nav-link-padding-x);padding-left:var(--bs-navbar-nav-link-padding-x)}.navbar-expand .navbar-nav-scroll{overflow:visible}.navbar-expand .navbar-collapse{display:flex !important;display:-webkit-flex !important;flex-basis:auto;-webkit-flex-basis:auto}.navbar-expand .navbar-toggler{display:none}.navbar-expand .offcanvas{position:static;z-index:auto;flex-grow:1;-webkit-flex-grow:1;width:auto !important;height:auto !important;visibility:visible !important;background-color:rgba(0,0,0,0) !important;border:0 !important;transform:none !important;transition:none}.navbar-expand .offcanvas .offcanvas-header{display:none}.navbar-expand .offcanvas .offcanvas-body{display:flex;display:-webkit-flex;flex-grow:0;-webkit-flex-grow:0;padding:0;overflow-y:visible}.navbar-dark,.navbar[data-bs-theme=dark]{--bs-navbar-color: #fdfefe;--bs-navbar-hover-color: rgba(253, 254, 255, 0.8);--bs-navbar-disabled-color: rgba(253, 254, 254, 0.75);--bs-navbar-active-color: #fdfeff;--bs-navbar-brand-color: #fdfefe;--bs-navbar-brand-hover-color: #fdfeff;--bs-navbar-toggler-border-color: rgba(253, 254, 254, 0);--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23fdfefe' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}[data-bs-theme=dark] .navbar-toggler-icon{--bs-navbar-toggler-icon-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 30 30'%3e%3cpath stroke='%23fdfefe' stroke-linecap='round' stroke-miterlimit='10' stroke-width='2' d='M4 7h22M4 15h22M4 23h22'/%3e%3c/svg%3e")}.card{--bs-card-spacer-y: 1rem;--bs-card-spacer-x: 1rem;--bs-card-title-spacer-y: 0.5rem;--bs-card-title-color: ;--bs-card-subtitle-color: ;--bs-card-border-width: 1px;--bs-card-border-color: rgba(0, 0, 0, 0.175);--bs-card-border-radius: 0.25rem;--bs-card-box-shadow: ;--bs-card-inner-border-radius: calc(0.25rem - 1px);--bs-card-cap-padding-y: 0.5rem;--bs-card-cap-padding-x: 1rem;--bs-card-cap-bg: rgba(52, 58, 64, 0.25);--bs-card-cap-color: ;--bs-card-height: ;--bs-card-color: ;--bs-card-bg: #ffffff;--bs-card-img-overlay-padding: 1rem;--bs-card-group-margin: 0.75rem;position:relative;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;min-width:0;height:var(--bs-card-height);color:var(--bs-body-color);word-wrap:break-word;background-color:var(--bs-card-bg);background-clip:border-box;border:var(--bs-card-border-width) solid var(--bs-card-border-color);border-radius:var(--bs-card-border-radius)}.card>hr{margin-right:0;margin-left:0}.card>.list-group{border-top:inherit;border-bottom:inherit}.card>.list-group:first-child{border-top-width:0;border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card>.list-group:last-child{border-bottom-width:0;border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card>.card-header+.list-group,.card>.list-group+.card-footer{border-top:0}.card-body{flex:1 1 auto;-webkit-flex:1 1 auto;padding:var(--bs-card-spacer-y) var(--bs-card-spacer-x);color:var(--bs-card-color)}.card-title{margin-bottom:var(--bs-card-title-spacer-y);color:var(--bs-card-title-color)}.card-subtitle{margin-top:calc(-0.5*var(--bs-card-title-spacer-y));margin-bottom:0;color:var(--bs-card-subtitle-color)}.card-text:last-child{margin-bottom:0}.card-link+.card-link{margin-left:var(--bs-card-spacer-x)}.card-header{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);margin-bottom:0;color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-bottom:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-header:first-child{border-radius:var(--bs-card-inner-border-radius) var(--bs-card-inner-border-radius) 0 0}.card-footer{padding:var(--bs-card-cap-padding-y) var(--bs-card-cap-padding-x);color:var(--bs-card-cap-color);background-color:var(--bs-card-cap-bg);border-top:var(--bs-card-border-width) solid var(--bs-card-border-color)}.card-footer:last-child{border-radius:0 0 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.nav-link.active{background-color:var(--bs-card-bg);border-bottom-color:var(--bs-card-bg)}.card-header-pills{margin-right:calc(-0.5*var(--bs-card-cap-padding-x));margin-left:calc(-0.5*var(--bs-card-cap-padding-x))}.card-img-overlay{position:absolute;top:0;right:0;bottom:0;left:0;padding:var(--bs-card-img-overlay-padding);border-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-top,.card-img-bottom{width:100%}.card-img,.card-img-top{border-top-left-radius:var(--bs-card-inner-border-radius);border-top-right-radius:var(--bs-card-inner-border-radius)}.card-img,.card-img-bottom{border-bottom-right-radius:var(--bs-card-inner-border-radius);border-bottom-left-radius:var(--bs-card-inner-border-radius)}.card-group>.card{margin-bottom:var(--bs-card-group-margin)}@media(min-width: 576px){.card-group{display:flex;display:-webkit-flex;flex-flow:row wrap;-webkit-flex-flow:row wrap}.card-group>.card{flex:1 0 0%;-webkit-flex:1 0 0%;margin-bottom:0}.card-group>.card+.card{margin-left:0;border-left:0}.card-group>.card:not(:last-child){border-top-right-radius:0;border-bottom-right-radius:0}.card-group>.card:not(:last-child) .card-img-top,.card-group>.card:not(:last-child) .card-header{border-top-right-radius:0}.card-group>.card:not(:last-child) .card-img-bottom,.card-group>.card:not(:last-child) .card-footer{border-bottom-right-radius:0}.card-group>.card:not(:first-child){border-top-left-radius:0;border-bottom-left-radius:0}.card-group>.card:not(:first-child) .card-img-top,.card-group>.card:not(:first-child) .card-header{border-top-left-radius:0}.card-group>.card:not(:first-child) .card-img-bottom,.card-group>.card:not(:first-child) .card-footer{border-bottom-left-radius:0}}.accordion{--bs-accordion-color: #212529;--bs-accordion-bg: #ffffff;--bs-accordion-transition: color 0.15s ease-in-out, background-color 0.15s ease-in-out, border-color 0.15s ease-in-out, box-shadow 0.15s ease-in-out, border-radius 0.15s ease;--bs-accordion-border-color: #dee2e6;--bs-accordion-border-width: 1px;--bs-accordion-border-radius: 0.25rem;--bs-accordion-inner-border-radius: calc(0.25rem - 1px);--bs-accordion-btn-padding-x: 1.25rem;--bs-accordion-btn-padding-y: 1rem;--bs-accordion-btn-color: #212529;--bs-accordion-btn-bg: #ffffff;--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23212529'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-icon-width: 1.25rem;--bs-accordion-btn-icon-transform: rotate(-180deg);--bs-accordion-btn-icon-transition: transform 0.2s ease-in-out;--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23052c65'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-focus-border-color: #86b7fe;--bs-accordion-btn-focus-box-shadow: 0 0 0 0.25rem rgba(13, 110, 253, 0.25);--bs-accordion-body-padding-x: 1.25rem;--bs-accordion-body-padding-y: 1rem;--bs-accordion-active-color: #052c65;--bs-accordion-active-bg: #cfe2ff}.accordion-button{position:relative;display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;width:100%;padding:var(--bs-accordion-btn-padding-y) var(--bs-accordion-btn-padding-x);font-size:1rem;color:var(--bs-accordion-btn-color);text-align:left;background-color:var(--bs-accordion-btn-bg);border:0;border-radius:0;overflow-anchor:none;transition:var(--bs-accordion-transition)}@media(prefers-reduced-motion: reduce){.accordion-button{transition:none}}.accordion-button:not(.collapsed){color:var(--bs-accordion-active-color);background-color:var(--bs-accordion-active-bg);box-shadow:inset 0 calc(-1*var(--bs-accordion-border-width)) 0 var(--bs-accordion-border-color)}.accordion-button:not(.collapsed)::after{background-image:var(--bs-accordion-btn-active-icon);transform:var(--bs-accordion-btn-icon-transform)}.accordion-button::after{flex-shrink:0;-webkit-flex-shrink:0;width:var(--bs-accordion-btn-icon-width);height:var(--bs-accordion-btn-icon-width);margin-left:auto;content:"";background-image:var(--bs-accordion-btn-icon);background-repeat:no-repeat;background-size:var(--bs-accordion-btn-icon-width);transition:var(--bs-accordion-btn-icon-transition)}@media(prefers-reduced-motion: reduce){.accordion-button::after{transition:none}}.accordion-button:hover{z-index:2}.accordion-button:focus{z-index:3;border-color:var(--bs-accordion-btn-focus-border-color);outline:0;box-shadow:var(--bs-accordion-btn-focus-box-shadow)}.accordion-header{margin-bottom:0}.accordion-item{color:var(--bs-accordion-color);background-color:var(--bs-accordion-bg);border:var(--bs-accordion-border-width) solid var(--bs-accordion-border-color)}.accordion-item:first-of-type{border-top-left-radius:var(--bs-accordion-border-radius);border-top-right-radius:var(--bs-accordion-border-radius)}.accordion-item:first-of-type .accordion-button{border-top-left-radius:var(--bs-accordion-inner-border-radius);border-top-right-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:not(:first-of-type){border-top:0}.accordion-item:last-of-type{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-item:last-of-type .accordion-button.collapsed{border-bottom-right-radius:var(--bs-accordion-inner-border-radius);border-bottom-left-radius:var(--bs-accordion-inner-border-radius)}.accordion-item:last-of-type .accordion-collapse{border-bottom-right-radius:var(--bs-accordion-border-radius);border-bottom-left-radius:var(--bs-accordion-border-radius)}.accordion-body{padding:var(--bs-accordion-body-padding-y) var(--bs-accordion-body-padding-x)}.accordion-flush .accordion-collapse{border-width:0}.accordion-flush .accordion-item{border-right:0;border-left:0;border-radius:0}.accordion-flush .accordion-item:first-child{border-top:0}.accordion-flush .accordion-item:last-child{border-bottom:0}.accordion-flush .accordion-item .accordion-button,.accordion-flush .accordion-item .accordion-button.collapsed{border-radius:0}[data-bs-theme=dark] .accordion-button::after{--bs-accordion-btn-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e");--bs-accordion-btn-active-icon: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%236ea8fe'%3e%3cpath fill-rule='evenodd' d='M1.646 4.646a.5.5 0 0 1 .708 0L8 10.293l5.646-5.647a.5.5 0 0 1 .708.708l-6 6a.5.5 0 0 1-.708 0l-6-6a.5.5 0 0 1 0-.708z'/%3e%3c/svg%3e")}.breadcrumb{--bs-breadcrumb-padding-x: 0;--bs-breadcrumb-padding-y: 0;--bs-breadcrumb-margin-bottom: 1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color: rgba(33, 37, 41, 0.75);--bs-breadcrumb-item-padding-x: 0.5rem;--bs-breadcrumb-item-active-color: rgba(33, 37, 41, 0.75);display:flex;display:-webkit-flex;flex-wrap:wrap;-webkit-flex-wrap:wrap;padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x);margin-bottom:var(--bs-breadcrumb-margin-bottom);font-size:var(--bs-breadcrumb-font-size);list-style:none;background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius)}.breadcrumb-item+.breadcrumb-item{padding-left:var(--bs-breadcrumb-item-padding-x)}.breadcrumb-item+.breadcrumb-item::before{float:left;padding-right:var(--bs-breadcrumb-item-padding-x);color:var(--bs-breadcrumb-divider-color);content:var(--bs-breadcrumb-divider, ">") /* rtl: var(--bs-breadcrumb-divider, ">") */}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.pagination{--bs-pagination-padding-x: 0.75rem;--bs-pagination-padding-y: 0.375rem;--bs-pagination-font-size:1rem;--bs-pagination-color: #0d6efd;--bs-pagination-bg: #ffffff;--bs-pagination-border-width: 1px;--bs-pagination-border-color: #dee2e6;--bs-pagination-border-radius: 0.25rem;--bs-pagination-hover-color: #0a58ca;--bs-pagination-hover-bg: #f8f9fa;--bs-pagination-hover-border-color: #dee2e6;--bs-pagination-focus-color: #0a58ca;--bs-pagination-focus-bg: #e9ecef;--bs-pagination-focus-box-shadow: 0 0 0 0.25rem rgba(13, 110, 253, 0.25);--bs-pagination-active-color: #ffffff;--bs-pagination-active-bg: #0d6efd;--bs-pagination-active-border-color: #0d6efd;--bs-pagination-disabled-color: rgba(33, 37, 41, 0.75);--bs-pagination-disabled-bg: #e9ecef;--bs-pagination-disabled-border-color: #dee2e6;display:flex;display:-webkit-flex;padding-left:0;list-style:none}.page-link{position:relative;display:block;padding:var(--bs-pagination-padding-y) var(--bs-pagination-padding-x);font-size:var(--bs-pagination-font-size);color:var(--bs-pagination-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-pagination-bg);border:var(--bs-pagination-border-width) solid var(--bs-pagination-border-color);transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out}@media(prefers-reduced-motion: reduce){.page-link{transition:none}}.page-link:hover{z-index:2;color:var(--bs-pagination-hover-color);background-color:var(--bs-pagination-hover-bg);border-color:var(--bs-pagination-hover-border-color)}.page-link:focus{z-index:3;color:var(--bs-pagination-focus-color);background-color:var(--bs-pagination-focus-bg);outline:0;box-shadow:var(--bs-pagination-focus-box-shadow)}.page-link.active,.active>.page-link{z-index:3;color:var(--bs-pagination-active-color);background-color:var(--bs-pagination-active-bg);border-color:var(--bs-pagination-active-border-color)}.page-link.disabled,.disabled>.page-link{color:var(--bs-pagination-disabled-color);pointer-events:none;background-color:var(--bs-pagination-disabled-bg);border-color:var(--bs-pagination-disabled-border-color)}.page-item:not(:first-child) .page-link{margin-left:calc(1px*-1)}.page-item:first-child .page-link{border-top-left-radius:var(--bs-pagination-border-radius);border-bottom-left-radius:var(--bs-pagination-border-radius)}.page-item:last-child .page-link{border-top-right-radius:var(--bs-pagination-border-radius);border-bottom-right-radius:var(--bs-pagination-border-radius)}.pagination-lg{--bs-pagination-padding-x: 1.5rem;--bs-pagination-padding-y: 0.75rem;--bs-pagination-font-size:1.25rem;--bs-pagination-border-radius: 0.5rem}.pagination-sm{--bs-pagination-padding-x: 0.5rem;--bs-pagination-padding-y: 0.25rem;--bs-pagination-font-size:0.875rem;--bs-pagination-border-radius: 0.2em}.badge{--bs-badge-padding-x: 0.65em;--bs-badge-padding-y: 0.35em;--bs-badge-font-size:0.75em;--bs-badge-font-weight: 700;--bs-badge-color: #ffffff;--bs-badge-border-radius: 0.25rem;display:inline-block;padding:var(--bs-badge-padding-y) var(--bs-badge-padding-x);font-size:var(--bs-badge-font-size);font-weight:var(--bs-badge-font-weight);line-height:1;color:var(--bs-badge-color);text-align:center;white-space:nowrap;vertical-align:baseline;border-radius:var(--bs-badge-border-radius)}.badge:empty{display:none}.btn .badge{position:relative;top:-1px}.alert{--bs-alert-bg: transparent;--bs-alert-padding-x: 1rem;--bs-alert-padding-y: 1rem;--bs-alert-margin-bottom: 1rem;--bs-alert-color: inherit;--bs-alert-border-color: transparent;--bs-alert-border: 1px solid var(--bs-alert-border-color);--bs-alert-border-radius: 0.25rem;--bs-alert-link-color: inherit;position:relative;padding:var(--bs-alert-padding-y) var(--bs-alert-padding-x);margin-bottom:var(--bs-alert-margin-bottom);color:var(--bs-alert-color);background-color:var(--bs-alert-bg);border:var(--bs-alert-border);border-radius:var(--bs-alert-border-radius)}.alert-heading{color:inherit}.alert-link{font-weight:700;color:var(--bs-alert-link-color)}.alert-dismissible{padding-right:3rem}.alert-dismissible .btn-close{position:absolute;top:0;right:0;z-index:2;padding:1.25rem 1rem}.alert-default{--bs-alert-color: var(--bs-default-text-emphasis);--bs-alert-bg: var(--bs-default-bg-subtle);--bs-alert-border-color: var(--bs-default-border-subtle);--bs-alert-link-color: var(--bs-default-text-emphasis)}.alert-primary{--bs-alert-color: var(--bs-primary-text-emphasis);--bs-alert-bg: var(--bs-primary-bg-subtle);--bs-alert-border-color: var(--bs-primary-border-subtle);--bs-alert-link-color: var(--bs-primary-text-emphasis)}.alert-secondary{--bs-alert-color: var(--bs-secondary-text-emphasis);--bs-alert-bg: var(--bs-secondary-bg-subtle);--bs-alert-border-color: var(--bs-secondary-border-subtle);--bs-alert-link-color: var(--bs-secondary-text-emphasis)}.alert-success{--bs-alert-color: var(--bs-success-text-emphasis);--bs-alert-bg: var(--bs-success-bg-subtle);--bs-alert-border-color: var(--bs-success-border-subtle);--bs-alert-link-color: var(--bs-success-text-emphasis)}.alert-info{--bs-alert-color: var(--bs-info-text-emphasis);--bs-alert-bg: var(--bs-info-bg-subtle);--bs-alert-border-color: var(--bs-info-border-subtle);--bs-alert-link-color: var(--bs-info-text-emphasis)}.alert-warning{--bs-alert-color: var(--bs-warning-text-emphasis);--bs-alert-bg: var(--bs-warning-bg-subtle);--bs-alert-border-color: var(--bs-warning-border-subtle);--bs-alert-link-color: var(--bs-warning-text-emphasis)}.alert-danger{--bs-alert-color: var(--bs-danger-text-emphasis);--bs-alert-bg: var(--bs-danger-bg-subtle);--bs-alert-border-color: var(--bs-danger-border-subtle);--bs-alert-link-color: var(--bs-danger-text-emphasis)}.alert-light{--bs-alert-color: var(--bs-light-text-emphasis);--bs-alert-bg: var(--bs-light-bg-subtle);--bs-alert-border-color: var(--bs-light-border-subtle);--bs-alert-link-color: var(--bs-light-text-emphasis)}.alert-dark{--bs-alert-color: var(--bs-dark-text-emphasis);--bs-alert-bg: var(--bs-dark-bg-subtle);--bs-alert-border-color: var(--bs-dark-border-subtle);--bs-alert-link-color: var(--bs-dark-text-emphasis)}@keyframes progress-bar-stripes{0%{background-position-x:1rem}}.progress,.progress-stacked{--bs-progress-height: 1rem;--bs-progress-font-size:0.75rem;--bs-progress-bg: #e9ecef;--bs-progress-border-radius: 0.25rem;--bs-progress-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.075);--bs-progress-bar-color: #ffffff;--bs-progress-bar-bg: #0d6efd;--bs-progress-bar-transition: width 0.6s ease;display:flex;display:-webkit-flex;height:var(--bs-progress-height);overflow:hidden;font-size:var(--bs-progress-font-size);background-color:var(--bs-progress-bg);border-radius:var(--bs-progress-border-radius)}.progress-bar{display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;justify-content:center;-webkit-justify-content:center;overflow:hidden;color:var(--bs-progress-bar-color);text-align:center;white-space:nowrap;background-color:var(--bs-progress-bar-bg);transition:var(--bs-progress-bar-transition)}@media(prefers-reduced-motion: reduce){.progress-bar{transition:none}}.progress-bar-striped{background-image:linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);background-size:var(--bs-progress-height) var(--bs-progress-height)}.progress-stacked>.progress{overflow:visible}.progress-stacked>.progress>.progress-bar{width:100%}.progress-bar-animated{animation:1s linear infinite progress-bar-stripes}@media(prefers-reduced-motion: reduce){.progress-bar-animated{animation:none}}.list-group{--bs-list-group-color: #212529;--bs-list-group-bg: #ffffff;--bs-list-group-border-color: #dee2e6;--bs-list-group-border-width: 1px;--bs-list-group-border-radius: 0.25rem;--bs-list-group-item-padding-x: 1rem;--bs-list-group-item-padding-y: 0.5rem;--bs-list-group-action-color: rgba(33, 37, 41, 0.75);--bs-list-group-action-hover-color: #000;--bs-list-group-action-hover-bg: #f8f9fa;--bs-list-group-action-active-color: #212529;--bs-list-group-action-active-bg: #e9ecef;--bs-list-group-disabled-color: rgba(33, 37, 41, 0.75);--bs-list-group-disabled-bg: #ffffff;--bs-list-group-active-color: #ffffff;--bs-list-group-active-bg: #0d6efd;--bs-list-group-active-border-color: #0d6efd;display:flex;display:-webkit-flex;flex-direction:column;-webkit-flex-direction:column;padding-left:0;margin-bottom:0;border-radius:var(--bs-list-group-border-radius)}.list-group-numbered{list-style-type:none;counter-reset:section}.list-group-numbered>.list-group-item::before{content:counters(section, ".") ". ";counter-increment:section}.list-group-item-action{width:100%;color:var(--bs-list-group-action-color);text-align:inherit}.list-group-item-action:hover,.list-group-item-action:focus{z-index:1;color:var(--bs-list-group-action-hover-color);text-decoration:none;background-color:var(--bs-list-group-action-hover-bg)}.list-group-item-action:active{color:var(--bs-list-group-action-active-color);background-color:var(--bs-list-group-action-active-bg)}.list-group-item{position:relative;display:block;padding:var(--bs-list-group-item-padding-y) var(--bs-list-group-item-padding-x);color:var(--bs-list-group-color);text-decoration:none;-webkit-text-decoration:none;-moz-text-decoration:none;-ms-text-decoration:none;-o-text-decoration:none;background-color:var(--bs-list-group-bg);border:var(--bs-list-group-border-width) solid var(--bs-list-group-border-color)}.list-group-item:first-child{border-top-left-radius:inherit;border-top-right-radius:inherit}.list-group-item:last-child{border-bottom-right-radius:inherit;border-bottom-left-radius:inherit}.list-group-item.disabled,.list-group-item:disabled{color:var(--bs-list-group-disabled-color);pointer-events:none;background-color:var(--bs-list-group-disabled-bg)}.list-group-item.active{z-index:2;color:var(--bs-list-group-active-color);background-color:var(--bs-list-group-active-bg);border-color:var(--bs-list-group-active-border-color)}.list-group-item+.list-group-item{border-top-width:0}.list-group-item+.list-group-item.active{margin-top:calc(-1*var(--bs-list-group-border-width));border-top-width:var(--bs-list-group-border-width)}.list-group-horizontal{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal>.list-group-item.active{margin-top:0}.list-group-horizontal>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}@media(min-width: 576px){.list-group-horizontal-sm{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-sm>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-sm>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-sm>.list-group-item.active{margin-top:0}.list-group-horizontal-sm>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-sm>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 768px){.list-group-horizontal-md{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-md>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-md>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-md>.list-group-item.active{margin-top:0}.list-group-horizontal-md>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-md>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 992px){.list-group-horizontal-lg{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-lg>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-lg>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-lg>.list-group-item.active{margin-top:0}.list-group-horizontal-lg>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-lg>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1200px){.list-group-horizontal-xl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xl>.list-group-item.active{margin-top:0}.list-group-horizontal-xl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}@media(min-width: 1400px){.list-group-horizontal-xxl{flex-direction:row;-webkit-flex-direction:row}.list-group-horizontal-xxl>.list-group-item:first-child:not(:last-child){border-bottom-left-radius:var(--bs-list-group-border-radius);border-top-right-radius:0}.list-group-horizontal-xxl>.list-group-item:last-child:not(:first-child){border-top-right-radius:var(--bs-list-group-border-radius);border-bottom-left-radius:0}.list-group-horizontal-xxl>.list-group-item.active{margin-top:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item{border-top-width:var(--bs-list-group-border-width);border-left-width:0}.list-group-horizontal-xxl>.list-group-item+.list-group-item.active{margin-left:calc(-1*var(--bs-list-group-border-width));border-left-width:var(--bs-list-group-border-width)}}.list-group-flush{border-radius:0}.list-group-flush>.list-group-item{border-width:0 0 var(--bs-list-group-border-width)}.list-group-flush>.list-group-item:last-child{border-bottom-width:0}.list-group-item-default{--bs-list-group-color: var(--bs-default-text-emphasis);--bs-list-group-bg: var(--bs-default-bg-subtle);--bs-list-group-border-color: var(--bs-default-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-default-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-default-border-subtle);--bs-list-group-active-color: var(--bs-default-bg-subtle);--bs-list-group-active-bg: var(--bs-default-text-emphasis);--bs-list-group-active-border-color: var(--bs-default-text-emphasis)}.list-group-item-primary{--bs-list-group-color: var(--bs-primary-text-emphasis);--bs-list-group-bg: var(--bs-primary-bg-subtle);--bs-list-group-border-color: var(--bs-primary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-primary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-primary-border-subtle);--bs-list-group-active-color: var(--bs-primary-bg-subtle);--bs-list-group-active-bg: var(--bs-primary-text-emphasis);--bs-list-group-active-border-color: var(--bs-primary-text-emphasis)}.list-group-item-secondary{--bs-list-group-color: var(--bs-secondary-text-emphasis);--bs-list-group-bg: var(--bs-secondary-bg-subtle);--bs-list-group-border-color: var(--bs-secondary-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-secondary-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-secondary-border-subtle);--bs-list-group-active-color: var(--bs-secondary-bg-subtle);--bs-list-group-active-bg: var(--bs-secondary-text-emphasis);--bs-list-group-active-border-color: var(--bs-secondary-text-emphasis)}.list-group-item-success{--bs-list-group-color: var(--bs-success-text-emphasis);--bs-list-group-bg: var(--bs-success-bg-subtle);--bs-list-group-border-color: var(--bs-success-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-success-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-success-border-subtle);--bs-list-group-active-color: var(--bs-success-bg-subtle);--bs-list-group-active-bg: var(--bs-success-text-emphasis);--bs-list-group-active-border-color: var(--bs-success-text-emphasis)}.list-group-item-info{--bs-list-group-color: var(--bs-info-text-emphasis);--bs-list-group-bg: var(--bs-info-bg-subtle);--bs-list-group-border-color: var(--bs-info-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-info-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-info-border-subtle);--bs-list-group-active-color: var(--bs-info-bg-subtle);--bs-list-group-active-bg: var(--bs-info-text-emphasis);--bs-list-group-active-border-color: var(--bs-info-text-emphasis)}.list-group-item-warning{--bs-list-group-color: var(--bs-warning-text-emphasis);--bs-list-group-bg: var(--bs-warning-bg-subtle);--bs-list-group-border-color: var(--bs-warning-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-warning-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-warning-border-subtle);--bs-list-group-active-color: var(--bs-warning-bg-subtle);--bs-list-group-active-bg: var(--bs-warning-text-emphasis);--bs-list-group-active-border-color: var(--bs-warning-text-emphasis)}.list-group-item-danger{--bs-list-group-color: var(--bs-danger-text-emphasis);--bs-list-group-bg: var(--bs-danger-bg-subtle);--bs-list-group-border-color: var(--bs-danger-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-danger-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-danger-border-subtle);--bs-list-group-active-color: var(--bs-danger-bg-subtle);--bs-list-group-active-bg: var(--bs-danger-text-emphasis);--bs-list-group-active-border-color: var(--bs-danger-text-emphasis)}.list-group-item-light{--bs-list-group-color: var(--bs-light-text-emphasis);--bs-list-group-bg: var(--bs-light-bg-subtle);--bs-list-group-border-color: var(--bs-light-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-light-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-light-border-subtle);--bs-list-group-active-color: var(--bs-light-bg-subtle);--bs-list-group-active-bg: var(--bs-light-text-emphasis);--bs-list-group-active-border-color: var(--bs-light-text-emphasis)}.list-group-item-dark{--bs-list-group-color: var(--bs-dark-text-emphasis);--bs-list-group-bg: var(--bs-dark-bg-subtle);--bs-list-group-border-color: var(--bs-dark-border-subtle);--bs-list-group-action-hover-color: var(--bs-emphasis-color);--bs-list-group-action-hover-bg: var(--bs-dark-border-subtle);--bs-list-group-action-active-color: var(--bs-emphasis-color);--bs-list-group-action-active-bg: var(--bs-dark-border-subtle);--bs-list-group-active-color: var(--bs-dark-bg-subtle);--bs-list-group-active-bg: var(--bs-dark-text-emphasis);--bs-list-group-active-border-color: var(--bs-dark-text-emphasis)}.btn-close{--bs-btn-close-color: #000;--bs-btn-close-bg: url("data:image/svg+xml,%3csvg xmlns='http://www.w3.org/2000/svg' viewBox='0 0 16 16' fill='%23000'%3e%3cpath d='M.293.293a1 1 0 0 1 1.414 0L8 6.586 14.293.293a1 1 0 1 1 1.414 1.414L9.414 8l6.293 6.293a1 1 0 0 1-1.414 1.414L8 9.414l-6.293 6.293a1 1 0 0 1-1.414-1.414L6.586 8 .293 1.707a1 1 0 0 1 0-1.414z'/%3e%3c/svg%3e");--bs-btn-close-opacity: 0.5;--bs-btn-close-hover-opacity: 0.75;--bs-btn-close-focus-shadow: 0 0 0 0.25rem rgba(13, 110, 253, 0.25);--bs-btn-close-focus-opacity: 1;--bs-btn-close-disabled-opacity: 0.25;--bs-btn-close-white-filter: invert(1) grayscale(100%) brightness(200%);box-sizing:content-box;width:1em;height:1em;padding:.25em .25em;color:var(--bs-btn-close-color);background:rgba(0,0,0,0) var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.25rem;opacity:var(--bs-btn-close-opacity)}.btn-close:hover{color:var(--bs-btn-close-color);text-decoration:none;opacity:var(--bs-btn-close-hover-opacity)}.btn-close:focus{outline:0;box-shadow:var(--bs-btn-close-focus-shadow);opacity:var(--bs-btn-close-focus-opacity)}.btn-close:disabled,.btn-close.disabled{pointer-events:none;user-select:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;-o-user-select:none;opacity:var(--bs-btn-close-disabled-opacity)}.btn-close-white{filter:var(--bs-btn-close-white-filter)}[data-bs-theme=dark] .btn-close{filter:var(--bs-btn-close-white-filter)}.toast{--bs-toast-zindex: 1090;--bs-toast-padding-x: 0.75rem;--bs-toast-padding-y: 0.5rem;--bs-toast-spacing: 1.5rem;--bs-toast-max-width: 350px;--bs-toast-font-size:0.875rem;--bs-toast-color: ;--bs-toast-bg: rgba(255, 255, 255, 0.85);--bs-toast-border-width: 1px;--bs-toast-border-color: rgba(0, 0, 0, 0.175);--bs-toast-border-radius: 0.25rem;--bs-toast-box-shadow: 0 0.5rem 1rem rgba(0, 0, 0, 0.15);--bs-toast-header-color: rgba(33, 37, 41, 0.75);--bs-toast-header-bg: rgba(255, 255, 255, 0.85);--bs-toast-header-border-color: rgba(0, 0, 0, 0.175);width:var(--bs-toast-max-width);max-width:100%;font-size:var(--bs-toast-font-size);color:var(--bs-toast-color);pointer-events:auto;background-color:var(--bs-toast-bg);background-clip:padding-box;border:var(--bs-toast-border-width) solid var(--bs-toast-border-color);box-shadow:var(--bs-toast-box-shadow);border-radius:var(--bs-toast-border-radius)}.toast.showing{opacity:0}.toast:not(.show){display:none}.toast-container{--bs-toast-zindex: 1090;position:absolute;z-index:var(--bs-toast-zindex);width:max-content;width:-webkit-max-content;width:-moz-max-content;width:-ms-max-content;width:-o-max-content;max-width:100%;pointer-events:none}.toast-container>:not(:last-child){margin-bottom:var(--bs-toast-spacing)}.toast-header{display:flex;display:-webkit-flex;align-items:center;-webkit-align-items:center;padding:var(--bs-toast-padding-y) var(--bs-toast-padding-x);color:var(--bs-toast-header-color);background-color:var(--bs-toast-header-bg);background-clip:padding-box;border-bottom:var(--bs-toast-border-width) solid var(--bs-toast-header-border-color);border-top-left-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width));border-top-right-radius:calc(var(--bs-toast-border-radius) - var(--bs-toast-border-width))}.toast-header .btn-close{margin-right:calc(-0.5*var(--bs-toast-padding-x));margin-left:var(--bs-toast-padding-x)}.toast-body{padding:var(--bs-toast-padding-x);word-wrap:break-word}.modal{--bs-modal-zindex: 1055;--bs-modal-width: 500px;--bs-modal-padding: 1rem;--bs-modal-margin: 0.5rem;--bs-modal-color: ;--bs-modal-bg: #ffffff;--bs-modal-border-color: rgba(0, 0, 0, 0.175);--bs-modal-border-width: 1px;--bs-modal-border-radius: 0.5rem;--bs-modal-box-shadow: 0 0.125rem 0.25rem rgba(0, 0, 0, 0.075);--bs-modal-inner-border-radius: calc(0.5rem - 1px);--bs-modal-header-padding-x: 1rem;--bs-modal-header-padding-y: 1rem;--bs-modal-header-padding: 1rem 1rem;--bs-modal-header-border-color: #dee2e6;--bs-modal-header-border-width: 1px;--bs-modal-title-line-height: 1.5;--bs-modal-footer-gap: 0.5rem;--bs-modal-footer-bg: ;--bs-modal-footer-border-color: #dee2e6;--bs-modal-footer-border-width: 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.value-box-grid{grid-template-columns:var(---bslib-value-box-showcase-w-fs, 1fr) auto}.bslib-value-box.showcase-left-center:not([data-fill-screen=true]) .value-box-grid .value-box-showcase{grid-area:left}.bslib-value-box.showcase-left-center:not([data-fill-screen=true]) .value-box-grid .value-box-area{grid-area:right}.bslib-value-box.showcase-bottom .value-box-grid{grid-template-columns:1fr;grid-template-rows:1fr var(---bslib-value-box-showcase-h, auto);grid-template-areas:"top" "bottom";overflow:hidden}.bslib-value-box.showcase-bottom .value-box-grid .value-box-showcase{grid-area:bottom;padding:0;margin:0}.bslib-value-box.showcase-bottom .value-box-grid .value-box-area{grid-area:top}.bslib-value-box.showcase-bottom[data-full-screen=true] .value-box-grid{grid-template-rows:1fr var(---bslib-value-box-showcase-h-fs, 2fr)}.bslib-value-box.showcase-bottom[data-full-screen=true] .value-box-grid .value-box-showcase{padding:1rem}[data-bs-theme=dark] .bslib-value-box{--bslib-value-box-shadow: 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J,Z="x"===y?zt:Vt,tt="x"===y?Rt:qt,et=A[w],it="y"===w?"height":"width",nt=et+g[Z],st=et-g[tt],ot=-1!==[zt,Vt].indexOf(_),rt=null!=(J=null==x?void 0:x[w])?J:0,at=ot?nt:et-E[it]-T[it]-rt+O.altAxis,lt=ot?et+E[it]+T[it]-rt-O.altAxis:st,ct=f&&ot?function(t,e,i){var n=Ne(t,e,i);return n>i?i:n}(at,et,lt):Ne(f?at:nt,et,f?lt:st);A[w]=ct,k[w]=ct-et}e.modifiersData[n]=k}},requiresIfExists:["offset"]};function di(t,e,i){void 0===i&&(i=!1);var n,s,o=me(e),r=me(e)&&function(t){var e=t.getBoundingClientRect(),i=we(e.width)/t.offsetWidth||1,n=we(e.height)/t.offsetHeight||1;return 1!==i||1!==n}(e),a=Le(e),l=Te(t,r,i),c={scrollLeft:0,scrollTop:0},h={x:0,y:0};return(o||!o&&!i)&&(("body"!==ue(e)||Ue(a))&&(c=(n=e)!==fe(n)&&me(n)?{scrollLeft:(s=n).scrollLeft,scrollTop:s.scrollTop}:Xe(n)),me(e)?((h=Te(e,!0)).x+=e.clientLeft,h.y+=e.clientTop):a&&(h.x=Ye(a))),{x:l.left+c.scrollLeft-h.x,y:l.top+c.scrollTop-h.y,width:l.width,height:l.height}}function ui(t){var e=new Map,i=new Set,n=[];function s(t){i.add(t.name),[].concat(t.requires||[],t.requiresIfExists||[]).forEach((function(t){if(!i.has(t)){var n=e.get(t);n&&s(n)}})),n.push(t)}return t.forEach((function(t){e.set(t.name,t)})),t.forEach((function(t){i.has(t.name)||s(t)})),n}var fi={placement:"bottom",modifiers:[],strategy:"absolute"};function pi(){for(var t=arguments.length,e=new Array(t),i=0;iNumber.parseInt(t,10))):"function"==typeof t?e=>t(e,this._element):t}_getPopperConfig(){const t={placement:this._getPlacement(),modifiers:[{name:"preventOverflow",options:{boundary:this._config.boundary}},{name:"offset",options:{offset:this._getOffset()}}]};return(this._inNavbar||"static"===this._config.display)&&(F.setDataAttribute(this._menu,"popper","static"),t.modifiers=[{name:"applyStyles",enabled:!1}]),{...t,...g(this._config.popperConfig,[t])}}_selectMenuItem({key:t,target:e}){const i=z.find(".dropdown-menu 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e=/input|textarea/i.test(t.target.tagName),i="Escape"===t.key,n=[Ei,Ti].includes(t.key);if(!n&&!i)return;if(e&&!i)return;t.preventDefault();const s=this.matches(Ii)?this:z.prev(this,Ii)[0]||z.next(this,Ii)[0]||z.findOne(Ii,t.delegateTarget.parentNode),o=qi.getOrCreateInstance(s);if(n)return t.stopPropagation(),o.show(),void o._selectMenuItem(t);o._isShown()&&(t.stopPropagation(),o.hide(),s.focus())}}N.on(document,Si,Ii,qi.dataApiKeydownHandler),N.on(document,Si,Pi,qi.dataApiKeydownHandler),N.on(document,Li,qi.clearMenus),N.on(document,Di,qi.clearMenus),N.on(document,Li,Ii,(function(t){t.preventDefault(),qi.getOrCreateInstance(this).toggle()})),m(qi);const Vi="backdrop",Ki="show",Qi=`mousedown.bs.${Vi}`,Xi={className:"modal-backdrop",clickCallback:null,isAnimated:!1,isVisible:!0,rootElement:"body"},Yi={className:"string",clickCallback:"(function|null)",isAnimated:"boolean",isVisible:"boolean",rootElement:"(element|string)"};class Ui extends H{constructor(t){super(),this._config=this._getConfig(t),this._isAppended=!1,this._element=null}static get Default(){return Xi}static get DefaultType(){return Yi}static get NAME(){return Vi}show(t){if(!this._config.isVisible)return void g(t);this._append();const e=this._getElement();this._config.isAnimated&&d(e),e.classList.add(Ki),this._emulateAnimation((()=>{g(t)}))}hide(t){this._config.isVisible?(this._getElement().classList.remove(Ki),this._emulateAnimation((()=>{this.dispose(),g(t)}))):g(t)}dispose(){this._isAppended&&(N.off(this._element,Qi),this._element.remove(),this._isAppended=!1)}_getElement(){if(!this._element){const t=document.createElement("div");t.className=this._config.className,this._config.isAnimated&&t.classList.add("fade"),this._element=t}return this._element}_configAfterMerge(t){return t.rootElement=r(t.rootElement),t}_append(){if(this._isAppended)return;const t=this._getElement();this._config.rootElement.append(t),N.on(t,Qi,(()=>{g(this._config.clickCallback)})),this._isAppended=!0}_emulateAnimation(t){_(t,this._getElement(),this._config.isAnimated)}}const Gi=".bs.focustrap",Ji=`focusin${Gi}`,Zi=`keydown.tab${Gi}`,tn="backward",en={autofocus:!0,trapElement:null},nn={autofocus:"boolean",trapElement:"element"};class sn extends H{constructor(t){super(),this._config=this._getConfig(t),this._isActive=!1,this._lastTabNavDirection=null}static get Default(){return en}static get DefaultType(){return nn}static get NAME(){return"focustrap"}activate(){this._isActive||(this._config.autofocus&&this._config.trapElement.focus(),N.off(document,Gi),N.on(document,Ji,(t=>this._handleFocusin(t))),N.on(document,Zi,(t=>this._handleKeydown(t))),this._isActive=!0)}deactivate(){this._isActive&&(this._isActive=!1,N.off(document,Gi))}_handleFocusin(t){const{trapElement:e}=this._config;if(t.target===document||t.target===e||e.contains(t.target))return;const i=z.focusableChildren(e);0===i.length?e.focus():this._lastTabNavDirection===tn?i[i.length-1].focus():i[0].focus()}_handleKeydown(t){"Tab"===t.key&&(this._lastTabNavDirection=t.shiftKey?tn:"forward")}}const on=".fixed-top, .fixed-bottom, .is-fixed, .sticky-top",rn=".sticky-top",an="padding-right",ln="margin-right";class cn{constructor(){this._element=document.body}getWidth(){const t=document.documentElement.clientWidth;return Math.abs(window.innerWidth-t)}hide(){const t=this.getWidth();this._disableOverFlow(),this._setElementAttributes(this._element,an,(e=>e+t)),this._setElementAttributes(on,an,(e=>e+t)),this._setElementAttributes(rn,ln,(e=>e-t))}reset(){this._resetElementAttributes(this._element,"overflow"),this._resetElementAttributes(this._element,an),this._resetElementAttributes(on,an),this._resetElementAttributes(rn,ln)}isOverflowing(){return this.getWidth()>0}_disableOverFlow(){this._saveInitialAttribute(this._element,"overflow"),this._element.style.overflow="hidden"}_setElementAttributes(t,e,i){const n=this.getWidth();this._applyManipulationCallback(t,(t=>{if(t!==this._element&&window.innerWidth>t.clientWidth+n)return;this._saveInitialAttribute(t,e);const s=window.getComputedStyle(t).getPropertyValue(e);t.style.setProperty(e,`${i(Number.parseFloat(s))}px`)}))}_saveInitialAttribute(t,e){const i=t.style.getPropertyValue(e);i&&F.setDataAttribute(t,e,i)}_resetElementAttributes(t,e){this._applyManipulationCallback(t,(t=>{const i=F.getDataAttribute(t,e);null!==i?(F.removeDataAttribute(t,e),t.style.setProperty(e,i)):t.style.removeProperty(e)}))}_applyManipulationCallback(t,e){if(o(t))e(t);else for(const i of z.find(t,this._element))e(i)}}const hn=".bs.modal",dn=`hide${hn}`,un=`hidePrevented${hn}`,fn=`hidden${hn}`,pn=`show${hn}`,mn=`shown${hn}`,gn=`resize${hn}`,_n=`click.dismiss${hn}`,bn=`mousedown.dismiss${hn}`,vn=`keydown.dismiss${hn}`,yn=`click${hn}.data-api`,wn="modal-open",An="show",En="modal-static",Tn={backdrop:!0,focus:!0,keyboard:!0},Cn={backdrop:"(boolean|string)",focus:"boolean",keyboard:"boolean"};class On extends W{constructor(t,e){super(t,e),this._dialog=z.findOne(".modal-dialog",this._element),this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._isShown=!1,this._isTransitioning=!1,this._scrollBar=new cn,this._addEventListeners()}static get Default(){return Tn}static get DefaultType(){return Cn}static get NAME(){return"modal"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||this._isTransitioning||N.trigger(this._element,pn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._isTransitioning=!0,this._scrollBar.hide(),document.body.classList.add(wn),this._adjustDialog(),this._backdrop.show((()=>this._showElement(t))))}hide(){this._isShown&&!this._isTransitioning&&(N.trigger(this._element,dn).defaultPrevented||(this._isShown=!1,this._isTransitioning=!0,this._focustrap.deactivate(),this._element.classList.remove(An),this._queueCallback((()=>this._hideModal()),this._element,this._isAnimated())))}dispose(){N.off(window,hn),N.off(this._dialog,hn),this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}handleUpdate(){this._adjustDialog()}_initializeBackDrop(){return new Ui({isVisible:Boolean(this._config.backdrop),isAnimated:this._isAnimated()})}_initializeFocusTrap(){return new sn({trapElement:this._element})}_showElement(t){document.body.contains(this._element)||document.body.append(this._element),this._element.style.display="block",this._element.removeAttribute("aria-hidden"),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.scrollTop=0;const e=z.findOne(".modal-body",this._dialog);e&&(e.scrollTop=0),d(this._element),this._element.classList.add(An),this._queueCallback((()=>{this._config.focus&&this._focustrap.activate(),this._isTransitioning=!1,N.trigger(this._element,mn,{relatedTarget:t})}),this._dialog,this._isAnimated())}_addEventListeners(){N.on(this._element,vn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():this._triggerBackdropTransition())})),N.on(window,gn,(()=>{this._isShown&&!this._isTransitioning&&this._adjustDialog()})),N.on(this._element,bn,(t=>{N.one(this._element,_n,(e=>{this._element===t.target&&this._element===e.target&&("static"!==this._config.backdrop?this._config.backdrop&&this.hide():this._triggerBackdropTransition())}))}))}_hideModal(){this._element.style.display="none",this._element.setAttribute("aria-hidden",!0),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._isTransitioning=!1,this._backdrop.hide((()=>{document.body.classList.remove(wn),this._resetAdjustments(),this._scrollBar.reset(),N.trigger(this._element,fn)}))}_isAnimated(){return this._element.classList.contains("fade")}_triggerBackdropTransition(){if(N.trigger(this._element,un).defaultPrevented)return;const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._element.style.overflowY;"hidden"===e||this._element.classList.contains(En)||(t||(this._element.style.overflowY="hidden"),this._element.classList.add(En),this._queueCallback((()=>{this._element.classList.remove(En),this._queueCallback((()=>{this._element.style.overflowY=e}),this._dialog)}),this._dialog),this._element.focus())}_adjustDialog(){const t=this._element.scrollHeight>document.documentElement.clientHeight,e=this._scrollBar.getWidth(),i=e>0;if(i&&!t){const t=p()?"paddingLeft":"paddingRight";this._element.style[t]=`${e}px`}if(!i&&t){const t=p()?"paddingRight":"paddingLeft";this._element.style[t]=`${e}px`}}_resetAdjustments(){this._element.style.paddingLeft="",this._element.style.paddingRight=""}static jQueryInterface(t,e){return this.each((function(){const i=On.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===i[t])throw new TypeError(`No method named "${t}"`);i[t](e)}}))}}N.on(document,yn,'[data-bs-toggle="modal"]',(function(t){const e=z.getElementFromSelector(this);["A","AREA"].includes(this.tagName)&&t.preventDefault(),N.one(e,pn,(t=>{t.defaultPrevented||N.one(e,fn,(()=>{a(this)&&this.focus()}))}));const i=z.findOne(".modal.show");i&&On.getInstance(i).hide(),On.getOrCreateInstance(e).toggle(this)})),R(On),m(On);const xn=".bs.offcanvas",kn=".data-api",Ln=`load${xn}${kn}`,Sn="show",Dn="showing",$n="hiding",In=".offcanvas.show",Nn=`show${xn}`,Pn=`shown${xn}`,Mn=`hide${xn}`,jn=`hidePrevented${xn}`,Fn=`hidden${xn}`,Hn=`resize${xn}`,Wn=`click${xn}${kn}`,Bn=`keydown.dismiss${xn}`,zn={backdrop:!0,keyboard:!0,scroll:!1},Rn={backdrop:"(boolean|string)",keyboard:"boolean",scroll:"boolean"};class qn extends W{constructor(t,e){super(t,e),this._isShown=!1,this._backdrop=this._initializeBackDrop(),this._focustrap=this._initializeFocusTrap(),this._addEventListeners()}static get Default(){return zn}static get DefaultType(){return Rn}static get NAME(){return"offcanvas"}toggle(t){return this._isShown?this.hide():this.show(t)}show(t){this._isShown||N.trigger(this._element,Nn,{relatedTarget:t}).defaultPrevented||(this._isShown=!0,this._backdrop.show(),this._config.scroll||(new cn).hide(),this._element.setAttribute("aria-modal",!0),this._element.setAttribute("role","dialog"),this._element.classList.add(Dn),this._queueCallback((()=>{this._config.scroll&&!this._config.backdrop||this._focustrap.activate(),this._element.classList.add(Sn),this._element.classList.remove(Dn),N.trigger(this._element,Pn,{relatedTarget:t})}),this._element,!0))}hide(){this._isShown&&(N.trigger(this._element,Mn).defaultPrevented||(this._focustrap.deactivate(),this._element.blur(),this._isShown=!1,this._element.classList.add($n),this._backdrop.hide(),this._queueCallback((()=>{this._element.classList.remove(Sn,$n),this._element.removeAttribute("aria-modal"),this._element.removeAttribute("role"),this._config.scroll||(new cn).reset(),N.trigger(this._element,Fn)}),this._element,!0)))}dispose(){this._backdrop.dispose(),this._focustrap.deactivate(),super.dispose()}_initializeBackDrop(){const t=Boolean(this._config.backdrop);return new Ui({className:"offcanvas-backdrop",isVisible:t,isAnimated:!0,rootElement:this._element.parentNode,clickCallback:t?()=>{"static"!==this._config.backdrop?this.hide():N.trigger(this._element,jn)}:null})}_initializeFocusTrap(){return new sn({trapElement:this._element})}_addEventListeners(){N.on(this._element,Bn,(t=>{"Escape"===t.key&&(this._config.keyboard?this.hide():N.trigger(this._element,jn))}))}static jQueryInterface(t){return this.each((function(){const e=qn.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t]||t.startsWith("_")||"constructor"===t)throw new TypeError(`No method named "${t}"`);e[t](this)}}))}}N.on(document,Wn,'[data-bs-toggle="offcanvas"]',(function(t){const e=z.getElementFromSelector(this);if(["A","AREA"].includes(this.tagName)&&t.preventDefault(),l(this))return;N.one(e,Fn,(()=>{a(this)&&this.focus()}));const i=z.findOne(In);i&&i!==e&&qn.getInstance(i).hide(),qn.getOrCreateInstance(e).toggle(this)})),N.on(window,Ln,(()=>{for(const t of z.find(In))qn.getOrCreateInstance(t).show()})),N.on(window,Hn,(()=>{for(const t of z.find("[aria-modal][class*=show][class*=offcanvas-]"))"fixed"!==getComputedStyle(t).position&&qn.getOrCreateInstance(t).hide()})),R(qn),m(qn);const Vn={"*":["class","dir","id","lang","role",/^aria-[\w-]*$/i],a:["target","href","title","rel"],area:[],b:[],br:[],col:[],code:[],div:[],em:[],hr:[],h1:[],h2:[],h3:[],h4:[],h5:[],h6:[],i:[],img:["src","srcset","alt","title","width","height"],li:[],ol:[],p:[],pre:[],s:[],small:[],span:[],sub:[],sup:[],strong:[],u:[],ul:[]},Kn=new Set(["background","cite","href","itemtype","longdesc","poster","src","xlink:href"]),Qn=/^(?!javascript:)(?:[a-z0-9+.-]+:|[^&:/?#]*(?:[/?#]|$))/i,Xn=(t,e)=>{const i=t.nodeName.toLowerCase();return e.includes(i)?!Kn.has(i)||Boolean(Qn.test(t.nodeValue)):e.filter((t=>t instanceof RegExp)).some((t=>t.test(i)))},Yn={allowList:Vn,content:{},extraClass:"",html:!1,sanitize:!0,sanitizeFn:null,template:"

"},Un={allowList:"object",content:"object",extraClass:"(string|function)",html:"boolean",sanitize:"boolean",sanitizeFn:"(null|function)",template:"string"},Gn={entry:"(string|element|function|null)",selector:"(string|element)"};class Jn extends H{constructor(t){super(),this._config=this._getConfig(t)}static get Default(){return Yn}static get DefaultType(){return Un}static get NAME(){return"TemplateFactory"}getContent(){return Object.values(this._config.content).map((t=>this._resolvePossibleFunction(t))).filter(Boolean)}hasContent(){return this.getContent().length>0}changeContent(t){return this._checkContent(t),this._config.content={...this._config.content,...t},this}toHtml(){const t=document.createElement("div");t.innerHTML=this._maybeSanitize(this._config.template);for(const[e,i]of Object.entries(this._config.content))this._setContent(t,i,e);const e=t.children[0],i=this._resolvePossibleFunction(this._config.extraClass);return i&&e.classList.add(...i.split(" ")),e}_typeCheckConfig(t){super._typeCheckConfig(t),this._checkContent(t.content)}_checkContent(t){for(const[e,i]of Object.entries(t))super._typeCheckConfig({selector:e,entry:i},Gn)}_setContent(t,e,i){const n=z.findOne(i,t);n&&((e=this._resolvePossibleFunction(e))?o(e)?this._putElementInTemplate(r(e),n):this._config.html?n.innerHTML=this._maybeSanitize(e):n.textContent=e:n.remove())}_maybeSanitize(t){return this._config.sanitize?function(t,e,i){if(!t.length)return t;if(i&&"function"==typeof i)return i(t);const n=(new window.DOMParser).parseFromString(t,"text/html"),s=[].concat(...n.body.querySelectorAll("*"));for(const t of s){const i=t.nodeName.toLowerCase();if(!Object.keys(e).includes(i)){t.remove();continue}const n=[].concat(...t.attributes),s=[].concat(e["*"]||[],e[i]||[]);for(const e of n)Xn(e,s)||t.removeAttribute(e.nodeName)}return n.body.innerHTML}(t,this._config.allowList,this._config.sanitizeFn):t}_resolvePossibleFunction(t){return g(t,[this])}_putElementInTemplate(t,e){if(this._config.html)return e.innerHTML="",void e.append(t);e.textContent=t.textContent}}const Zn=new Set(["sanitize","allowList","sanitizeFn"]),ts="fade",es="show",is=".modal",ns="hide.bs.modal",ss="hover",os="focus",rs={AUTO:"auto",TOP:"top",RIGHT:p()?"left":"right",BOTTOM:"bottom",LEFT:p()?"right":"left"},as={allowList:Vn,animation:!0,boundary:"clippingParents",container:!1,customClass:"",delay:0,fallbackPlacements:["top","right","bottom","left"],html:!1,offset:[0,6],placement:"top",popperConfig:null,sanitize:!0,sanitizeFn:null,selector:!1,template:'',title:"",trigger:"hover focus"},ls={allowList:"object",animation:"boolean",boundary:"(string|element)",container:"(string|element|boolean)",customClass:"(string|function)",delay:"(number|object)",fallbackPlacements:"array",html:"boolean",offset:"(array|string|function)",placement:"(string|function)",popperConfig:"(null|object|function)",sanitize:"boolean",sanitizeFn:"(null|function)",selector:"(string|boolean)",template:"string",title:"(string|element|function)",trigger:"string"};class cs extends W{constructor(t,e){if(void 0===vi)throw new TypeError("Bootstrap's tooltips require Popper (https://popper.js.org)");super(t,e),this._isEnabled=!0,this._timeout=0,this._isHovered=null,this._activeTrigger={},this._popper=null,this._templateFactory=null,this._newContent=null,this.tip=null,this._setListeners(),this._config.selector||this._fixTitle()}static get Default(){return as}static get DefaultType(){return ls}static get 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i=this._getTipElement();this._element.setAttribute("aria-describedby",i.getAttribute("id"));const{container:n}=this._config;if(this._element.ownerDocument.documentElement.contains(this.tip)||(n.append(i),N.trigger(this._element,this.constructor.eventName("inserted"))),this._popper=this._createPopper(i),i.classList.add(es),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))N.on(t,"mouseover",h);this._queueCallback((()=>{N.trigger(this._element,this.constructor.eventName("shown")),!1===this._isHovered&&this._leave(),this._isHovered=!1}),this.tip,this._isAnimated())}hide(){if(this._isShown()&&!N.trigger(this._element,this.constructor.eventName("hide")).defaultPrevented){if(this._getTipElement().classList.remove(es),"ontouchstart"in document.documentElement)for(const t of[].concat(...document.body.children))N.off(t,"mouseover",h);this._activeTrigger.click=!1,this._activeTrigger[os]=!1,this._activeTrigger[ss]=!1,this._isHovered=null,this._queueCallback((()=>{this._isWithActiveTrigger()||(this._isHovered||this._disposePopper(),this._element.removeAttribute("aria-describedby"),N.trigger(this._element,this.constructor.eventName("hidden")))}),this.tip,this._isAnimated())}}update(){this._popper&&this._popper.update()}_isWithContent(){return Boolean(this._getTitle())}_getTipElement(){return this.tip||(this.tip=this._createTipElement(this._newContent||this._getContentForTemplate())),this.tip}_createTipElement(t){const e=this._getTemplateFactory(t).toHtml();if(!e)return null;e.classList.remove(ts,es),e.classList.add(`bs-${this.constructor.NAME}-auto`);const i=(t=>{do{t+=Math.floor(1e6*Math.random())}while(document.getElementById(t));return t})(this.constructor.NAME).toString();return e.setAttribute("id",i),this._isAnimated()&&e.classList.add(ts),e}setContent(t){this._newContent=t,this._isShown()&&(this._disposePopper(),this.show())}_getTemplateFactory(t){return this._templateFactory?this._templateFactory.changeContent(t):this._templateFactory=new Jn({...this._config,content:t,extraClass:this._resolvePossibleFunction(this._config.customClass)}),this._templateFactory}_getContentForTemplate(){return{".tooltip-inner":this._getTitle()}}_getTitle(){return this._resolvePossibleFunction(this._config.title)||this._element.getAttribute("data-bs-original-title")}_initializeOnDelegatedTarget(t){return this.constructor.getOrCreateInstance(t.delegateTarget,this._getDelegateConfig())}_isAnimated(){return this._config.animation||this.tip&&this.tip.classList.contains(ts)}_isShown(){return this.tip&&this.tip.classList.contains(es)}_createPopper(t){const e=g(this._config.placement,[this,t,this._element]),i=rs[e.toUpperCase()];return 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t=this._element.getAttribute("title");t&&(this._element.getAttribute("aria-label")||this._element.textContent.trim()||this._element.setAttribute("aria-label",t),this._element.setAttribute("data-bs-original-title",t),this._element.removeAttribute("title"))}_enter(){this._isShown()||this._isHovered?this._isHovered=!0:(this._isHovered=!0,this._setTimeout((()=>{this._isHovered&&this.show()}),this._config.delay.show))}_leave(){this._isWithActiveTrigger()||(this._isHovered=!1,this._setTimeout((()=>{this._isHovered||this.hide()}),this._config.delay.hide))}_setTimeout(t,e){clearTimeout(this._timeout),this._timeout=setTimeout(t,e)}_isWithActiveTrigger(){return Object.values(this._activeTrigger).includes(!0)}_getConfig(t){const e=F.getDataAttributes(this._element);for(const t of Object.keys(e))Zn.has(t)&&delete e[t];return t={...e,..."object"==typeof t&&t?t:{}},t=this._mergeConfigObj(t),t=this._configAfterMerge(t),this._typeCheckConfig(t),t}_configAfterMerge(t){return t.container=!1===t.container?document.body:r(t.container),"number"==typeof t.delay&&(t.delay={show:t.delay,hide:t.delay}),"number"==typeof t.title&&(t.title=t.title.toString()),"number"==typeof t.content&&(t.content=t.content.toString()),t}_getDelegateConfig(){const t={};for(const[e,i]of Object.entries(this._config))this.constructor.Default[e]!==i&&(t[e]=i);return t.selector=!1,t.trigger="manual",t}_disposePopper(){this._popper&&(this._popper.destroy(),this._popper=null),this.tip&&(this.tip.remove(),this.tip=null)}static jQueryInterface(t){return this.each((function(){const e=cs.getOrCreateInstance(this,t);if("string"==typeof t){if(void 0===e[t])throw new TypeError(`No method named "${t}"`);e[t]()}}))}}m(cs);const hs={...cs.Default,content:"",offset:[0,8],placement:"right",template:'',trigger:"click"},ds={...cs.DefaultType,content:"(null|string|element|function)"};class us extends cs{static get Default(){return hs}static get DefaultType(){return ds}static get 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method named "${t}"`);e[t]()}}))}}N.on(window,gs,(()=>{for(const t of z.find('[data-bs-spy="scroll"]'))Es.getOrCreateInstance(t)})),m(Es);const Ts=".bs.tab",Cs=`hide${Ts}`,Os=`hidden${Ts}`,xs=`show${Ts}`,ks=`shown${Ts}`,Ls=`click${Ts}`,Ss=`keydown${Ts}`,Ds=`load${Ts}`,$s="ArrowLeft",Is="ArrowRight",Ns="ArrowUp",Ps="ArrowDown",Ms="Home",js="End",Fs="active",Hs="fade",Ws="show",Bs=":not(.dropdown-toggle)",zs='[data-bs-toggle="tab"], [data-bs-toggle="pill"], [data-bs-toggle="list"]',Rs=`.nav-link${Bs}, .list-group-item${Bs}, [role="tab"]${Bs}, ${zs}`,qs=`.${Fs}[data-bs-toggle="tab"], .${Fs}[data-bs-toggle="pill"], .${Fs}[data-bs-toggle="list"]`;class Vs extends W{constructor(t){super(t),this._parent=this._element.closest('.list-group, .nav, [role="tablist"]'),this._parent&&(this._setInitialAttributes(this._parent,this._getChildren()),N.on(this._element,Ss,(t=>this._keydown(t))))}static get NAME(){return"tab"}show(){const t=this._element;if(this._elemIsActive(t))return;const e=this._getActiveElem(),i=e?N.trigger(e,Cs,{relatedTarget:t}):null;N.trigger(t,xs,{relatedTarget:e}).defaultPrevented||i&&i.defaultPrevented||(this._deactivate(e,t),this._activate(t,e))}_activate(t,e){t&&(t.classList.add(Fs),this._activate(z.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.removeAttribute("tabindex"),t.setAttribute("aria-selected",!0),this._toggleDropDown(t,!0),N.trigger(t,ks,{relatedTarget:e})):t.classList.add(Ws)}),t,t.classList.contains(Hs)))}_deactivate(t,e){t&&(t.classList.remove(Fs),t.blur(),this._deactivate(z.getElementFromSelector(t)),this._queueCallback((()=>{"tab"===t.getAttribute("role")?(t.setAttribute("aria-selected",!1),t.setAttribute("tabindex","-1"),this._toggleDropDown(t,!1),N.trigger(t,Os,{relatedTarget:e})):t.classList.remove(Ws)}),t,t.classList.contains(Hs)))}_keydown(t){if(![$s,Is,Ns,Ps,Ms,js].includes(t.key))return;t.stopPropagation(),t.preventDefault();const e=this._getChildren().filter((t=>!l(t)));let 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NAME(){return"toast"}show(){N.trigger(this._element,Zs).defaultPrevented||(this._clearTimeout(),this._config.animation&&this._element.classList.add("fade"),this._element.classList.remove(eo),d(this._element),this._element.classList.add(io,no),this._queueCallback((()=>{this._element.classList.remove(no),N.trigger(this._element,to),this._maybeScheduleHide()}),this._element,this._config.animation))}hide(){this.isShown()&&(N.trigger(this._element,Gs).defaultPrevented||(this._element.classList.add(no),this._queueCallback((()=>{this._element.classList.add(eo),this._element.classList.remove(no,io),N.trigger(this._element,Js)}),this._element,this._config.animation)))}dispose(){this._clearTimeout(),this.isShown()&&this._element.classList.remove(io),super.dispose()}isShown(){return 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index 00000000..d9fd98f0 --- /dev/null +++ b/session-survival/docs/site_libs/quarto-html/quarto-syntax-highlighting.css @@ -0,0 +1,203 @@ +/* quarto syntax highlight colors */ +:root { + --quarto-hl-ot-color: #003B4F; + --quarto-hl-at-color: #657422; + --quarto-hl-ss-color: #20794D; + --quarto-hl-an-color: #5E5E5E; + --quarto-hl-fu-color: #4758AB; + --quarto-hl-st-color: #20794D; + --quarto-hl-cf-color: #003B4F; + --quarto-hl-op-color: #5E5E5E; + --quarto-hl-er-color: #AD0000; + --quarto-hl-bn-color: #AD0000; + --quarto-hl-al-color: #AD0000; + --quarto-hl-va-color: #111111; + --quarto-hl-bu-color: inherit; + --quarto-hl-ex-color: inherit; + --quarto-hl-pp-color: #AD0000; + --quarto-hl-in-color: #5E5E5E; + --quarto-hl-vs-color: #20794D; + --quarto-hl-wa-color: #5E5E5E; + --quarto-hl-do-color: #5E5E5E; + --quarto-hl-im-color: #00769E; + --quarto-hl-ch-color: #20794D; + --quarto-hl-dt-color: #AD0000; + --quarto-hl-fl-color: #AD0000; + --quarto-hl-co-color: #5E5E5E; + --quarto-hl-cv-color: #5E5E5E; + --quarto-hl-cn-color: #8f5902; + --quarto-hl-sc-color: #5E5E5E; + --quarto-hl-dv-color: #AD0000; + --quarto-hl-kw-color: #003B4F; +} + +/* other quarto variables */ +:root { + --quarto-font-monospace: SFMono-Regular, Menlo, Monaco, Consolas, "Liberation Mono", "Courier New", monospace; +} + +pre > code.sourceCode > span { + color: #003B4F; +} + +code span { + color: #003B4F; +} + +code.sourceCode > span { + color: #003B4F; +} + +div.sourceCode, +div.sourceCode pre.sourceCode { + color: #003B4F; +} + +code span.ot { + color: #003B4F; + font-style: inherit; +} + +code span.at { + color: #657422; + font-style: inherit; +} + +code span.ss { + color: #20794D; + font-style: inherit; +} + +code span.an { + color: #5E5E5E; + font-style: inherit; +} + +code span.fu { + color: #4758AB; + font-style: inherit; +} + +code span.st { + color: #20794D; + font-style: inherit; +} + +code span.cf { + color: #003B4F; + font-style: inherit; +} + +code span.op { + color: #5E5E5E; + font-style: inherit; +} + +code span.er { + color: #AD0000; + font-style: inherit; +} + +code span.bn { + color: #AD0000; + font-style: inherit; +} + +code span.al { + color: #AD0000; + font-style: inherit; +} + +code span.va { + color: #111111; + font-style: inherit; +} + +code span.bu { + font-style: inherit; +} + +code span.ex { + font-style: inherit; +} + +code span.pp { + color: #AD0000; + font-style: inherit; +} + +code span.in { + color: #5E5E5E; + font-style: inherit; +} + +code span.vs { + color: #20794D; + font-style: inherit; +} + +code span.wa { + color: #5E5E5E; + font-style: italic; +} + +code span.do { + color: #5E5E5E; + font-style: italic; +} + +code span.im { + color: #00769E; + font-style: inherit; +} + +code span.ch { + color: #20794D; + font-style: inherit; +} + +code span.dt { + color: #AD0000; + font-style: inherit; +} + +code span.fl { + color: #AD0000; + font-style: inherit; +} + +code span.co { + color: #5E5E5E; + font-style: inherit; +} + +code span.cv { + color: #5E5E5E; + font-style: italic; +} + +code span.cn { + color: #8f5902; + font-style: inherit; +} + +code span.sc { + color: #5E5E5E; + font-style: inherit; +} + +code span.dv { + color: #AD0000; + font-style: inherit; +} + +code span.kw { + color: #003B4F; + font-style: inherit; +} + +.prevent-inlining { + content: " { + // Find any conflicting margin elements and add margins to the + // top to prevent overlap + const marginChildren = window.document.querySelectorAll( + ".column-margin.column-container > *, .margin-caption, .aside" + ); + + let lastBottom = 0; + for (const marginChild of marginChildren) { + if (marginChild.offsetParent !== null) { + // clear the top margin so we recompute it + marginChild.style.marginTop = null; + const top = marginChild.getBoundingClientRect().top + window.scrollY; + if (top < lastBottom) { + const marginChildStyle = window.getComputedStyle(marginChild); + const marginBottom = parseFloat(marginChildStyle["marginBottom"]); + const margin = lastBottom - top + marginBottom; + marginChild.style.marginTop = `${margin}px`; + } + const styles = window.getComputedStyle(marginChild); + const marginTop = parseFloat(styles["marginTop"]); + lastBottom = top + marginChild.getBoundingClientRect().height + marginTop; + } + } +}; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Recompute the position of margin elements anytime the body size changes + if (window.ResizeObserver) { + const resizeObserver = new window.ResizeObserver( + throttle(() => { + layoutMarginEls(); + if ( + window.document.body.getBoundingClientRect().width < 990 && + isReaderMode() + ) { + quartoToggleReader(); + } + }, 50) + ); + resizeObserver.observe(window.document.body); + } + + const tocEl = window.document.querySelector('nav.toc-active[role="doc-toc"]'); + const sidebarEl = window.document.getElementById("quarto-sidebar"); + const leftTocEl = window.document.getElementById("quarto-sidebar-toc-left"); + const marginSidebarEl = window.document.getElementById( + "quarto-margin-sidebar" + ); + // function to determine whether the element has a previous sibling that is active + const prevSiblingIsActiveLink = (el) => { + const sibling = el.previousElementSibling; + if (sibling && sibling.tagName === "A") { + return sibling.classList.contains("active"); + } else { + return false; + } + }; + + // fire slideEnter for bootstrap tab activations (for htmlwidget resize behavior) + function fireSlideEnter(e) { + const event = window.document.createEvent("Event"); + event.initEvent("slideenter", true, true); + window.document.dispatchEvent(event); + } + const tabs = window.document.querySelectorAll('a[data-bs-toggle="tab"]'); + tabs.forEach((tab) => { + tab.addEventListener("shown.bs.tab", fireSlideEnter); + }); + + // fire slideEnter for tabby tab activations (for htmlwidget resize behavior) + document.addEventListener("tabby", fireSlideEnter, false); + + // Track scrolling and mark TOC links as active + // get table of contents and sidebar (bail if we don't have at least one) + const tocLinks = tocEl + ? [...tocEl.querySelectorAll("a[data-scroll-target]")] + : []; + const makeActive = (link) => tocLinks[link].classList.add("active"); + const removeActive = (link) => tocLinks[link].classList.remove("active"); + const removeAllActive = () => + [...Array(tocLinks.length).keys()].forEach((link) => removeActive(link)); + + // activate the anchor for a section associated with this TOC entry + tocLinks.forEach((link) => { + link.addEventListener("click", () => { + if (link.href.indexOf("#") !== -1) { + const anchor = link.href.split("#")[1]; + const heading = window.document.querySelector( + `[data-anchor-id=${anchor}]` + ); + if (heading) { + // Add the class + heading.classList.add("reveal-anchorjs-link"); + + // function to show the anchor + const handleMouseout = () => { + heading.classList.remove("reveal-anchorjs-link"); + heading.removeEventListener("mouseout", handleMouseout); + }; + + // add a function to clear the anchor when the user mouses out of it + heading.addEventListener("mouseout", handleMouseout); + } + } + }); + }); + + const sections = tocLinks.map((link) => { + const target = link.getAttribute("data-scroll-target"); + if (target.startsWith("#")) { + return window.document.getElementById(decodeURI(`${target.slice(1)}`)); + } else { + return window.document.querySelector(decodeURI(`${target}`)); + } + }); + + const sectionMargin = 200; + let currentActive = 0; + // track whether we've initialized state the first time + let init = false; + + const updateActiveLink = () => { + // The index from bottom to top (e.g. reversed list) + let sectionIndex = -1; + if ( + window.innerHeight + window.pageYOffset >= + window.document.body.offsetHeight + ) { + sectionIndex = 0; + } else { + sectionIndex = [...sections].reverse().findIndex((section) => { + if (section) { + return window.pageYOffset >= section.offsetTop - sectionMargin; + } else { + return false; + } + }); + } + if (sectionIndex > -1) { + const current = sections.length - sectionIndex - 1; + if (current !== currentActive) { + removeAllActive(); + currentActive = current; + makeActive(current); + if (init) { + window.dispatchEvent(sectionChanged); + } + init = true; + } + } + }; + + const inHiddenRegion = (top, bottom, hiddenRegions) => { + for (const region of hiddenRegions) { + if (top <= region.bottom && bottom >= region.top) { + return true; + } + } + return false; + }; + + const categorySelector = "header.quarto-title-block .quarto-category"; + const activateCategories = (href) => { + // Find any categories + // Surround them with a link pointing back to: + // #category=Authoring + try { + const categoryEls = window.document.querySelectorAll(categorySelector); + for (const categoryEl of categoryEls) { + const categoryText = categoryEl.textContent; + if (categoryText) { + const link = `${href}#category=${encodeURIComponent(categoryText)}`; + const linkEl = window.document.createElement("a"); + linkEl.setAttribute("href", link); + for (const child of categoryEl.childNodes) { + linkEl.append(child); + } + categoryEl.appendChild(linkEl); + } + } + } catch { + // Ignore errors + } + }; + function hasTitleCategories() { + return window.document.querySelector(categorySelector) !== null; + } + + function offsetRelativeUrl(url) { + const offset = getMeta("quarto:offset"); + return offset ? offset + url : url; + } + + function offsetAbsoluteUrl(url) { + const offset = getMeta("quarto:offset"); + const baseUrl = new URL(offset, window.location); + + const projRelativeUrl = url.replace(baseUrl, ""); + if (projRelativeUrl.startsWith("/")) { + return projRelativeUrl; + } else { + return "/" + projRelativeUrl; + } + } + + // read a meta tag value + function getMeta(metaName) { + const metas = window.document.getElementsByTagName("meta"); + for (let i = 0; i < metas.length; i++) { + if (metas[i].getAttribute("name") === metaName) { + return metas[i].getAttribute("content"); + } + } + return ""; + } + + async function findAndActivateCategories() { + const currentPagePath = offsetAbsoluteUrl(window.location.href); + const response = await fetch(offsetRelativeUrl("listings.json")); + if (response.status == 200) { + return response.json().then(function (listingPaths) { + const listingHrefs = []; + for (const listingPath of listingPaths) { + const pathWithoutLeadingSlash = listingPath.listing.substring(1); + for (const item of listingPath.items) { + if ( + item === currentPagePath || + item === currentPagePath + "index.html" + ) { + // Resolve this path against the offset to be sure + // we already are using the correct path to the listing + // (this adjusts the listing urls to be rooted against + // whatever root the page is actually running against) + const relative = offsetRelativeUrl(pathWithoutLeadingSlash); + const baseUrl = window.location; + const resolvedPath = new URL(relative, baseUrl); + listingHrefs.push(resolvedPath.pathname); + break; + } + } + } + + // Look up the tree for a nearby linting and use that if we find one + const nearestListing = findNearestParentListing( + offsetAbsoluteUrl(window.location.pathname), + listingHrefs + ); + if (nearestListing) { + activateCategories(nearestListing); + } else { + // See if the referrer is a listing page for this item + const referredRelativePath = offsetAbsoluteUrl(document.referrer); + const referrerListing = listingHrefs.find((listingHref) => { + const isListingReferrer = + listingHref === referredRelativePath || + listingHref === referredRelativePath + "index.html"; + return isListingReferrer; + }); + + if (referrerListing) { + // Try to use the referrer if possible + activateCategories(referrerListing); + } else if (listingHrefs.length > 0) { + // Otherwise, just fall back to the first listing + activateCategories(listingHrefs[0]); + } + } + }); + } + } + if (hasTitleCategories()) { + findAndActivateCategories(); + } + + const findNearestParentListing = (href, listingHrefs) => { + if (!href || !listingHrefs) { + return undefined; + } + // Look up the tree for a nearby linting and use that if we find one + const relativeParts = href.substring(1).split("/"); + while (relativeParts.length > 0) { + const path = relativeParts.join("/"); + for (const listingHref of listingHrefs) { + if (listingHref.startsWith(path)) { + return listingHref; + } + } + relativeParts.pop(); + } + + return undefined; + }; + + const manageSidebarVisiblity = (el, placeholderDescriptor) => { + let isVisible = true; + let elRect; + + return (hiddenRegions) => { + if (el === null) { + return; + } + + // Find the last element of the TOC + const lastChildEl = el.lastElementChild; + + if (lastChildEl) { + // Converts the sidebar to a menu + const convertToMenu = () => { + for (const child of el.children) { + child.style.opacity = 0; + child.style.overflow = "hidden"; + } + + nexttick(() => { + const toggleContainer = window.document.createElement("div"); + toggleContainer.style.width = "100%"; + toggleContainer.classList.add("zindex-over-content"); + toggleContainer.classList.add("quarto-sidebar-toggle"); + toggleContainer.classList.add("headroom-target"); // Marks this to be managed by headeroom + toggleContainer.id = placeholderDescriptor.id; + toggleContainer.style.position = "fixed"; + + const toggleIcon = window.document.createElement("i"); + toggleIcon.classList.add("quarto-sidebar-toggle-icon"); + toggleIcon.classList.add("bi"); + toggleIcon.classList.add("bi-caret-down-fill"); + + const toggleTitle = window.document.createElement("div"); + const titleEl = window.document.body.querySelector( + placeholderDescriptor.titleSelector + ); + if (titleEl) { + toggleTitle.append( + titleEl.textContent || titleEl.innerText, + toggleIcon + ); + } + toggleTitle.classList.add("zindex-over-content"); + toggleTitle.classList.add("quarto-sidebar-toggle-title"); + toggleContainer.append(toggleTitle); + + const toggleContents = window.document.createElement("div"); + toggleContents.classList = el.classList; + toggleContents.classList.add("zindex-over-content"); + toggleContents.classList.add("quarto-sidebar-toggle-contents"); + for (const child of el.children) { + if (child.id === "toc-title") { + continue; + } + + const clone = child.cloneNode(true); + clone.style.opacity = 1; + clone.style.display = null; + toggleContents.append(clone); + } + toggleContents.style.height = "0px"; + const positionToggle = () => { + // position the element (top left of parent, same width as parent) + if (!elRect) { + elRect = el.getBoundingClientRect(); + } + toggleContainer.style.left = `${elRect.left}px`; + toggleContainer.style.top = `${elRect.top}px`; + toggleContainer.style.width = `${elRect.width}px`; + }; + positionToggle(); + + toggleContainer.append(toggleContents); + el.parentElement.prepend(toggleContainer); + + // Process clicks + let tocShowing = false; + // Allow the caller to control whether this is dismissed + // when it is clicked (e.g. sidebar navigation supports + // opening and closing the nav tree, so don't dismiss on click) + const clickEl = placeholderDescriptor.dismissOnClick + ? toggleContainer + : toggleTitle; + + const closeToggle = () => { + if (tocShowing) { + toggleContainer.classList.remove("expanded"); + toggleContents.style.height = "0px"; + tocShowing = false; + } + }; + + // Get rid of any expanded toggle if the user scrolls + window.document.addEventListener( + "scroll", + throttle(() => { + closeToggle(); + }, 50) + ); + + // Handle positioning of the toggle + window.addEventListener( + "resize", + throttle(() => { + elRect = undefined; + positionToggle(); + }, 50) + ); + + window.addEventListener("quarto-hrChanged", () => { + elRect = undefined; + }); + + // Process the click + clickEl.onclick = () => { + if (!tocShowing) { + toggleContainer.classList.add("expanded"); + toggleContents.style.height = null; + tocShowing = true; + } else { + closeToggle(); + } + }; + }); + }; + + // Converts a sidebar from a menu back to a sidebar + const convertToSidebar = () => { + for (const child of el.children) { + child.style.opacity = 1; + child.style.overflow = null; + } + + const placeholderEl = window.document.getElementById( + placeholderDescriptor.id + ); + if (placeholderEl) { + placeholderEl.remove(); + } + + el.classList.remove("rollup"); + }; + + if (isReaderMode()) { + convertToMenu(); + isVisible = false; + } else { + // Find the top and bottom o the element that is being managed + const elTop = el.offsetTop; + const elBottom = + elTop + lastChildEl.offsetTop + lastChildEl.offsetHeight; + + if (!isVisible) { + // If the element is current not visible reveal if there are + // no conflicts with overlay regions + if (!inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToSidebar(); + isVisible = true; + } + } else { + // If the element is visible, hide it if it conflicts with overlay regions + // and insert a placeholder toggle (or if we're in reader mode) + if (inHiddenRegion(elTop, elBottom, hiddenRegions)) { + convertToMenu(); + isVisible = false; + } + } + } + } + }; + }; + + const tabEls = document.querySelectorAll('a[data-bs-toggle="tab"]'); + for (const tabEl of tabEls) { + const id = tabEl.getAttribute("data-bs-target"); + if (id) { + const columnEl = document.querySelector( + `${id} .column-margin, .tabset-margin-content` + ); + if (columnEl) + tabEl.addEventListener("shown.bs.tab", function (event) { + const el = event.srcElement; + if (el) { + const visibleCls = `${el.id}-margin-content`; + // walk up until we find a parent tabset + let panelTabsetEl = el.parentElement; + while (panelTabsetEl) { + if (panelTabsetEl.classList.contains("panel-tabset")) { + break; + } + panelTabsetEl = panelTabsetEl.parentElement; + } + + if (panelTabsetEl) { + const prevSib = panelTabsetEl.previousElementSibling; + if ( + prevSib && + prevSib.classList.contains("tabset-margin-container") + ) { + const childNodes = prevSib.querySelectorAll( + ".tabset-margin-content" + ); + for (const childEl of childNodes) { + if (childEl.classList.contains(visibleCls)) { + childEl.classList.remove("collapse"); + } else { + childEl.classList.add("collapse"); + } + } + } + } + } + + layoutMarginEls(); + }); + } + } + + // Manage the visibility of the toc and the sidebar + const marginScrollVisibility = manageSidebarVisiblity(marginSidebarEl, { + id: "quarto-toc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + const sidebarScrollVisiblity = manageSidebarVisiblity(sidebarEl, { + id: "quarto-sidebarnav-toggle", + titleSelector: ".title", + dismissOnClick: false, + }); + let tocLeftScrollVisibility; + if (leftTocEl) { + tocLeftScrollVisibility = manageSidebarVisiblity(leftTocEl, { + id: "quarto-lefttoc-toggle", + titleSelector: "#toc-title", + dismissOnClick: true, + }); + } + + // Find the first element that uses formatting in special columns + const conflictingEls = window.document.body.querySelectorAll( + '[class^="column-"], [class*=" column-"], aside, [class*="margin-caption"], [class*=" margin-caption"], [class*="margin-ref"], [class*=" margin-ref"]' + ); + + // Filter all the possibly conflicting elements into ones + // the do conflict on the left or ride side + const arrConflictingEls = Array.from(conflictingEls); + const leftSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return false; + } + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + className.startsWith("column-") && + !className.endsWith("right") && + !className.endsWith("container") && + className !== "column-margin" + ); + }); + }); + const rightSideConflictEls = arrConflictingEls.filter((el) => { + if (el.tagName === "ASIDE") { + return true; + } + + const hasMarginCaption = Array.from(el.classList).find((className) => { + return className == "margin-caption"; + }); + if (hasMarginCaption) { + return true; + } + + return Array.from(el.classList).find((className) => { + return ( + className !== "column-body" && + !className.endsWith("container") && + className.startsWith("column-") && + !className.endsWith("left") + ); + }); + }); + + const kOverlapPaddingSize = 10; + function toRegions(els) { + return els.map((el) => { + const boundRect = el.getBoundingClientRect(); + const top = + boundRect.top + + document.documentElement.scrollTop - + kOverlapPaddingSize; + return { + top, + bottom: top + el.scrollHeight + 2 * kOverlapPaddingSize, + }; + }); + } + + let hasObserved = false; + const visibleItemObserver = (els) => { + let visibleElements = [...els]; + const intersectionObserver = new IntersectionObserver( + (entries, _observer) => { + entries.forEach((entry) => { + if (entry.isIntersecting) { + if (visibleElements.indexOf(entry.target) === -1) { + visibleElements.push(entry.target); + } + } else { + visibleElements = visibleElements.filter((visibleEntry) => { + return visibleEntry !== entry; + }); + } + }); + + if (!hasObserved) { + hideOverlappedSidebars(); + } + hasObserved = true; + }, + {} + ); + els.forEach((el) => { + intersectionObserver.observe(el); + }); + + return { + getVisibleEntries: () => { + return visibleElements; + }, + }; + }; + + const rightElementObserver = visibleItemObserver(rightSideConflictEls); + const leftElementObserver = visibleItemObserver(leftSideConflictEls); + + const hideOverlappedSidebars = () => { + marginScrollVisibility(toRegions(rightElementObserver.getVisibleEntries())); + sidebarScrollVisiblity(toRegions(leftElementObserver.getVisibleEntries())); + if (tocLeftScrollVisibility) { + tocLeftScrollVisibility( + toRegions(leftElementObserver.getVisibleEntries()) + ); + } + }; + + window.quartoToggleReader = () => { + // Applies a slow class (or removes it) + // to update the transition speed + const slowTransition = (slow) => { + const manageTransition = (id, slow) => { + const el = document.getElementById(id); + if (el) { + if (slow) { + el.classList.add("slow"); + } else { + el.classList.remove("slow"); + } + } + }; + + manageTransition("TOC", slow); + manageTransition("quarto-sidebar", slow); + }; + const readerMode = !isReaderMode(); + setReaderModeValue(readerMode); + + // If we're entering reader mode, slow the transition + if (readerMode) { + slowTransition(readerMode); + } + highlightReaderToggle(readerMode); + hideOverlappedSidebars(); + + // If we're exiting reader mode, restore the non-slow transition + if (!readerMode) { + slowTransition(!readerMode); + } + }; + + const highlightReaderToggle = (readerMode) => { + const els = document.querySelectorAll(".quarto-reader-toggle"); + if (els) { + els.forEach((el) => { + if (readerMode) { + el.classList.add("reader"); + } else { + el.classList.remove("reader"); + } + }); + } + }; + + const setReaderModeValue = (val) => { + if (window.location.protocol !== "file:") { + window.localStorage.setItem("quarto-reader-mode", val); + } else { + localReaderMode = val; + } + }; + + const isReaderMode = () => { + if (window.location.protocol !== "file:") { + return window.localStorage.getItem("quarto-reader-mode") === "true"; + } else { + return localReaderMode; + } + }; + let localReaderMode = null; + + const tocOpenDepthStr = tocEl?.getAttribute("data-toc-expanded"); + const tocOpenDepth = tocOpenDepthStr ? Number(tocOpenDepthStr) : 1; + + // Walk the TOC and collapse/expand nodes + // Nodes are expanded if: + // - they are top level + // - they have children that are 'active' links + // - they are directly below an link that is 'active' + const walk = (el, depth) => { + // Tick depth when we enter a UL + if (el.tagName === "UL") { + depth = depth + 1; + } + + // It this is active link + let isActiveNode = false; + if (el.tagName === "A" && el.classList.contains("active")) { + isActiveNode = true; + } + + // See if there is an active child to this element + let hasActiveChild = false; + for (child of el.children) { + hasActiveChild = walk(child, depth) || hasActiveChild; + } + + // Process the collapse state if this is an UL + if (el.tagName === "UL") { + if (tocOpenDepth === -1 && depth > 1) { + el.classList.add("collapse"); + } else if ( + depth <= tocOpenDepth || + hasActiveChild || + prevSiblingIsActiveLink(el) + ) { + el.classList.remove("collapse"); + } else { + el.classList.add("collapse"); + } + + // untick depth when we leave a UL + depth = depth - 1; + } + return hasActiveChild || isActiveNode; + }; + + // walk the TOC and expand / collapse any items that should be shown + + if (tocEl) { + walk(tocEl, 0); + updateActiveLink(); + } + + // Throttle the scroll event and walk peridiocally + window.document.addEventListener( + "scroll", + throttle(() => { + if (tocEl) { + updateActiveLink(); + walk(tocEl, 0); + } + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 5) + ); + window.addEventListener( + "resize", + throttle(() => { + if (!isReaderMode()) { + hideOverlappedSidebars(); + } + }, 10) + ); + hideOverlappedSidebars(); + highlightReaderToggle(isReaderMode()); +}); + +// grouped tabsets +window.addEventListener("pageshow", (_event) => { + function getTabSettings() { + const data = localStorage.getItem("quarto-persistent-tabsets-data"); + if (!data) { + localStorage.setItem("quarto-persistent-tabsets-data", "{}"); + return {}; + } + if (data) { + return JSON.parse(data); + } + } + + function setTabSettings(data) { + localStorage.setItem( + "quarto-persistent-tabsets-data", + JSON.stringify(data) + ); + } + + function setTabState(groupName, groupValue) { + const data = getTabSettings(); + data[groupName] = groupValue; + setTabSettings(data); + } + + function toggleTab(tab, active) { + const tabPanelId = tab.getAttribute("aria-controls"); + const tabPanel = document.getElementById(tabPanelId); + if (active) { + tab.classList.add("active"); + tabPanel.classList.add("active"); + } else { + tab.classList.remove("active"); + tabPanel.classList.remove("active"); + } + } + + function toggleAll(selectedGroup, selectorsToSync) { + for (const [thisGroup, tabs] of Object.entries(selectorsToSync)) { + const active = selectedGroup === thisGroup; + for (const tab of tabs) { + toggleTab(tab, active); + } + } + } + + function findSelectorsToSyncByLanguage() { + const result = {}; + const tabs = Array.from( + document.querySelectorAll(`div[data-group] a[id^='tabset-']`) + ); + for (const item of tabs) { + const div = item.parentElement.parentElement.parentElement; + const group = div.getAttribute("data-group"); + if (!result[group]) { + result[group] = {}; + } + const selectorsToSync = result[group]; + const value = item.innerHTML; + if (!selectorsToSync[value]) { + selectorsToSync[value] = []; + } + selectorsToSync[value].push(item); + } + return result; + } + + function setupSelectorSync() { + const selectorsToSync = findSelectorsToSyncByLanguage(); + Object.entries(selectorsToSync).forEach(([group, tabSetsByValue]) => { + Object.entries(tabSetsByValue).forEach(([value, items]) => { + items.forEach((item) => { + item.addEventListener("click", (_event) => { + setTabState(group, value); + toggleAll(value, selectorsToSync[group]); + }); + }); + }); + }); + return selectorsToSync; + } + + const selectorsToSync = setupSelectorSync(); + for (const [group, selectedName] of 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b/session-survival/docs/site_libs/quarto-nav/headroom.min.js new file mode 100644 index 00000000..b08f1dff --- /dev/null +++ b/session-survival/docs/site_libs/quarto-nav/headroom.min.js @@ -0,0 +1,7 @@ +/*! + * headroom.js v0.12.0 - Give your page some headroom. 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index 00000000..f6a53b10 --- /dev/null +++ b/session-survival/docs/site_libs/quarto-nav/quarto-nav.js @@ -0,0 +1,289 @@ +const headroomChanged = new CustomEvent("quarto-hrChanged", { + detail: {}, + bubbles: true, + cancelable: false, + composed: false, +}); + +window.document.addEventListener("DOMContentLoaded", function () { + let init = false; + + // Manage the back to top button, if one is present. + let lastScrollTop = window.pageYOffset || document.documentElement.scrollTop; + const scrollDownBuffer = 5; + const scrollUpBuffer = 35; + const btn = document.getElementById("quarto-back-to-top"); + const hideBackToTop = () => { + btn.style.display = "none"; + }; + const showBackToTop = () => { + btn.style.display = "inline-block"; + }; + if (btn) { + window.document.addEventListener( + "scroll", + function () { + const currentScrollTop = + window.pageYOffset || document.documentElement.scrollTop; + + // Shows and hides the button 'intelligently' as the user scrolls + if (currentScrollTop - scrollDownBuffer > lastScrollTop) { + hideBackToTop(); + lastScrollTop = currentScrollTop <= 0 ? 0 : currentScrollTop; + } else if (currentScrollTop < lastScrollTop - scrollUpBuffer) { + showBackToTop(); + lastScrollTop = currentScrollTop <= 0 ? 0 : currentScrollTop; + } + + // Show the button at the bottom, hides it at the top + if (currentScrollTop <= 0) { + hideBackToTop(); + } else if ( + window.innerHeight + currentScrollTop >= + document.body.offsetHeight + ) { + showBackToTop(); + } + }, + false + ); + } + + function throttle(func, wait) { + var timeout; + return function () { + const context = this; + const args = arguments; + const later = function () { + clearTimeout(timeout); + timeout = null; + func.apply(context, args); + }; + + if (!timeout) { + timeout = setTimeout(later, wait); + } + }; + } + + function headerOffset() { + // Set an offset if there is are fixed top navbar + const headerEl = window.document.querySelector("header.fixed-top"); + if (headerEl) { + return headerEl.clientHeight; + } else { + return 0; + } + } + + function footerOffset() { + const footerEl = window.document.querySelector("footer.footer"); + if (footerEl) { + return footerEl.clientHeight; + } else { + return 0; + } + } + + function dashboardOffset() { + const dashboardNavEl = window.document.getElementById( + "quarto-dashboard-header" + ); + if (dashboardNavEl !== null) { + return dashboardNavEl.clientHeight; + } else { + return 0; + } + } + + function updateDocumentOffsetWithoutAnimation() { + updateDocumentOffset(false); + } + + function updateDocumentOffset(animated) { + // set body offset + const topOffset = headerOffset(); + const bodyOffset = topOffset + footerOffset() + dashboardOffset(); + const bodyEl = window.document.body; + bodyEl.setAttribute("data-bs-offset", topOffset); + bodyEl.style.paddingTop = topOffset + "px"; + + // deal with sidebar offsets + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + if (!animated) { + sidebar.classList.add("notransition"); + // Remove the no transition class after the animation has time to complete + setTimeout(function () { + sidebar.classList.remove("notransition"); + }, 201); + } + + if (window.Headroom && sidebar.classList.contains("sidebar-unpinned")) { + sidebar.style.top = "0"; + sidebar.style.maxHeight = "100vh"; + } else { + sidebar.style.top = topOffset + "px"; + sidebar.style.maxHeight = "calc(100vh - " + topOffset + "px)"; + } + }); + + // allow space for footer + const mainContainer = window.document.querySelector(".quarto-container"); + if (mainContainer) { + mainContainer.style.minHeight = "calc(100vh - " + bodyOffset + "px)"; + } + + // link offset + let linkStyle = window.document.querySelector("#quarto-target-style"); + if (!linkStyle) { + linkStyle = window.document.createElement("style"); + linkStyle.setAttribute("id", "quarto-target-style"); + window.document.head.appendChild(linkStyle); + } + while (linkStyle.firstChild) { + linkStyle.removeChild(linkStyle.firstChild); + } + if (topOffset > 0) { + linkStyle.appendChild( + window.document.createTextNode(` + section:target::before { + content: ""; + display: block; + height: ${topOffset}px; + margin: -${topOffset}px 0 0; + }`) + ); + } + if (init) { + window.dispatchEvent(headroomChanged); + } + init = true; + } + + // initialize headroom + var header = window.document.querySelector("#quarto-header"); + if (header && window.Headroom) { + const headroom = new window.Headroom(header, { + tolerance: 5, + onPin: function () { + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + sidebar.classList.remove("sidebar-unpinned"); + }); + updateDocumentOffset(); + }, + onUnpin: function () { + const sidebars = window.document.querySelectorAll( + ".sidebar, .headroom-target" + ); + sidebars.forEach((sidebar) => { + sidebar.classList.add("sidebar-unpinned"); + }); + 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setTimeout(updateDocumentOffsetWithoutAnimation, 250); + + // fixup index.html links if we aren't on the filesystem + if (window.location.protocol !== "file:") { + const links = window.document.querySelectorAll("a"); + for (let i = 0; i < links.length; i++) { + if (links[i].href) { + links[i].dataset.originalHref = links[i].href; + links[i].href = links[i].href.replace(/\/index\.html/, "/"); + } + } + + // Fixup any sharing links that require urls + // Append url to any sharing urls + const sharingLinks = window.document.querySelectorAll( + "a.sidebar-tools-main-item, a.quarto-navigation-tool, a.quarto-navbar-tools, a.quarto-navbar-tools-item" + ); + for (let i = 0; i < sharingLinks.length; i++) { + const sharingLink = sharingLinks[i]; + const href = sharingLink.getAttribute("href"); + if (href) { + sharingLink.setAttribute( + "href", + href.replace("|url|", window.location.href) + ); + } + } + + // Scroll the active navigation item into view, if necessary + const navSidebar = window.document.querySelector("nav#quarto-sidebar"); + if (navSidebar) { + // Find the active item + const activeItem = navSidebar.querySelector("li.sidebar-item a.active"); + if (activeItem) { + // Wait for the scroll height and height to resolve by observing size changes on the + // nav element that is scrollable + const resizeObserver = new ResizeObserver((_entries) => { + // The bottom of the element + const elBottom = activeItem.offsetTop; + const viewBottom = navSidebar.scrollTop + navSidebar.clientHeight; + + // The element height and scroll height are the same, then we are still loading + if (viewBottom !== navSidebar.scrollHeight) { + // Determine if the item isn't visible and scroll to it + if (elBottom >= viewBottom) { + navSidebar.scrollTop = elBottom; + } + + // stop observing now since we've completed the scroll + resizeObserver.unobserve(navSidebar); + } + }); + resizeObserver.observe(navSidebar); + } + } + } +}); diff --git 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Apache Software License 2.0 + * + * http://www.apache.org/licenses/LICENSE-2.0 + */ +var e,t;e=this,t=function(){"use strict";function e(e,t){var n=Object.keys(e);if(Object.getOwnPropertySymbols){var r=Object.getOwnPropertySymbols(e);t&&(r=r.filter((function(t){return Object.getOwnPropertyDescriptor(e,t).enumerable}))),n.push.apply(n,r)}return n}function t(t){for(var n=1;ne.length)&&(t=e.length);for(var n=0,r=new Array(t);n0&&void 0!==arguments[0]?arguments[0]:1,t=arguments.length>1&&void 0!==arguments[1]?arguments[1]:3,n=new Map,r=Math.pow(10,t);return{get:function(t){var i=t.match(C).length;if(n.has(i))return n.get(i);var o=1/Math.pow(i,.5*e),c=parseFloat(Math.round(o*r)/r);return n.set(i,c),c},clear:function(){n.clear()}}}var $=function(){function e(){var t=arguments.length>0&&void 0!==arguments[0]?arguments[0]:{},n=t.getFn,i=void 0===n?I.getFn:n,o=t.fieldNormWeight,c=void 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a/session-survival/docs/site_libs/quarto-search/quarto-search.js b/session-survival/docs/site_libs/quarto-search/quarto-search.js new file mode 100644 index 00000000..5f723d72 --- /dev/null +++ b/session-survival/docs/site_libs/quarto-search/quarto-search.js @@ -0,0 +1,1286 @@ +const kQueryArg = "q"; +const kResultsArg = "show-results"; + +// If items don't provide a URL, then both the navigator and the onSelect +// function aren't called (and therefore, the default implementation is used) +// +// We're using this sentinel URL to signal to those handlers that this +// item is a more item (along with the type) and can be handled appropriately +const kItemTypeMoreHref = "0767FDFD-0422-4E5A-BC8A-3BE11E5BBA05"; + +window.document.addEventListener("DOMContentLoaded", function (_event) { + // Ensure that search is available on this page. If it isn't, + // should return early and not do anything + var searchEl = window.document.getElementById("quarto-search"); + if (!searchEl) return; + + const { autocomplete } = window["@algolia/autocomplete-js"]; + + let quartoSearchOptions = {}; + let language = {}; + const searchOptionEl = window.document.getElementById( + "quarto-search-options" + ); + if (searchOptionEl) { + const jsonStr = searchOptionEl.textContent; + quartoSearchOptions = JSON.parse(jsonStr); + language = quartoSearchOptions.language; + } + + // note the search mode + if (quartoSearchOptions.type === "overlay") { + searchEl.classList.add("type-overlay"); + } else { + searchEl.classList.add("type-textbox"); + } + + // Used to determine highlighting behavior for this page + // A `q` query param is expected when the user follows a search + // to this page + const currentUrl = new URL(window.location); + const query = currentUrl.searchParams.get(kQueryArg); + const showSearchResults = currentUrl.searchParams.get(kResultsArg); + const mainEl = window.document.querySelector("main"); + + // highlight matches on the page + if (query && mainEl) { + // perform any highlighting + highlight(escapeRegExp(query), mainEl); + + // fix up the URL to remove the q query param + const replacementUrl = new URL(window.location); + replacementUrl.searchParams.delete(kQueryArg); + window.history.replaceState({}, "", replacementUrl); + } + + // function to clear highlighting on the page when the search query changes + // (e.g. if the user edits the query or clears it) + let highlighting = true; + const resetHighlighting = (searchTerm) => { + if (mainEl && highlighting && query && searchTerm !== query) { + clearHighlight(query, mainEl); + highlighting = false; + } + }; + + // Clear search highlighting when the user scrolls sufficiently + const resetFn = () => { + resetHighlighting(""); + window.removeEventListener("quarto-hrChanged", resetFn); + window.removeEventListener("quarto-sectionChanged", resetFn); + }; + + // Register this event after the initial scrolling and settling of events + // on the page + window.addEventListener("quarto-hrChanged", resetFn); + window.addEventListener("quarto-sectionChanged", resetFn); + + // Responsively switch to overlay mode if the search is present on the navbar + // Note that switching the sidebar to overlay mode requires more coordinate (not just + // the media query since we generate different HTML for sidebar overlays than we do + // for sidebar input UI) + const detachedMediaQuery = + quartoSearchOptions.type === "overlay" ? "all" : "(max-width: 991px)"; + + // If configured, include the analytics client to send insights + const plugins = configurePlugins(quartoSearchOptions); + + let lastState = null; + const { setIsOpen, setQuery, setCollections } = autocomplete({ + container: searchEl, + detachedMediaQuery: detachedMediaQuery, + defaultActiveItemId: 0, + panelContainer: "#quarto-search-results", + panelPlacement: quartoSearchOptions["panel-placement"], + debug: false, + openOnFocus: true, + plugins, + classNames: { + form: "d-flex", + }, + placeholder: language["search-text-placeholder"], + translations: { + clearButtonTitle: language["search-clear-button-title"], + detachedCancelButtonText: language["search-detached-cancel-button-title"], + submitButtonTitle: language["search-submit-button-title"], + }, + initialState: { + query, + }, + getItemUrl({ item }) { + return item.href; + }, + onStateChange({ state }) { + // If this is a file URL, note that + + // Perhaps reset highlighting + resetHighlighting(state.query); + + // If the panel just opened, ensure the panel is positioned properly + if (state.isOpen) { + if (lastState && !lastState.isOpen) { + setTimeout(() => { + positionPanel(quartoSearchOptions["panel-placement"]); + }, 150); + } + } + + // Perhaps show the copy link + showCopyLink(state.query, quartoSearchOptions); + + lastState = state; + }, + reshape({ sources, state }) { + return sources.map((source) => { + try { + const items = source.getItems(); + + // Validate the items + validateItems(items); + + // group the items by document + const groupedItems = new Map(); + items.forEach((item) => { + const hrefParts = item.href.split("#"); + const baseHref = hrefParts[0]; + const isDocumentItem = hrefParts.length === 1; + + const items = groupedItems.get(baseHref); + if (!items) { + groupedItems.set(baseHref, [item]); + } else { + // If the href for this item matches the document + // exactly, place this item first as it is the item that represents + // the document itself + if (isDocumentItem) { + items.unshift(item); + } else { + items.push(item); + } + groupedItems.set(baseHref, items); + } + }); + + const reshapedItems = []; + let count = 1; + for (const [_key, value] of groupedItems) { + const firstItem = value[0]; + reshapedItems.push({ + ...firstItem, + type: kItemTypeDoc, + }); + + const collapseMatches = quartoSearchOptions["collapse-after"]; + const collapseCount = + typeof collapseMatches === "number" ? collapseMatches : 1; + + if (value.length > 1) { + const target = `search-more-${count}`; + const isExpanded = + state.context.expanded && + state.context.expanded.includes(target); + + const remainingCount = value.length - collapseCount; + + for (let i = 1; i < value.length; i++) { + if (collapseMatches && i === collapseCount) { + reshapedItems.push({ + target, + title: isExpanded + ? language["search-hide-matches-text"] + : remainingCount === 1 + ? `${remainingCount} ${language["search-more-match-text"]}` + : `${remainingCount} ${language["search-more-matches-text"]}`, + type: kItemTypeMore, + href: kItemTypeMoreHref, + }); + } + + if (isExpanded || !collapseMatches || i < collapseCount) { + reshapedItems.push({ + ...value[i], + type: kItemTypeItem, + target, + }); + } + } + } + count += 1; + } + + return { + ...source, + getItems() { + return reshapedItems; + }, + }; + } catch (error) { + // Some form of error occurred + return { + ...source, + getItems() { + return [ + { + title: error.name || "An Error Occurred While Searching", + text: + error.message || + "An unknown error occurred while attempting to perform the requested search.", + type: kItemTypeError, + }, + ]; + }, + }; + } + }); + }, + navigator: { + navigate({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + window.location.assign(itemUrl); + } + }, + navigateNewTab({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + const windowReference = window.open(itemUrl, "_blank", "noopener"); + if (windowReference) { + windowReference.focus(); + } + } + }, + navigateNewWindow({ itemUrl }) { + if (itemUrl !== offsetURL(kItemTypeMoreHref)) { + window.open(itemUrl, "_blank", "noopener"); + } + }, + }, + getSources({ state, setContext, setActiveItemId, refresh }) { + return [ + { + sourceId: "documents", + getItemUrl({ item }) { + if (item.href) { + return offsetURL(item.href); + } else { + return undefined; + } + }, + onSelect({ + item, + state, + setContext, + setIsOpen, + setActiveItemId, + refresh, + }) { + if (item.type === kItemTypeMore) { + toggleExpanded(item, state, setContext, setActiveItemId, refresh); + + // Toggle more + setIsOpen(true); + } + }, + getItems({ query }) { + if (query === null || query === "") { + return []; + } + + const limit = quartoSearchOptions.limit; + if (quartoSearchOptions.algolia) { + return algoliaSearch(query, limit, quartoSearchOptions.algolia); + } else { + // Fuse search options + const fuseSearchOptions = { + isCaseSensitive: false, + shouldSort: true, + minMatchCharLength: 2, + limit: limit, + }; + + return readSearchData().then(function (fuse) { + return fuseSearch(query, fuse, fuseSearchOptions); + }); + } + }, + templates: { + noResults({ createElement }) { + const hasQuery = lastState.query; + + return createElement( + "div", + { + class: `quarto-search-no-results${ + hasQuery ? "" : " no-query" + }`, + }, + language["search-no-results-text"] + ); + }, + header({ items, createElement }) { + // count the documents + const count = items.filter((item) => { + return item.type === kItemTypeDoc; + }).length; + + if (count > 0) { + return createElement( + "div", + { class: "search-result-header" }, + `${count} ${language["search-matching-documents-text"]}` + ); + } else { + return createElement( + "div", + { class: "search-result-header-no-results" }, + `` + ); + } + }, + footer({ _items, createElement }) { + if ( + quartoSearchOptions.algolia && + quartoSearchOptions.algolia["show-logo"] + ) { + const libDir = quartoSearchOptions.algolia["libDir"]; + const logo = createElement("img", { + src: offsetURL( + `${libDir}/quarto-search/search-by-algolia.svg` + ), + class: "algolia-search-logo", + }); + return createElement( + "a", + { href: "http://www.algolia.com/" }, + logo + ); + } + }, + + item({ item, createElement }) { + return renderItem( + item, + createElement, + state, + setActiveItemId, + setContext, + refresh, + quartoSearchOptions + ); + }, + }, + }, + ]; + }, + }); + + window.quartoOpenSearch = () => { + setIsOpen(false); + setIsOpen(true); + focusSearchInput(); + }; + + document.addEventListener("keyup", (event) => { + const { key } = event; + const kbds = quartoSearchOptions["keyboard-shortcut"]; + const focusedEl = document.activeElement; + + const isFormElFocused = [ + "input", + "select", + "textarea", + "button", + "option", + ].find((tag) => { + return focusedEl.tagName.toLowerCase() === tag; + }); + + if ( + kbds && + kbds.includes(key) && + !isFormElFocused && + !document.activeElement.isContentEditable + ) { + event.preventDefault(); + window.quartoOpenSearch(); + } + }); + + // Remove the labeleledby attribute since it is pointing + // to a non-existent label + if (quartoSearchOptions.type === "overlay") { + const inputEl = window.document.querySelector( + "#quarto-search .aa-Autocomplete" + ); + if (inputEl) { + inputEl.removeAttribute("aria-labelledby"); + } + } + + function throttle(func, wait) { + let waiting = false; + return function () { + if (!waiting) { + func.apply(this, arguments); + waiting = true; + setTimeout(function () { + waiting = false; + }, wait); + } + }; + } + + // If the main document scrolls dismiss the search results + // (otherwise, since they're floating in the document they can scroll with the document) + window.document.body.onscroll = throttle(() => { + // Only do this if we're not detached + // Bug #7117 + // This will happen when the keyboard is shown on ios (resulting in a scroll) + // which then closed the search UI + if (!window.matchMedia(detachedMediaQuery).matches) { + setIsOpen(false); + } + }, 50); + + if (showSearchResults) { + setIsOpen(true); + focusSearchInput(); + } +}); + +function configurePlugins(quartoSearchOptions) { + const autocompletePlugins = []; + const algoliaOptions = quartoSearchOptions.algolia; + if ( + algoliaOptions && + algoliaOptions["analytics-events"] && + algoliaOptions["search-only-api-key"] && + algoliaOptions["application-id"] + ) { + const apiKey = algoliaOptions["search-only-api-key"]; + const appId = algoliaOptions["application-id"]; + + // Aloglia insights may not be loaded because they require cookie consent + // Use deferred loading so events will start being recorded when/if consent + // is granted. + const algoliaInsightsDeferredPlugin = deferredLoadPlugin(() => { + if ( + window.aa && + window["@algolia/autocomplete-plugin-algolia-insights"] + ) { + window.aa("init", { + appId, + apiKey, + useCookie: true, + }); + + const { createAlgoliaInsightsPlugin } = + window["@algolia/autocomplete-plugin-algolia-insights"]; + // Register the insights client + const algoliaInsightsPlugin = createAlgoliaInsightsPlugin({ + insightsClient: window.aa, + onItemsChange({ insights, insightsEvents }) { + const events = insightsEvents.flatMap((event) => { + // This API limits the number of items per event to 20 + const chunkSize = 20; + const itemChunks = []; + const eventItems = event.items; + for (let i = 0; i < eventItems.length; i += chunkSize) { + itemChunks.push(eventItems.slice(i, i + chunkSize)); + } + // Split the items into multiple events that can be sent + const events = itemChunks.map((items) => { + return { + ...event, + items, + }; + }); + return events; + }); + + for (const event of events) { + insights.viewedObjectIDs(event); + } + }, + }); + return algoliaInsightsPlugin; + } + }); + + // Add the plugin + autocompletePlugins.push(algoliaInsightsDeferredPlugin); + return autocompletePlugins; + } +} + +// For plugins that may not load immediately, create a wrapper +// plugin and forward events and plugin data once the plugin +// is initialized. This is useful for cases like cookie consent +// which may prevent the analytics insights event plugin from initializing +// immediately. +function deferredLoadPlugin(createPlugin) { + let plugin = undefined; + let subscribeObj = undefined; + const wrappedPlugin = () => { + if (!plugin && subscribeObj) { + plugin = createPlugin(); + if (plugin && plugin.subscribe) { + plugin.subscribe(subscribeObj); + } + } + return plugin; + }; + + return { + subscribe: (obj) => { + subscribeObj = obj; + }, + onStateChange: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onStateChange) { + plugin.onStateChange(obj); + } + }, + onSubmit: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onSubmit) { + plugin.onSubmit(obj); + } + }, + onReset: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.onReset) { + plugin.onReset(obj); + } + }, + getSources: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.getSources) { + return plugin.getSources(obj); + } else { + return Promise.resolve([]); + } + }, + data: (obj) => { + const plugin = wrappedPlugin(); + if (plugin && plugin.data) { + plugin.data(obj); + } + }, + }; +} + +function validateItems(items) { + // Validate the first item + if (items.length > 0) { + const item = items[0]; + const missingFields = []; + if (item.href == undefined) { + missingFields.push("href"); + } + if (!item.title == undefined) { + missingFields.push("title"); + } + if (!item.text == undefined) { + missingFields.push("text"); + } + + if (missingFields.length === 1) { + throw { + name: `Error: Search index is missing the ${missingFields[0]} field.`, + message: `The items being returned for this search do not include all the required fields. Please ensure that your index items include the ${missingFields[0]} field or use index-fields in your _quarto.yml file to specify the field names.`, + }; + } else if (missingFields.length > 1) { + const missingFieldList = missingFields + .map((field) => { + return `${field}`; + }) + .join(", "); + + throw { + name: `Error: Search index is missing the following fields: ${missingFieldList}.`, + message: `The items being returned for this search do not include all the required fields. Please ensure that your index items includes the following fields: ${missingFieldList}, or use index-fields in your _quarto.yml file to specify the field names.`, + }; + } + } +} + +let lastQuery = null; +function showCopyLink(query, options) { + const language = options.language; + lastQuery = query; + // Insert share icon + const inputSuffixEl = window.document.body.querySelector( + ".aa-Form .aa-InputWrapperSuffix" + ); + + if (inputSuffixEl) { + let copyButtonEl = window.document.body.querySelector( + ".aa-Form .aa-InputWrapperSuffix .aa-CopyButton" + ); + + if (copyButtonEl === null) { + copyButtonEl = window.document.createElement("button"); + copyButtonEl.setAttribute("class", "aa-CopyButton"); + copyButtonEl.setAttribute("type", "button"); + copyButtonEl.setAttribute("title", language["search-copy-link-title"]); + copyButtonEl.onmousedown = (e) => { + e.preventDefault(); + e.stopPropagation(); + }; + + const linkIcon = "bi-clipboard"; + const checkIcon = "bi-check2"; + + const shareIconEl = window.document.createElement("i"); + shareIconEl.setAttribute("class", `bi ${linkIcon}`); + copyButtonEl.appendChild(shareIconEl); + inputSuffixEl.prepend(copyButtonEl); + + const clipboard = new window.ClipboardJS(".aa-CopyButton", { + text: function (_trigger) { + const copyUrl = new URL(window.location); + copyUrl.searchParams.set(kQueryArg, lastQuery); + copyUrl.searchParams.set(kResultsArg, "1"); + return copyUrl.toString(); + }, + }); + clipboard.on("success", function (e) { + // Focus the input + + // button target + const button = e.trigger; + const icon = button.querySelector("i.bi"); + + // flash "checked" + icon.classList.add(checkIcon); + icon.classList.remove(linkIcon); + setTimeout(function () { + icon.classList.remove(checkIcon); + icon.classList.add(linkIcon); + }, 1000); + }); + } + + // If there is a query, show the link icon + if (copyButtonEl) { + if (lastQuery && options["copy-button"]) { + copyButtonEl.style.display = "flex"; + } else { + copyButtonEl.style.display = "none"; + } + } + } +} + +/* Search Index Handling */ +// create the index +var fuseIndex = undefined; +var shownWarning = false; + +// fuse index options +const kFuseIndexOptions = { + keys: [ + { name: "title", weight: 20 }, + { name: "section", weight: 20 }, + { name: "text", weight: 10 }, + ], + ignoreLocation: true, + threshold: 0.1, +}; + +async function readSearchData() { + // Initialize the search index on demand + if (fuseIndex === undefined) { + if (window.location.protocol === "file:" && !shownWarning) { + window.alert( + "Search requires JavaScript features disabled when running in file://... URLs. In order to use search, please run this document in a web server." + ); + shownWarning = true; + return; + } + const fuse = new window.Fuse([], kFuseIndexOptions); + + // fetch the main search.json + const response = await fetch(offsetURL("search.json")); + if (response.status == 200) { + return response.json().then(function (searchDocs) { + searchDocs.forEach(function (searchDoc) { + fuse.add(searchDoc); + }); + fuseIndex = fuse; + return fuseIndex; + }); + } else { + return Promise.reject( + new Error( + "Unexpected status from search index request: " + response.status + ) + ); + } + } + + return fuseIndex; +} + +function inputElement() { + return window.document.body.querySelector(".aa-Form .aa-Input"); +} + +function focusSearchInput() { + setTimeout(() => { + const inputEl = inputElement(); + if (inputEl) { + inputEl.focus(); + } + }, 50); +} + +/* Panels */ +const kItemTypeDoc = "document"; +const kItemTypeMore = "document-more"; +const kItemTypeItem = "document-item"; +const kItemTypeError = "error"; + +function renderItem( + item, + createElement, + state, + setActiveItemId, + setContext, + refresh, + quartoSearchOptions +) { + switch (item.type) { + case kItemTypeDoc: + return createDocumentCard( + createElement, + "file-richtext", + item.title, + item.section, + item.text, + item.href, + item.crumbs, + quartoSearchOptions + ); + case kItemTypeMore: + return createMoreCard( + createElement, + item, + state, + setActiveItemId, + setContext, + refresh + ); + case kItemTypeItem: + return createSectionCard( + createElement, + item.section, + item.text, + item.href + ); + case kItemTypeError: + return createErrorCard(createElement, item.title, item.text); + default: + return undefined; + } +} + +function createDocumentCard( + createElement, + icon, + title, + section, + text, + href, + crumbs, + quartoSearchOptions +) { + const iconEl = createElement("i", { + class: `bi bi-${icon} search-result-icon`, + }); + const titleEl = createElement("p", { class: "search-result-title" }, title); + const titleContents = [iconEl, titleEl]; + const showParent = quartoSearchOptions["show-item-context"]; + if (crumbs && showParent) { + let crumbsOut = undefined; + const crumbClz = ["search-result-crumbs"]; + if (showParent === "root") { + crumbsOut = crumbs.length > 1 ? crumbs[0] : undefined; + } else if (showParent === "parent") { + crumbsOut = crumbs.length > 1 ? crumbs[crumbs.length - 2] : undefined; + } else { + crumbsOut = crumbs.length > 1 ? crumbs.join(" > ") : undefined; + crumbClz.push("search-result-crumbs-wrap"); + } + + const crumbEl = createElement( + "p", + { class: crumbClz.join(" ") }, + crumbsOut + ); + titleContents.push(crumbEl); + } + + const titleContainerEl = createElement( + "div", + { class: "search-result-title-container" }, + titleContents + ); + + const textEls = []; + if (section) { + const sectionEl = createElement( + "p", + { class: "search-result-section" }, + section + ); + textEls.push(sectionEl); + } + const descEl = createElement("p", { + class: "search-result-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + textEls.push(descEl); + + const textContainerEl = createElement( + "div", + { class: "search-result-text-container" }, + textEls + ); + + const containerEl = createElement( + "div", + { + class: "search-result-container", + }, + [titleContainerEl, textContainerEl] + ); + + const linkEl = createElement( + "a", + { + href: offsetURL(href), + class: "search-result-link", + }, + containerEl + ); + + const classes = ["search-result-doc", "search-item"]; + if (!section) { + classes.push("document-selectable"); + } + + return createElement( + "div", + { + class: classes.join(" "), + }, + linkEl + ); +} + +function createMoreCard( + createElement, + item, + state, + setActiveItemId, + setContext, + refresh +) { + const moreCardEl = createElement( + "div", + { + class: "search-result-more search-item", + onClick: (e) => { + // Handle expanding the sections by adding the expanded + // section to the list of expanded sections + toggleExpanded(item, state, setContext, setActiveItemId, refresh); + e.stopPropagation(); + }, + }, + item.title + ); + + return moreCardEl; +} + +function toggleExpanded(item, state, setContext, setActiveItemId, refresh) { + const expanded = state.context.expanded || []; + if (expanded.includes(item.target)) { + setContext({ + expanded: expanded.filter((target) => target !== item.target), + }); + } else { + setContext({ expanded: [...expanded, item.target] }); + } + + refresh(); + setActiveItemId(item.__autocomplete_id); +} + +function createSectionCard(createElement, section, text, href) { + const sectionEl = createSection(createElement, section, text, href); + return createElement( + "div", + { + class: "search-result-doc-section search-item", + }, + sectionEl + ); +} + +function createSection(createElement, title, text, href) { + const descEl = createElement("p", { + class: "search-result-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + + const titleEl = createElement("p", { class: "search-result-section" }, title); + const linkEl = createElement( + "a", + { + href: offsetURL(href), + class: "search-result-link", + }, + [titleEl, descEl] + ); + return linkEl; +} + +function createErrorCard(createElement, title, text) { + const descEl = createElement("p", { + class: "search-error-text", + dangerouslySetInnerHTML: { + __html: text, + }, + }); + + const titleEl = createElement("p", { + class: "search-error-title", + dangerouslySetInnerHTML: { + __html: ` ${title}`, + }, + }); + const errorEl = createElement("div", { class: "search-error" }, [ + titleEl, + descEl, + ]); + return errorEl; +} + +function positionPanel(pos) { + const panelEl = window.document.querySelector( + "#quarto-search-results .aa-Panel" + ); + const inputEl = window.document.querySelector( + "#quarto-search .aa-Autocomplete" + ); + + if (panelEl && inputEl) { + panelEl.style.top = `${Math.round(panelEl.offsetTop)}px`; + if (pos === "start") { + panelEl.style.left = `${Math.round(inputEl.left)}px`; + } else { + panelEl.style.right = `${Math.round(inputEl.offsetRight)}px`; + } + } +} + +/* Highlighting */ +// highlighting functions +function highlightMatch(query, text) { + if (text) { + const start = text.toLowerCase().indexOf(query.toLowerCase()); + if (start !== -1) { + const startMark = ""; + const endMark = ""; + + const end = start + query.length; + text = + text.slice(0, start) + + startMark + + text.slice(start, end) + + endMark + + text.slice(end); + const startInfo = clipStart(text, start); + const endInfo = clipEnd( + text, + startInfo.position + startMark.length + endMark.length + ); + text = + startInfo.prefix + + text.slice(startInfo.position, endInfo.position) + + endInfo.suffix; + + return text; + } else { + return text; + } + } else { + return text; + } +} + +function clipStart(text, pos) { + const clipStart = pos - 50; + if (clipStart < 0) { + // This will just return the start of the string + return { + position: 0, + prefix: "", + }; + } else { + // We're clipping before the start of the string, walk backwards to the first space. + const spacePos = findSpace(text, pos, -1); + return { + position: spacePos.position, + prefix: "", + }; + } +} + +function clipEnd(text, pos) { + const clipEnd = pos + 200; + if (clipEnd > text.length) { + return { + position: text.length, + suffix: "", + }; + } else { + const spacePos = findSpace(text, clipEnd, 1); + return { + position: spacePos.position, + suffix: spacePos.clipped ? "…" : "", + }; + } +} + +function findSpace(text, start, step) { + let stepPos = start; + while (stepPos > -1 && stepPos < text.length) { + const char = text[stepPos]; + if (char === " " || char === "," || char === ":") { + return { + position: step === 1 ? stepPos : stepPos - step, + clipped: stepPos > 1 && stepPos < text.length, + }; + } + stepPos = stepPos + step; + } + + return { + position: stepPos - step, + clipped: false, + }; +} + +// removes highlighting as implemented by the mark tag +function clearHighlight(searchterm, el) { + const childNodes = el.childNodes; + for (let i = childNodes.length - 1; i >= 0; i--) { + const node = childNodes[i]; + if (node.nodeType === Node.ELEMENT_NODE) { + if ( + node.tagName === "MARK" && + node.innerText.toLowerCase() === searchterm.toLowerCase() + ) { + el.replaceChild(document.createTextNode(node.innerText), node); + } else { + clearHighlight(searchterm, node); + } + } + } +} + +function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string +} + +// highlight matches +function highlight(term, el) { + const termRegex = new RegExp(term, "ig"); + const childNodes = el.childNodes; + + // walk back to front avoid mutating elements in front of us + for (let i = childNodes.length - 1; i >= 0; i--) { + const node = childNodes[i]; + + if (node.nodeType === Node.TEXT_NODE) { + // Search text nodes for text to highlight + const text = node.nodeValue; + + let startIndex = 0; + let matchIndex = text.search(termRegex); + if (matchIndex > -1) { + const markFragment = document.createDocumentFragment(); + while (matchIndex > -1) { + const prefix = text.slice(startIndex, matchIndex); + markFragment.appendChild(document.createTextNode(prefix)); + + const mark = document.createElement("mark"); + mark.appendChild( + document.createTextNode( + text.slice(matchIndex, matchIndex + term.length) + ) + ); + markFragment.appendChild(mark); + + startIndex = matchIndex + term.length; + matchIndex = text.slice(startIndex).search(new RegExp(term, "ig")); + if (matchIndex > -1) { + matchIndex = startIndex + matchIndex; + } + } + if (startIndex < text.length) { + markFragment.appendChild( + document.createTextNode(text.slice(startIndex, text.length)) + ); + } + + el.replaceChild(markFragment, node); + } + } else if (node.nodeType === Node.ELEMENT_NODE) { + // recurse through elements + highlight(term, node); + } + } +} + +/* Link Handling */ +// get the offset from this page for a given site root relative url +function offsetURL(url) { + var offset = getMeta("quarto:offset"); + return offset ? offset + url : url; +} + +// read a meta tag value +function getMeta(metaName) { + var metas = window.document.getElementsByTagName("meta"); + for (let i = 0; i < metas.length; i++) { + if (metas[i].getAttribute("name") === metaName) { + return metas[i].getAttribute("content"); + } + } + return ""; +} + +function algoliaSearch(query, limit, algoliaOptions) { + const { getAlgoliaResults } = window["@algolia/autocomplete-preset-algolia"]; + + const applicationId = algoliaOptions["application-id"]; + const searchOnlyApiKey = algoliaOptions["search-only-api-key"]; + const indexName = algoliaOptions["index-name"]; + const indexFields = algoliaOptions["index-fields"]; + const searchClient = window.algoliasearch(applicationId, searchOnlyApiKey); + const searchParams = algoliaOptions["params"]; + const searchAnalytics = !!algoliaOptions["analytics-events"]; + + return getAlgoliaResults({ + searchClient, + queries: [ + { + indexName: indexName, + query, + params: { + hitsPerPage: limit, + clickAnalytics: searchAnalytics, + ...searchParams, + }, + }, + ], + transformResponse: (response) => { + if (!indexFields) { + return response.hits.map((hit) => { + return hit.map((item) => { + return { + ...item, + text: highlightMatch(query, item.text), + }; + }); + }); + } else { + const remappedHits = response.hits.map((hit) => { + return hit.map((item) => { + const newItem = { ...item }; + ["href", "section", "title", "text", "crumbs"].forEach( + (keyName) => { + const mappedName = indexFields[keyName]; + if ( + mappedName && + item[mappedName] !== undefined && + mappedName !== keyName + ) { + newItem[keyName] = item[mappedName]; + delete newItem[mappedName]; + } + } + ); + newItem.text = highlightMatch(query, newItem.text); + return newItem; + }); + }); + return remappedHits; + } + }, + }); +} + +let subSearchTerm = undefined; +let subSearchFuse = undefined; +const kFuseMaxWait = 125; + +async function fuseSearch(query, fuse, fuseOptions) { + let index = fuse; + // Fuse.js using the Bitap algorithm for text matching which runs in + // O(nm) time (no matter the structure of the text). In our case this + // means that long search terms mixed with large index gets very slow + // + // This injects a subIndex that will be used once the terms get long enough + // Usually making this subindex is cheap since there will typically be + // a subset of results matching the existing query + if (subSearchFuse !== undefined && query.startsWith(subSearchTerm)) { + // Use the existing subSearchFuse + index = subSearchFuse; + } else if (subSearchFuse !== undefined) { + // The term changed, discard the existing fuse + subSearchFuse = undefined; + subSearchTerm = undefined; + } + + // Search using the active fuse + const then = performance.now(); + const resultsRaw = await index.search(query, fuseOptions); + const now = performance.now(); + + const results = resultsRaw.map((result) => { + const addParam = (url, name, value) => { + const anchorParts = url.split("#"); + const baseUrl = anchorParts[0]; + const sep = baseUrl.search("\\?") > 0 ? "&" : "?"; + anchorParts[0] = baseUrl + sep + name + "=" + value; + return anchorParts.join("#"); + }; + + return { + title: result.item.title, + section: result.item.section, + href: addParam(result.item.href, kQueryArg, query), + text: highlightMatch(query, result.item.text), + crumbs: result.item.crumbs, + }; + }); + + // If we don't have a subfuse and the query is long enough, go ahead + // and create a subfuse to use for subsequent queries + if (now - then > kFuseMaxWait && subSearchFuse === undefined) { + subSearchTerm = query; + subSearchFuse = new window.Fuse([], kFuseIndexOptions); + resultsRaw.forEach((rr) => { + subSearchFuse.add(rr.item); + }); + } + return results; +} diff --git a/session-survival/index.qmd b/session-survival/index.qmd new file mode 100644 index 00000000..83b89099 --- /dev/null +++ b/session-survival/index.qmd @@ -0,0 +1,19 @@ +# Preface {.unnumbered} + +**Aims** + +- to introduce survival analysis + +**Learning outcomes** + +- TBD + +Do you see a mistake or a typo? We would be grateful if you let us know via [edu.ml-biostats@nbis.se](edu.ml-biostats@nbis.se) + +*This repository contains teaching and learning materials prepared for and used during “Introduction to biostatistics and Machine Learning” and "Biostatistics and Machine Learning II" courses, organized by NBIS, National Bioinformatics Infrastructure Sweden. The courses are open for PhD students, postdoctoral researcher and other employees within Swedish universities. The materials are geared towards life scientists wanting to be able to understand and use the basic statistical and machine learning methods. More about the course [https://nbisweden.github.io/workshop-mlbiostatistics/](https://nbisweden.github.io/workshop-mlbiostatistics/) +* + + + + + diff --git a/session-survival/intro.qmd b/session-survival/intro.qmd new file mode 100644 index 00000000..f76d6587 --- /dev/null +++ b/session-survival/intro.qmd @@ -0,0 +1,424 @@ +--- +output: html_document +editor_options: + chunk_output_type: console +--- + +# Introduction + +```{r} +#| include: false +#| message: false +#| warning: false + +library(ISLR2) +library(janitor) +library(skimr) +library(tidyverse) +library(RColorBrewer) +library(survival) +library(survminer) + +mycols <- brewer.pal(6, "Set1") +``` + + +## Why survival analysis? + +- The term **survival analysis** refers to statistical methods developed to study time-to-event data, originally applied to time until death in 17th-century mortality studies, which is where the term "survival" originates. + +- Over the decades its scope has broadened to include a wide range of applications in both medical studies and beyond, in fields such as marketing, engineering or criminology and many more. + +- The common factor in survival analysis is the study of **censored data** that arise when studying a unique kind of outcome variable: **time until an event occurs**. + +- This event, can be death, as previously mentioned, but also disease onset or relapse in medical studies. In other fields, we can study time until next purchase (marketing), time until machine failure (engineering), time until a released prisoner commits another crime (criminology), and many more. + +- In fact, survival analysis is applicable whenever censored data occurs, and these cases are **not limited to studying time-based events**. We can use survival analysis methods to study the number of treatment sessions until a patient recovers or the number of attempts until a student passes an exam. + +- **Censoring** occurs when the event of interest has not yet been observed by the end of the study. This makes survival analysis especially useful for incomplete data sets, where we can still make use of the available information to estimate the time to event. + +- With survival analysis, we can compare groups (e.g., treatment vs. control) and determine which factors increase or decrease the hazard (risk) of the event occurring over time, even when adjusting for other variables. + +- We can incorporate time-dependent variables, allowing the analysis to account for changing risk factors, such as fluctuating health conditions or environmental factors, and include multiple states and/or events. For example, a patient may progress through different stages of a disease (multiple states), or a patient dying from different causes may prevent the occurrence of the primary event being studied (multiple events, competing risks). + +## Survival and censoring times + +Let's imagine a mortality study of terminally ill patients receiving a certain medical treatment. We follow a group of patients over 24 months follow-up period and record the time until death or until the end of the study. +For each patient, we suppose there is: + +- a true **survival time** $T$ and +- a **censoring time** $C$. + +The survival time represents the time at which the event of interest occurs, here death. The censoring time represents the time at which the patient is lost to follow-up (e.g. withdraws from the study) or the study ends. + +We observe the **random variable** $$Y = min (T, C)$$ + +If the event occurs before censoring (i.e. $T < C$) then we observe the true survival time $T$. Otherwise, if censoring occurs before the event (i.e. $T > C$), we observe the censoring time $C$. + +We also observe a **status indicator** + +\begin{equation} + \sigma = + \left\{ + \begin{array}{cc} + 1 & \mathrm{if\ } T \le C \\ + 0 & \mathrm{if\ } T>C \\ + \end{array} + \right. +\end{equation} + +A figure below illustrates the data for 10 patients in this ficitious study. + +```{r} +#| label: fig-censoring +#| fig-cap: Example of right-censored data in a mortatily study lasting 24 months. The horizontal lines represent the time to event or censoring for each patient. The points indicate whether the event occurred (event) or not (censored). +#| fig-cap-pos: margin + +# Example data for 10 patients +data <- data.frame( + patient = factor(1:10), # Patient IDs + time = c(23, 24, 9, 7, 14, 16, 19, 20, 24, 24), # Time to event or censoring + status = c(1, 0, 1, 0, 1, 1, 0, 1, 0, 1) # Event status (1 = event, 0 = censored) +) + +# Plot the data with horizontal lines +ggplot(data, aes(x = time, y = patient)) + + geom_segment(aes(x = 0, xend = time, y = patient, yend = patient), color = mycols[2]) + # + geom_point(aes(shape = factor(status)), size = 3) + + geom_vline(xintercept = 24, linetype = "dashed") + + scale_shape_manual(values = c(1, 16), + labels = c("Censored", "Event")) + + labs(x = "Time (months)", y = "Patient", shape = "Status") + + theme_minimal() + + theme(legend.position = "top") +``` + + +We we observe 10 $(Y, \sigma)$ pairs, which we denote as $(y_1, \sigma_1, \dots, (y_n, \sigma_n))$, where $n = 10$. From @fig-censoring, we can see that: + +- for patient #1 we get $(y_1= t_1, \space \sigma_1 = 1)$ since we observe event (death) as ca. 23 months, before the end of the follow-up at 24 months. +- for patient #2 we get $(y_2= c_1, \space \sigma_1 = 0)$ since the patient is still alive at the end of the study at 24 moths, meaning that the patient has survied at least 24 months (censored) +- for patient #3 we get $(y_3= t_3, \space \sigma_3 = 1)$ since we observe event (death) at 9 months +- for patients #4, we get $(y_4= c_4, \space \sigma_4 = 0)$, anther censored data point. The study is not over yet, the patient has not diet but is **lost to follow-up**, e.g. due to withdrawing from the study +- analogously, for patients #5, #6, #8 and #10 we observe survival times, and for patients #7 and #9 we observe censoring times. + +## Independent censoring + +In order to analyze survival data, we need to make some **assumptions** about the censoring process. One of the key assumptions in survival analysis is **independent censoring**. This assumption states that the reason for **censoring is unrelated to the likelihood of the event occurring**. In other words, censored individuals should have the same probability of experiencing the event as those who remain in the study. + +For example, in a cancer study, if a number of patients leave the study for reasons unrelated to their health status, their data can be treated as censored without biasing the results, since censoring does not provide any information about the outcome. However, if a number of patients drop out early because they are very sick, this would violate the assumption of the independent censoring. + +## Left and interval censoring + +In addition to right-censored data, we may encounter left and/or interval-censored data. **Left censoring** occurs when the event happened before the observation period started, whereas **interval censoring** occurs when the event is known to have happened between two specific time points within the study. + +Imagine an age study in which **left censoring** occurs when a patient was diagnosed with a disease before the study began, but the exact age of diagnosis is unknown. For example, if a patient is known to have been diagnosed before age 25, but the exact age is not recorded, they are left-censored. **Interval censoring** occurs when a patient’s diagnosis happens between two observed ages, but the exact age of diagnosis is unknown. For instance, if a patient was healthy at age 45 and diagnosed by age 55, but the precise age of diagnosis isn't known, their data is interval-censored. + +```{r} +#| label: fig-left-interval +#| fig-cap: "Example of left and interval censoring in an age study. Left-censored patients were diagnosed before the study, and interval-censored patients were diagnosed between visits." + +# Create a data frame for 8 patients +# Left-censored patients were diagnosed before the study, and interval-censored patients were diagnosed between visits +data <- data.frame( + patient_id = 1:8, + start_age = c(NA, 30, 40, 35, NA, 25, 45, 50), # Start of observation or known age for interval censoring (NA for left-censored) + end_age = c(25, 35, 45, 50, 20, 40, 55, 60), # Age of diagnosis or end of observation + type = c("Left Censored", "Fully Observed", "Fully Observed", "Fully Observed", + "Left Censored", "Interval Censored", "Interval Censored", "Fully Observed") +) + +# Plot the data +ggplot(data, aes(x = end_age, y = factor(patient_id), color = type)) + + # Add points for observed ages (diagnosis age or censoring) + geom_point(size = 4) + + + # For interval-censored patients, add a horizontal line to show the interval + geom_segment(data = subset(data, type == "Interval Censored"), aes(x = start_age, xend = end_age, y = factor(patient_id), yend = factor(patient_id)), + color = mycols[2], linetype = "dashed") + + + # For left-censored patients, add an arrow showing the uncertainty before diagnosis + geom_segment(data = subset(data, type == "Left Censored"), aes(x = 0, xend = end_age, y = factor(patient_id), yend = factor(patient_id)), + color = mycols[1], linetype = "dotted", arrow = arrow(length = unit(0.2, "cm"))) + + + # Customize the plot labels and appearance + labs(x = "Age", y = "Patient ID") + + scale_color_manual(values = c("Left Censored" = mycols[1], "Fully Observed" = "black", "Interval Censored" = mycols[2])) + + theme_minimal() + + theme(legend.position = "top", legend.title = element_blank()) + +``` + +In this session, we will focus on right censored data. + +## Kaplan-Meier estimator + +The **survival curve**, or survival function, is a decreasing function that quantifies the probability of surviving past time $t$ and is defined as $$S(t) = Pr(T > t)$$ {#eq-surv} + +The **Kaplan-Meier (KM) estimator** is a non-parametric statistic used to estimate the survival function from lifetime data and is given by: $$\hat{S}(t) = \prod_{j=1}^{t} \left( 1 - \frac{d_j}{n_j} \right)$$ {#eq-KM} + +where: + +- $d_j$: number of failures at time $t_j$ +- $n_j$: number of patients at risk just before time $t_ij$ +- and the product is taken over all time intervals in which a death occurred, up to and including $t$ + +Kapalan-Meier estimator is also known as **product-limit** estimator. + + +### Tooth filling example + +#### Complete follow-up data + +Suppose we have 10 patients who receive a tooth filling, and we want to track how long these fillings last before they fail (fall out or need replacement). The failure times (in months) for each patient are recorded: + +```{r} +#| label: tooth-filling + +data_tooth <- data.frame( + patientID = 1:10, + time = c(10, 8, 11, 5, 4, 3, 7, 6, 12, 8), + surv = 1 +) + +str(data_tooth) + +``` + +Note that the `surv` column is a binary indicator of survival status (1 = event, 0 = censored). In this example, we assume all patients have experienced the event (failure) and have complete follow-up data. + +To calculate the Kaplan-Meier Curve: + +1. We **sort** the data by time to failure.2. +2. For each time, we calculate the **survival probabilities**, i.e. the probability the filling lasting beyond that time. +3. We apply the Kaplan-Meier formula (@eq-KM) to calculate the probability of surviving up to time $t$, as the product of all individual survival probabilities at each time up to $t$: + +In our example: + +| Time
(months) | Patients at Risk
$n_i$ | Failures
$d_i$ | Survival Probability
at $t_i$ | Cumulative Survival
$S(t)$ | +|---------------|-----------------------------|--------------------|----------------------------------|--------------------------------| +| 3 | 10 | 1 | 1 - $\frac{1}{10}$ = 0.9 | 0.9 | +| 4 | 9 | 1 | 1 - $\frac{1}{9}$ = 0.888 | 0.9 $\times$ 0.888 = 0.8 | +| 5 | 8 | 1 | 1 - $\frac{1}{8}$ = 0.875 | 0.8 $\times$ 0.875 = 0.7 | +| 6 | 7 | 1 | 1 - $\frac{1}{7}$ = 0.857 | 0.7 $\times$ 0.857 = 0.6 | +| 7 | 6 | 1 | 1 - $\frac{1}{6}$ = 0.833 | 0.6 $\times$ 0.833 = 0.5 | +| 8 | 5 | 2 | 1 - $\frac{2}{5}$ = 0.6 | 0.5 $\times$ 0.6 = 0.3 | +| 10 | 3 | 1 | 1 - $\frac{1}{3}$ = 0.667 | 0.3 $\times$ 0.667 = 0.2 | +| 11 | 2 | 1 | 1 - $\frac{1}{2}$ = 0.5 | 0.2 $\times$ 0.5 = 0.1 | +| 12 | 1 | 1 | 1 - $\frac{1}{1}$ = 0 | 0.1 $\times$ 0 = 0 | + + +We can see that: + +- At 3 months: 10 patients are at risk, and 1 failure occurs, so the survival probability for this period is 0.9. +- For 4 months: 9 patients are at risk (since one patient failed at 3 months), and the survival probability is now cumulative (0.9 from the first interval multiplied by 0.888 from this interval). +- At time 8 months: two failures occur and the survival probability drops more sharply. +- At time 12 months: the survival probability reaches 0, indicating that all patients experienced the event (tooth filling failure) by the end of the study. + +We can visualize the Kaplan-Meier curve for this data: + +```{r} +#| label: fig-tooth-filling-1 +#| fig.cap: Kaplan-Meier survival curve for tooth filling data. The curve shows the estimated probability of survival (filling intact) over time. + +fit.surv <- survfit(Surv(time, surv) ~ 1, data = data_tooth) +plot(fit.surv, conf.int=FALSE, xlab = "Months", ylab = "Estiamted Probabiliy of Survival") + +``` + +#### Including censored data + +In real life, not all patients may have complete follow-up data. Some patients might drop out of the study (e.g., move away or lose contact). This is called **censoring**, and these patients are still considered "alive" (their fillings intact) at the time they are censored, but we do not know what happens to them afterward. The filling can also be intact at the end of the study, but we do not know how long it will last. + +Let’s update the data with censoring for two patients: + +| Patient | Time to Failure (months) | Censored | +|---------|--------------------------|-----------| +| 1 | 3 | No | +| 2 | 4 | No | +| 3 | 5 | Yes | +| 4 | 6 | No | +| 5 | 7 | No | +| 6 | 8 | No | +| 7 | 8 | No | +| 8 | 10 | Yes | +| 9 | 11 | No | +| 10 | 12 | No | + +Now, we can adjust the KM calculations to account for censored data. For censored patients, we do not count them in the number of events (failures), but they are still included in the "at-risk" population until they are censored. + +| Time
(months) | Patients at Risk
$n_i$ | Failures
$d_i$ | C | Survival Probability
at $t_i$ | Cumulative Survival
$S(t)$ | +|---------------|-----------------------------|--------------------|----------|----------------------------------|--------------------------------| +| 3 | 10 | 1 | 0 | $1 - \frac{1}{10} = 0.9$ | 0.9 | +| 4 | 9 | 1 | 0 | $1 - \frac{1}{9} = 0.888$ | 0.9 $\times$ 0.888 = 0.8 | +| 5 | 8 | 0 | 1 | | 0.8 | +| 6 | 7 | 1 | 0 | $1 - \frac{1}{7} = 0.857$ | 0.8 $\times$ 0.857 = 0.686 | +| 7 | 6 | 1 | 0 | $1 - \frac{1}{6} = 0.833$ | 0.686 $\times$ 0.833 = 0.571| +| 8 | 5 | 2 | 0 | $1 - \frac{2}{5} = 0.0.6$ | 0.571 $\times$ 0.6 = 0.343 | +| 10 | 3 | 0 | 1 | | 0.343 | +| 11 | 2 | 1 | 0 | $1 - \frac{1}{2} = 0.5$ | 0.343 $\times$ 0.5 = 0.171| +| 12 | 1 | 1 | 0 | $1 - \frac{1}{1} = 0$ | 0.171 $\times$ 0 = 0| + + +Censoring affects the calculations: + +- At 5 months, one patient is censored, so we do not record a failure, and the cumulative survival rate remains unchanged. +- At 10 months, one more patient is censored. Again, there is no effect on the failure probability, but the number of at-risk patients decreases. +- In other words, if a patient is censored, they contribute to the number of at-risk patients until their censoring time, but they do not contribute to the event (failure) counts. + +We can see how the censoring data affect the KM curve: +```{r} +#| label: fig-tooth-filling-2 +#| fig.cap: Kaplan-Meier survival curve for tooth filling data. The curve shows the estimated probability of survival (filling intact) over time. + +data_tooth[1, "surv"] <- 0 +data_tooth[4, "surv"] <- 0 +fit.surv <- survfit(Surv(time, surv) ~ 1, data = data_tooth) +plot(fit.surv, mark.time=TRUE, conf.int=FALSE, xlab = "Months", ylab = "Estiamted Probabiliy of Survival") + +``` + +### Interpretations + +The Kaplan-Meier curve shows the cumulative survival probabilities. A steeper slope indicates a higher event rate (death rate) and therefore a worse survival prognosis. A flatter slope indicates a lower event rate and therefore a better survival prognosis. The curve may have plateaus or flat areas, indicating periods of relatively stable survival. + +At specific time points, you can estimate the survival probability by locating the time point on the horizontal axis and dropping a vertical line to the curve. Then, read the corresponding survival probability from the vertical axis. + +- So, what is the probability that a tooth filling will last 8 months? We can estimate this by locating 8 months on the x-axis and dropping a vertical line to the curve. The estimated survival probability at 10 months is approximately 0.343, meaning there is a 34.3.7% chance that a tooth filling will last 10 months. + +```{r} +#| label: fig-tooth-filling-2b +#| fig.cap: Kaplan-Meier survival curve for tooth filling data. Red lines indicate the estimated survival probability (tooth being intact) at 10 months. + +plot(fit.surv, conf.int=FALSE, xlab = "Months", ylab = "Estiamted Probabiliy of Survival") +abline(v = 10, h = 0.343, col = "red", lty = 2) + +``` + +If the interest is the average length of survival, or **life expectancy**, following the start of treatment, this may be crudely estimated from the survival curve as the time corresponding to a cumulative probability of survival at 0.5. Alternatively, it can be calculated as: $$\textrm{Life expectancy} = 0.5 + \sum(\textrm{length of interval} \times \textrm{cumulative chance of survival})$$ + +If there are multiple curves representing different groups, you can compare their shapes and patterns. If the curves are parallel, it suggests that the groups have similar survival experiences. If the curves diverge or cross, it indicates differences in survival between the groups. + +```{r} +#| label: fig-tooth-filling-20-part +#| fig.cap: Kaplan-Meier survival curves for a new extended study of tooth filling data, including 20 participants (10 men and 10 women). +#| fig-height: 6 + +# Create data frame for survival data +data_tooth_ext <- data.frame( + time = c(3, 5, 6, 7, 9, 10, 12, 15, 18, 20, + 4, 6, 7, 9, 11, 13, 14, 16, 18, 22), + surv = c(1, 1, 1, 1, 1, 0, 1, 1, 0, 1, + 1, 1, 0, 1, 1, 1, 1, 0, 1, 1), + gender = c(rep("M", 10), rep("F", 10)) +) + +# Create a survival object and +surv_object <- Surv(time = data_tooth_ext$time, event = data_tooth_ext$surv) + +# Fit Kaplan-Meier curves for gender +fit <- survfit(surv_object ~ gender, data = data_tooth_ext) + +# Plot Kaplan-Meier survival curves +ggsurvplot(fit, + data = data_tooth_ext, + pval = FALSE, # Show p-value for difference in survival + conf.int = FALSE, # Add confidence interval + risk.table = TRUE, # Show risk table at the bottom + ggtheme = theme_minimal(), # Use a minimal theme + xlab = "Time (Months)", + ylab = "Survival Probability", + palette = mycols[1:2]) + +``` + +## Comparisons of hazards + +Often, in survival analysis we would like to compare the survival patterns of different groups, e.g. is there a group difference between survival times (tooth filling intact) between men and women in the tooth filling study? + +The differences between the survival curves are not constant. For example, both groups start at 1 and the survival probability deceases, with lower survival probability for men at 10 months, but then again higher survival probability at 15 months. + +We solve the problem of allowing for differences in survival time by comparing **hazards** in the two groups over the duration of the follow-up. We also assume that the **ratio of the hazards** in the two groups remains constant over time, even if the underlying hazards change. We assume, that at all times $t$: $$\frac{h_1(t)}{h_0(t)} = constant$$ + +where: + +- $h_1(t)$ and $h_0(t)$ are the hazards in the exposed and unexposed groups at time $t$ (or in one group and the other). + +This assumption is called the **proportional hazards assumption**. It is difficult to estimate the hazard directly from the data, since this would give a series of "spikes" when an even occurs, interspersed with zeros when there is no disease event. Instead we use the **cumulative hazard function**, $H(t)$. This is the total hazard experienced up to time $t$, and is estimated by the sum of the risks at each time $t$ at which an event occurs $$H(t) = \sum_{i=1}^{t}\frac{d_i}{n_i}$$ + +If the ratio of the hazards in both groups is constant over time, if follows that the ratio of the cumulative hazard functions must also equal this constant $$\frac{H_1(t)}{H_0(t)} = \frac{h_1(t)}{h_0(t)} = constant$$ +and applying the rules of logarithms: $$log(H_1(t) - log(H_0(t)) = log(constant)$$ + +Therefore, if the proportional hazards assumptions is met, then graphs of the log of the cumulative hazard functions should be parallel. + +```{r} +#| label: fig-tooth-filling-cumhaz +#| fig.cap: Cumulative hazard functions for the two groups in the extended tooth filling study. +#| fig-height: 5 + +# Plot Kaplan-Meier survival curves +ggsurvplot(fit, + data = data_tooth_ext, + fun = "cumhaz", # Add cumulative hazard function + pval = FALSE, # Show p-value for difference in survival + conf.int = FALSE, # Add confidence interval + risk.table = FALSE, # Show risk table at the bottom + ggtheme = theme_minimal(), # Use a minimal theme + xlab = "Time (Months)", + ylab = "Cumulative hazard (log)", + palette = mycols[1:2]) + +``` + +Note that it can be shown mathematically that the cumulative hazard is related to the survival function by $H(t) = -log(S(t))$. + +## Mantel-Cox (log-rank test) + +Assuming the proportional hazards assumption holds, we can test for differences in survival between groups using the **Mantel-Cox test**, also known as the **log-rank test**. It tests the null hypothesis of no difference in survival between two or more independent groups. The test compares the entire survival experience between groups and can be thought of as a test of whether the survival curves are identical (overlapping). + +The **log-rank test statistic** is constructed by comparing the **observed** number of events (e.g. deaths, failures) in each group to the **expected** number of events, assuming that the survival experiences of the groups are the same. The test statistic is based on a **chi-square distribution**, which tests whether the difference between the observed and expected numbers of events is larger than what we would expect by chance. + +Let's start with two groups, A and B. If at time $t_j$ there were $d_j$ deaths and there were $n'_{jA}$ and $n'_{jB}$ subjects alive just before $t_j$ in groups A and B respectively, then the data can be arranged in $2\times2$ table: + +| | **Died** | **Survived** | **Total** | +|------------|----------------------------|-------------------------------|----------------------------| +| **Group A** | $d_{jA}$ | $r_{jA} = n'_{_jA} - d_{jA}$ | $n'_{jA}$ | +| **Group B** | $d_{jB}$ | $r_{jB} = n_{_jB} - d_{jB}$ | $n'_{jB}$ | +| **Total** | $d_j = d_{jA} + d_{jB}$ | $r_j = r_{jA} + r_{jB}$ | $n'_j = n'_{jA} + n'_{jB}$ | + +where: + +- **Group A** and **Group B** represent the two groups being compared (e.g., two different treatments, genders, etc.). +- **Died** ($d_{jA}$, $d_{jB}$): The number of individuals who experienced the event (e.g., death, tooth filling falling out) in each group. +- **Survived** ($r_{jA}$, $r_{jB}$): The number of individuals who did not experience the event (survivors, with tooth filling intact) in each group. This is calculated as the total number in the group minus the number of deaths. +- **Total** ($n_A$, $n_B$): The total number of individuals in each group at risk just before the time of an event + +The log-rank test compares the **observed** number of events to the **expected** number of events if the survival curves were the same for all groups. The **expected number of events** in group A at each event time is denoted by: $$E(d_{jA}) = \frac{n'_{jA}d_i}{n'_j}$$ and analogously for group B $$E(d_{jB}) = \frac{n'_{jB}d_j}{n'_j}$$ + +The difference between observed and expected number of events is evidence against the null hypothesis. The Mantel-Cox chi-squared (log-rank) its simple for is the combination of these differences **over all the times at which deaths (events) occurred**: $$\chi^2_{MC} = \frac{(O_A - E_A)^2}{E_A} + \frac{(O_B - E_B)^2}{E_B}$$ + +with 1 $df$. + +The test can be easily expanded to more than two groups, by covering more groups in the summation and having $k-1$ degrees of freedom, where $k$ is the number of groups. + +In more advanced version, the log-rank test incorporates variance into the test statistics. This ensures more accurate and reliable comparisons of survival distributions, particularly when groups differ in size, have varying numbers of individuals at risk, or experience uneven censoring: + $$\chi^2_{MC} = \frac{(O_A - E_A)^2}{V_A} + \frac{(O_B - E_B)^2}{V_B}$$ where $V$ stands for total variance across all times at which event occurs, with variance at these points calculated as $$V(d_{jA}) = \frac{d_j(n'_j - d_j)n'_{jA}n'_{jB}}{n'^2_j(n'_j -1)}$$ for group A (and analogously for other groups in the study). + +In our extended tooth filling study, we can compare the survival experiences between men and women: +```{r} +#| label: tooth-filling-logrank +#| code-collapse: false + +# Perform the log-rank test +log_rank_test <- survdiff(surv_object ~ gender, data = data_tooth_ext) + +# Display the result of the log-rank test +print(log_rank_test) + +``` + +In this example, the chi-square values is `r round(log_rank_test$chisq, 2)`, with $1$ degrees of freedom. The critical value for $\chi^2$ at $\alpha = 0.05$ is approximately $3.84$ and since `r round(log_rank_test$chisq, 2)` < $3.84$ we would conclude that we cannot reject the null hypothesis of no difference in survival between the groups (indicated also by the obtained p-value `r round(log_rank_test$p,2)`). + + + + diff --git a/session-survival/lab.qmd b/session-survival/lab.qmd new file mode 100644 index 00000000..2c5174b4 --- /dev/null +++ b/session-survival/lab.qmd @@ -0,0 +1,250 @@ +--- +output: html_document +editor_options: + chunk_output_type: console +knitr: + opts_chunk: + message: false + warning: false + code-fold: false + include: true + collapse: true + eval: true +--- + +# R examples + +## KM, log-rank, Cox proportional hazards model + +```{r} +#| label: KM-logrank-Cox + +rm(list=ls()) + +# load libraries +library(tidyverse) +library(skimr) +library(survival) + +# load data +data_brain <- readRDS("data/data_cancer_brain.rds") + +# preview data +skim(data_brain) +# status = 0, indicates a censored observation +# status = 1, indicates an uncensored observation + +# Kaplan-Meier survival curve +fit.surv <- survfit(Surv(time, status) ~ 1, data = data_brain) +plot(fit.surv, xlab = "Months", ylab = "Estiamted Probabiliy of Survival") + +# KM stratify by sex +fit.surv <- survfit(Surv(time, status) ~ sex, data = data_brain) +plot(fit.surv, mark.time = TRUE, col = c(2, 4), xlab = "Months", ylab = "Estiamted Probabiliy of Survival") + +# Log-rank test to compare survival of men and women +logrank_test <- survdiff(Surv(time, status) ~ sex, data = data_brain) +print(logrank_test) + +# Fit Cox proportional hazards model +fit.cox <- coxph(Surv(time, status) ~ sex, data = data_brain) +summary(fit.cox) + +# Test the proportional hazards assumption in the two gender groups +ph_test <- cox.zph(fit.cox) +print(ph_test) + +# Fit Cox model with multiple predictors +fit.cox_multi <- coxph(Surv(time, status) ~ sex + diagnosis + loc + ki + gtv + stereo, data = data_brain) +summary(fit.cox_multi) + +# Plot survival curves for each diagnosis category, adjusting for other predictors +# we set the values of the other predictors to the mean for quantitative predictors, +# and to the modal value for factors + +modeldata <- data.frame( + diagnosis = levels(data_brain$diagnosis), + sex = rep("Female", 4), + loc = rep("Supratentorial", 4), + ki = rep(mean(data_brain$ki), 4), + gtv = rep(mean(data_brain$gtv), 4), + stereo = rep("SRT", 4) +) + +survplots <- survfit(fit.cox_multi, newdata = modeldata) +plot(survplots, mark.time = FALSE, xlab = "Months", ylab = "Estiamted Probabiliy of Survival", col = 2:5) +legend("bottomleft", legend = levels(data_brain$diagnosis), col = 2:5, lty = 1) + +``` + + +## Time-varying predictors + +adpoted from [https://www.bookdown.org/rwnahhas/RMPH/](adapoted from https://www.bookdown.org/rwnahhas/RMPH/) + +A dataset with time-varying predictors will have multiple rows per individual, with different rows having different values for the time-varying predictors, reflecting how they change over time. Additionally, rather than having a single event time variable, each row will have two time variables indicating the beginning and end of the time interval represented by that row of data. + +```{r} +#| label: laod-data-opioid +data_opioid <- readRDS("data/data-opioid.rds") + +data_opioid %>% + filter(RANDID == 10) %>% + select(RANDID, wave, START, STOP, heroin, age_at_init, sex, dep_lifetime) +``` + +Here, we see a preview of the `data_opioid` dataset containing longitudinal information for 362 individuals, who at baseline had used non-prescribed pharmaceutical opioids (NPPO, “pain pills”), but were not dependent on NPPOs and had never used heroin. + +- Each row contains the time variables START and STOP which define the time interval (years from initiation of NPPO use) associated with that row. +- Time-invariant variables in the dataset are constant over all rows for the same individual, while time-varying variables can change between rows. +- Each (START, STOP] interval defines a period of time during which no variables changed. +- Two time-varying variables in the dataset are heroin use (heroin) (the event indicator variable) and lifetime opioid dependence (dep_lifetime) +- For instance, for this particular individual, male, started using NPPOs at age 19 (time invariant variables). He did not meet the criteria for lifetime opioid dependence at baseline (wave = 0), but did at the next interview (wave = 1) and he first reported using heroin at his 6th interview. + +To model the heroin usage using both time-invariant and time-varying predictors, we can use the `coxph` function: +```{r} +#| label: cox-timevar + +cox.timevar <- coxph(Surv(START, STOP, heroin) ~ + age_at_init + sex + dep_lifetime, + data = data_opioid) + +summary(cox.timevar) +``` + +Whether a predictor is time-varying or time-invariant, its HR can be interpreted as a comparison of the hazard between groups of individuals with different values of that predictor. Thus, in this example, we could conclude that after adjusting for age at NPPO initiation and sex, those with lifetime opioid dependence have 2.88 times the hazard of using heroin as those who do not (AHR = 2.88; 95% CI = 1.32, 6.30; p = 0.008). The HR for a time-varying predictor, however, can also be interpreted as the effect of within-individual change on the hazard. If an individual without opioid dependence transitions to dependence, their hazard of transitioning to heroin is multiplied by 2.88. + +## Competing risks + +### Comparing CIFs + +Let's demonstrate how to analyze competing risks using the **`cmprsk`** package in R. This package includes functions for estimating the cumulative incidence function (CIF) in the presence of competing risks. + +Bone marrow transplant (BMT) data from the `cmprsk` package includes information for 35 leukemia cancer patients who underwent bone marrow transplantation. The data includes the following variables: + +- dis: disease; 0 = ALL; 1 = AML (ALL, Acute lymphoblastic leukemia; AML, Acute myeloid leukemia) +- ftime: follow-up time +- status: 0 = censored (survival); 1 = Transplant-related mortality; 2 = relapse + +```{r} +#| label: cmpr-bmt + +library(cmprsk) +library(survminer) + +# preview data +data_bmt <- readRDS("data/data-bmt.rds") +str(data_bmt) + +# label levels +data_bmt <- data_bmt %>% + mutate(dis = factor(dis, levels = c(0,1), labels = c("ALL", "AML")), + status = factor(status, levels = c(0,1,2), labels = c("Censored","Mortality","Relapse"))) + +str(data_bmt) + +# Estimate the cumulative incidence functions for each event type +cif <- cuminc(ftime = data_bmt$ftime, # Failure time variable + fstatus = data_bmt$status) # Codes for different causes of failure + +# Print the CIF object +print(cif) + +# Plot the CIFs for all even types +plot(cif, lty = 1, col = c("green", "red", "blue"), xlab = "Time", ylab = "Cumulative Incidence", + main = "Cumulative Incidence Functions for Competing Risks", curvlab = c("Censored", "Mortality", "Relapse")) + +# Estimate the cumulative incidence functions fo each even type per group +cif <- cuminc(ftime = data_bmt$ftime, + fstatus = data_bmt$status, + group = data_bmt$dis) # Estimates will calculated within groups + +# Plot the CIFs for both event types using survminer package +ggcompetingrisks(cif, multiple_panels = FALSE, legend = "right") + +# Show Gray's test for equality of CIFs +print(cif$Tests) + +``` + +Gray's test for equality of CIFs across groups (ALL vs. AML) is split across different types of events, here (Censored, Mortality, and Relapse). + +- For **Censored**, the p-value is small and suggest that there isa **statistically significant difference** in the CIF for censored events between the groups. +- For **Mortality**, the p-value is greater than 0.05, suggesting that there is **no statistically significant difference** in the CIF for mortality between the groups. +- For **Relapse**, the p-value is less than 0.05, indicating a **statistically significant difference** in the CIF for relapse between the groups. +- The above suggest that the groups are behaving differently in terms of the time to relapse and censored events, but not for mortality. + +### Competing Risks Regression + +adopted from [https://www.nature.com/articles/bmt2009359](https://www.nature.com/articles/bmt2009359) + +Suppose that the BMT study was extended to include more participants and additional covariates and now includes: + +- 177 observations +- Sex: gender of the individual +- D: disease; 0 = ALL; 1 = AML +- Phase: phase at transplant (Relapse, CR1, CR2, CR3) +- Age: age at the beginning of follow-up +- Status: 0 = censored; 1 = Transplant-related mortality; 2 = relapse +- Source: source of stem cells (BM+PB, PB) +- ftime: failure time + +We are interested in modeling time to relapse in the presence of transplant-related death (competing event). We want to sutdy the effect on relapse of sex, disease type, phase at transplant, source of stem cells and age. + +```{r} +#| label: cmpr-bmtcrr + +# load additional libraries +library(fastDummies) + +# load data +data_bmtcrr <- readRDS("data/data-bmtcrr.rds") + +# preview data +str(data_bmtcrr) + +# prepare matrix of covariates +# factor variables need to be coded as dummy variables +x_sex <- dummy_cols(data_bmtcrr$Sex) %>% dplyr::select(.data_F) +x_phase <- dummy_cols(data_bmtcrr$Phase) %>% dplyr::select(.data_CR1, .data_CR2, .data_CR3) +x_d <- dummy_cols(data_bmtcrr$D) %>% dplyr::select(.data_AML) +x_source <- dummy_cols(data_bmtcrr$Source) %>% dplyr::select(.data_PB) +x <- data.frame(age = data_bmtcrr$Age, sex = x_sex, x_d, x_phase, x_source) +colnames(x) <- c("age", "sex_F", "D_AML", "P:CR1", "P:CR2", "P:CR3", "source:PB") + +# The first regression model for relapse can be produced by typing +mod1 <- crr(ftime = data_bmtcrr$ftime, + fstatus = data_bmtcrr$Status, + cov1 = x) + +summary(mod1) + +``` + + +The first part of the output shows for each term in the design matrix the estimated coefficient $\hat{\beta}_j$, the relative risk $\mathrm{exp}(\hat{\beta}_j)$, the standard error, the z-value and the corresponding P-value for assessing significance. + +Here, `Sex` is not significant, followed by `Age` and `D` (disease type), whereas `Source` is only marginally significant. `Phase` is a factor with relapse as baseline, so each P-value provides a test for the difference of each level with respect to the baseline. + +An overall P-value for Phase can be obtained through the Wald test via `aod` R package: +```{r} +#| label: "cmpr-wald-test" + +library(aod) +wald.test(mod1$var, mod1$coef, Terms = 4:6) + +``` + +The first argument to the function wald.test() is the estimated covariance matrix for the coefficients, followed by the vector of coefficients estimates, and the position of coefficients for which we want to assess significance. In our case, the P-value indicates that Phase is statistically significant. + +The second part of the output for competing risks regression shows the relative risk for each term, and a 95% confidence interval. The relative risk or subdistribution hazard ratio for a categorical covariate is the ratio of subdistribution hazards for the actual group with respect to the baseline, with all other covariates being equal. If the covariate is continuous then the relative risk refers to the effect of a one unit increase in the covariate, with all other covariates being equal. In our data, exp(−0.0352)=0.965 is the relative risk of a woman with respect to a man, and exp(−0.0185)=0.982 is the relative risk for a 1 year increase in age. + + +The last part of the output shows the pseudo log-likelihood at maximum and the pseudo likelihood ratio test, that is, the difference in the objective function at the global null and at the final estimates. As this objective function is not a true likelihood, this test statistic is not asymptotically distributed as a χ2. As a consequence, model comparison based on likelihood ratio approach cannot be performed directly, but significance must be evaluated through simulations. However, a model selection criterion can be easily adopted as described in the following section. + +For an example on model selection, model diagnostics or adding time-varying covariates in presence of competing events, please refer to the original tutorial article @Scrucca2010 [https://www.nature.com/articles/bmt2009359](https://www.nature.com/articles/bmt2009359). + + + + + diff --git a/session-survival/notes.qmd b/session-survival/notes.qmd new file mode 100644 index 00000000..c0a29ac7 --- /dev/null +++ b/session-survival/notes.qmd @@ -0,0 +1,49 @@ +--- +output: html_document +editor_options: + chunk_output_type: console +--- + +# Notes + +## ToDos +- update preface with learning outcomes (once session is completed) +- make presentation slides + +## Presentation + +Part I (ca. 20 min) + +- Introduction: why bother with survival analysis (1 slide) +- Censoring data: 2 - 3 slides, include more examples and check understanding +- Assumption: independent censoring +- Kaplan-Meier: 2 - 5 slides +- Hazards: 2 - 3 slides +- Log-rank test: 2 + +Part II (ca. 20 min) + +- Cox model: 5 slides + - construction + - interpretations of coefficients + - predictions + - global p-values? + - run few examples and check for understanding + +Part III (ca. 10 min) + +- Mention about things beyond + - time-dependent covariates + - competing risks + - machine learning models + +## Quiz question + +- Interpretation of KM plot +- Interpretation of Cox model +- Predicting survival probability for a given individual + + + + + diff --git a/session-survival/references.bib b/session-survival/references.bib new file mode 100644 index 00000000..0e5699f7 --- /dev/null +++ b/session-survival/references.bib @@ -0,0 +1,137 @@ +@article{Wilcoxon1945, + author = {Frank Wilcoxon}, + issue = {6}, + journal = {Biometrics Bulletin}, + pages = {80-83}, + title = {Individual Comparisions by Ranking Methods}, + volume = {1}, + year = {1945}, +} +@article{Ghaderyan2020, + abstract = {Gait rhythm fluctuations are of great importance for automatic neurodegenerative diseases (NDDs) detection. They provide a cost-effective and noninvasive monitoring tool in which their parameters are related to neuromuscular function. This study investigated a new solution based on a set of new symmetric features and sparse non-negative least squares (NNLS) coding classifier. Dynamic gait series warping (DGSW), Euclidean, Manhattan, Minkowski, Chebyshev, Canberra distances, and cosine function were used to quantify the amount of divergence between the left and right stride, swing, and stance intervals. The algorithm was evaluated using the gait signals of 20 healthy control subjects, 20 patients with amyotrophic lateral sclerosis (ALS), 15 patients with Parkinson's disease (PD) and 20 patients with Huntington's disease (HD). The proposed new approach using symmetric features and NNLS technique achieved outstanding accuracies of 98%, 97%, and 95% on the patients with PD, ALS, and HD, respectively. The findings also suggested that the new DGSW, cosine function, and Chebyshev distance, which are designed to dynamically, geometrically, or nonlinearly quantify the similarity between two time series, provide the discriminatory measures to describe how NDDs alter the gait symmetry. In comparison with other studies, combining symmetric features with a sparse NNLS coding classifier can improve the detection accuracy providing an efficient and cost-effective framework for the development of a NDDs detection system.}, + author = {Peyvand Ghaderyan and Seyede Marziyeh Ghoreshi Beyrami}, + doi = {10.1016/j.compbiomed.2020.103736}, + issn = {18790534}, + journal = {Computers in Biology and Medicine}, + title = {Neurodegenerative diseases detection using distance metrics and sparse coding: A new perspective on gait symmetric features}, + volume = {120}, + year = {2020}, +} +@article{Blanco2023, + abstract = {Many machine learning and data mining tasks are based on distance measures, so a large amount of literature addresses this aspect somehow. Due to the broad scope of the topic, this paper aims to provide an overview of the use of these measures in the most common machine learning problems, pointing out those aspects to consider to choose the most appropriate measure for a particular task. For this purpose, the most recent works addressing the subject were reviewed and seven of the most commonly used measures were analyzed, investigating in detail their main properties and applications. Different experiments were carried out to study their relationships and compare their performance. The degradation of the results in the presence of noise was also considered, as well as the execution time required by each measure.}, + author = {Eva Blanco-Mallo and Laura Morán-Fernández and Beatriz Remeseiro and Verónica Bolón-Canedo}, + doi = {10.1016/j.patcog.2023.109646}, + issn = {00313203}, + journal = {Pattern Recognition}, + title = {Do all roads lead to Rome? Studying distance measures in the context of machine learning}, + volume = {141}, + year = {2023}, +} +@article{Preud2021, + abstract = {The choice of the most appropriate unsupervised machine-learning method for “heterogeneous” or “mixed” data, i.e. with both continuous and categorical variables, can be challenging. Our aim was to examine the performance of various clustering strategies for mixed data using both simulated and real-life data. We conducted a benchmark analysis of “ready-to-use” tools in R comparing 4 model-based (Kamila algorithm, Latent Class Analysis, Latent Class Model [LCM] and Clustering by Mixture Modeling) and 5 distance/dissimilarity-based (Gower distance or Unsupervised Extra Trees dissimilarity followed by hierarchical clustering or Partitioning Around Medoids, K-prototypes) clustering methods. Clustering performances were assessed by Adjusted Rand Index (ARI) on 1000 generated virtual populations consisting of mixed variables using 7 scenarios with varying population sizes, number of clusters, number of continuous and categorical variables, proportions of relevant (non-noisy) variables and degree of variable relevance (low, mild, high). Clustering methods were then applied on the EPHESUS randomized clinical trial data (a heart failure trial evaluating the effect of eplerenone) allowing to illustrate the differences between different clustering techniques. The simulations revealed the dominance of K-prototypes, Kamila and LCM models over all other methods. Overall, methods using dissimilarity matrices in classical algorithms such as Partitioning Around Medoids and Hierarchical Clustering had a lower ARI compared to model-based methods in all scenarios. When applying clustering methods to a real-life clinical dataset, LCM showed promising results with regard to differences in (1) clinical profiles across clusters, (2) prognostic performance (highest C-index) and (3) identification of patient subgroups with substantial treatment benefit. The present findings suggest key differences in clustering performance between the tested algorithms (limited to tools readily available in R). In most of the tested scenarios, model-based methods (in particular the Kamila and LCM packages) and K-prototypes typically performed best in the setting of heterogeneous data.}, + author = {Gregoire Preud’homme and Kevin Duarte and Kevin Dalleau and Claire Lacomblez and Emmanuel Bresso and Malika Smaïl-Tabbone and Miguel Couceiro and Marie Dominique Devignes and Masatake Kobayashi and Olivier Huttin and João Pedro Ferreira and Faiez Zannad and Patrick Rossignol and Nicolas Girerd}, + doi = {10.1038/s41598-021-83340-8}, + issn = {20452322}, + issue = {1}, + journal = {Scientific Reports}, + title = {Head-to-head comparison of clustering methods for heterogeneous data: a simulation-driven benchmark}, + volume = {11}, + year = {2021}, +} +@article{Forgy1965, + abstract = {In recent years a number of procedures to cluster, group, or classify a sample of data points have been advanced, but such methods are still not generally accepted as useful and dependable tools for data analysis. Much of the reluctance to rely on these methods may be due to uncertainty as to just what the end result of a cluster analysis means. This of course depends on the particular method used. In the literature two rather different purposes of clustering methods may be discerned. (1) To establish a maximally efficient way of partitioning the sample into given numbers of classes. The classical measure of efficiency is the pooled variance within classes, which is to be minimized. (2) To discover and describe the 'natural' way of classifying the sample. Here the data determine the number of classes, so there is the possibility of there being no partitioning at all. Examples of methods advanced for each purpose are given, and experimental results of several of these upon samples from known multivariate populations with and without 'natural' partitionings are presented. The most striking finding is that some procedures having the second purpose will almost always suggest 'natural' (i.e. interpretable) classifications when in fact there are none (e.g. in data from a joint normal population). A user would thus be exposed to virtual certainty of Type I error. Improved clustering procedures are suggested for both purposes.}, + author = {E. Forgy}, + issue = {3}, + journal = {Biometrics}, + title = {Cluster analysis of multivariate data: Efficiency vs. interpretability of classifications}, + volume = {21}, + year = {1965}, +} +@article{Hartigan1979, + abstract = {The K-means clustering algorithm is described indetail by Hartigan(1975). An efficient version of the algorithm is presented here.\nThe aim of the K-means algorithm is to divide M points in N dimensions into K clusters so that the within-cluster sum of squares is minimized. It is not practical to require that the solution has minimal sum of squares against all partitions except when M,N are small and K = 2. We seek instead "local" optima, solution such that no movement of a point from one cluster to another will reduce the within cluster sum of squares.}, + author = {J. A. Hartigan and M. A. Wong}, + doi = {10.2307/2346830}, + issn = {00359254}, + issue = {1}, + journal = {Applied Statistics}, + title = {Algorithm AS 136: A K-Means Clustering Algorithm}, + volume = {28}, + year = {1979}, +} +@article{Lloyd1982, + abstract = {It has long been realized that in pulse-code modulation (PCM), with a given ensemble of signals to handle, the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall. It has been shown by Panter and Dite that, in the limit as the number of quanta becomes infinite, the asymptotic fractional density of quanta per unit voltage should vary as the one-third power of the probability density per unit voltage of signal amplitudes. In this paper the corresponding result for any finite number of quanta is derived; that is, necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy. The optimization criterion used is that the average quantization noise power be a minimum. It is shown that the result obtained here goes over into the Panter and Dite result as the number of quanta become large. The optimum quantization schemes for 2b quanta, b = 1,2, ···, 7, are given numerically for Gaussian and for Laplacian distribution of signal amplitudes. ©1982 IEEE}, + author = {Stuart P. Lloyd}, + doi = {10.1109/TIT.1982.1056489}, + issn = {15579654}, + issue = {2}, + journal = {IEEE Transactions on Information Theory}, + title = {Least Squares Quantization in PCM}, + volume = {28}, + year = {1982}, +} +@misc{Gao2023, + abstract = {Cluster analyzes have been widely used in mental health research to decompose inter-individual heterogeneity by identifying more homogeneous subgroups of individuals. However, despite advances in new algorithms and increasing popularity, there is little guidance on model choice, analytical framework and reporting requirements. In this paper, we aimed to address this gap by introducing the philosophy, design, advantages/disadvantages and implementation of major algorithms that are particularly relevant in mental health research. Extensions of basic models, such as kernel methods, deep learning, semi-supervised clustering, and clustering ensembles are subsequently introduced. How to choose algorithms to address common issues as well as methods for pre-clustering data processing, clustering evaluation and validation are then discussed. Importantly, we also provide general guidance on clustering workflow and reporting requirements. To facilitate the implementation of different algorithms, we provide information on R functions and libraries.}, + author = {Caroline X. Gao and Dominic Dwyer and Ye Zhu and Catherine L. Smith and Lan Du and Kate M. Filia and Johanna Bayer and Jana M. Menssink and Teresa Wang and Christoph Bergmeir and Stephen Wood and Sue M. Cotton}, + doi = {10.1016/j.psychres.2023.115265}, + issn = {18727123}, + journal = {Psychiatry Research}, + title = {An overview of clustering methods with guidelines for application in mental health research}, + volume = {327}, + year = {2023}, +} +@article{Suzuki2006, + abstract = {Summary: Pvclust is an add-on package for a statistical software R to assess the uncertainty in hierarchical cluster analysis. Pvclust can be used easily for general statistical problems, such as DNA microarray analysis, to perform the bootstrap analysis of clustering, which has been popular in phylogenetic analysis. Pvclust calculates probability values (p-values) for each cluster using bootstrap resampling techniques. Two types of p-values are available: approximately unbiased (AU) p-value and bootstrap probability (BP) value. Multiscale bootstrap resampling is used for the calculation of AU p-value, which has superiority in bias over BP value calculated by the ordinary bootstrap resampling. In addition the computation time can be enormously decreased with parallel computing option. © 2006 Oxford University Press.}, + author = {Ryota Suzuki and Hidetoshi Shimodaira}, + doi = {10.1093/bioinformatics/btl117}, + issn = {13674811}, + issue = {12}, + journal = {Bioinformatics}, + title = {Pvclust: An R package for assessing the uncertainty in hierarchical clustering}, + volume = {22}, + year = {2006}, +} +@article{Traag2019, + abstract = {Community detection is often used to understand the structure of large and complex networks. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. To address this problem, we introduce the Leiden algorithm. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. In addition, we prove that, when the Leiden algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are locally optimally assigned. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees.}, + author = {V. A. Traag and L. Waltman and N. J. van Eck}, + doi = {10.1038/s41598-019-41695-z}, + issn = {20452322}, + issue = {1}, + journal = {Scientific Reports}, + title = {From Louvain to Leiden: guaranteeing well-connected communities}, + volume = {9}, + year = {2019}, +} +@article{Gray1998, + abstract = {In this paper, for right censored competing risks data, a class of tests developed for comparing the cumulative incidence of a particular type of failure among different groups. The tests are based on comparing weighted averages of the hazards of the subdistribution for the failure type of interest. Asymptotic results are derived by expressing the statistics in terms of counting processes and using martingale central limit theory. It is proposed that weight functions very similar to those for the GP tests from ordinary survival analysis be used. Simulation results indicate that the asymptotic distributions provide adequate approximations in moderate sized samples.}, + author = {Robert J. Gray}, + doi = {10.1214/aos/1176350951}, + issue = {3}, + journal = {The Annals of Statistics}, + title = {A Class of K-Sample Tests for Comparing the Cumulative Incidence of a Competing Risk}, + volume = {16}, + year = {1998}, +} +@article{Fine1999, + abstract = {With explanatory covariates, the standard analysis for competing risks data involves modeling the cause-specific hazard functions via a proportional hazards assumption. Unfortunately, the cause-specific hazard function does not have a direct interpretation in terms of survival probabilities for the particular failure type. In recent years many clinicians have begun using the cumulative incidence function, the marginal failure probabilities for a particular cause, which is intuitively appealing and more easily explained to the nonstatistician. The cumulative incidence is especially relevant in cost-effectiveness analyses in which the survival probabilities are needed to determine treatment utility. Previously, authors have considered methods for combining estimates of the cause-specific hazard functions under the proportional hazards formulation. However, these methods do not allow the analyst to directly assess the effect of a covariate on the marginal probability function. In this article we propose a novel semiparametric proportional hazards model for the subdistribution. Using the partial likelihood principle and weighting techniques, we derive estimation and inference procedures for the finite-dimensional regression parameter under a variety of censoring scenarios. We give a uniformly consistent estimator for the predicted cumulative incidence for an individual with certain covariates; confidence intervals and bands can be obtained analytically or with an easy-to-implement simulation technique. To contrast the two approaches, we analyze a dataset from a breast cancer clinical trial under both models. © 1999 Taylor & Francis Group, LLC.}, + author = {Jason P. Fine and Robert J. Gray}, + doi = {10.1080/01621459.1999.10474144}, + issn = {1537274X}, + issue = {446}, + journal = {Journal of the American Statistical Association}, + title = {A Proportional Hazards Model for the Subdistribution of a Competing Risk}, + volume = {94}, + year = {1999}, +} +@article{Scrucca2010, + abstract = {We describe how to conduct a regression analysis for competing risks data. The use of an add-on package for the R statistical software is described, which allows for the estimation of the semiparametric proportional hazards model for the subdistribution of a competing risk analysis as proposed by Fine and Gray. J Am Stat Assoc 1999; 94: 496–509.}, + author = {L Scrucca and A Santucci and F Aversa}, + doi = {10.1038/bmt.2009.359}, + issn = {1476-5365}, + issue = {9}, + journal = {Bone Marrow Transplantation}, + pages = {1388-1395}, + title = {Regression modeling of competing risk using R: an in depth guide for clinicians}, + volume = {45}, + url = {https://doi.org/10.1038/bmt.2009.359}, + year = {2010}, +} diff --git a/session-survival/references.qmd b/session-survival/references.qmd new file mode 100644 index 00000000..925f7c49 --- /dev/null +++ b/session-survival/references.qmd @@ -0,0 +1,4 @@ +# References {.unnumbered} + +::: {#refs} +::: diff --git a/session-survival/regression.qmd b/session-survival/regression.qmd new file mode 100644 index 00000000..08beeb33 --- /dev/null +++ b/session-survival/regression.qmd @@ -0,0 +1,270 @@ +--- +output: html_document +editor_options: + chunk_output_type: console +--- + +# Regression with survival response + +```{r} +#| include: false +#| message: false +#| warning: false + +library(ISLR2) +library(janitor) +library(skimr) +library(tidyverse) +library(RColorBrewer) +library(survival) +library(survminer) + +mycols <- brewer.pal(6, "Set1") +``` + + +## Cox proportional hazards model + +If survival time is of interest, such as in mortality studies, a common approach is to postulate a distribution for survival time and estimate the parameters of this distribution from the data. For instance, the exponential distribution can be used if the death rate is independent of time, the Weibull distribution allows for increasing or decreasing hazard rates, and the Gompertz distribution models hazard rates that exponentially increase or decrease over time. + +More commonly, it is the relationship between survival time and one or more predictor variables (covariates) that is of interest. The Cox proportional hazards model is a widely used semi-parametric model to study this relationship. It is used to estimate the hazard ratio for individuals based on their covariates, without needing to specify the baseline hazard function. + +Estimation of $\boldsymbol{\beta}$ and inferences are developed by considering the information supplied at each time that a death (event) occurred. + +Consider: + +- a death occurring at time $t_j$, and +- suppose that there were $n_j$ subjects alive just before $t_j$, +- that the values of $\boldsymbol{x}$ for these subjects are $\boldsymbol{x_1}, \boldsymbol{x_2}, \cdots, \boldsymbol{x_{n'j}}$ +- and that the subject that dies is denoted, by the subscript 1 + +Then: + +- The set of $n'_j$ subjects at risk is referred to as the *risk set*. +- The risk of death at time $t_j$ for each subject in the risk set is given by: $$h(t) = h_0(t)\textrm{exp}(\boldsymbol{\beta}^T\boldsymbol{x})$$ + +where: + +- $\boldsymbol{\beta}^T\boldsymbol{x}$ is the matrix representation of the regression function, $\beta_1x_1 + \beta_2x_2 + \cdots + \beta_px_p$ +- and $h_0(t)$ is **baseline hazard function** + +The risk of death at time $t_j$ in the risk set does not supply absolute measures of risk, but does supply the relative risks for each subject, since, although $h_0(t)$ is unknown, it is the same for each subject. Thus the probability that the death observed at $t_j$ was of the subject who did die at that time is: $$p_j = \frac{\textrm{exp}(\boldsymbol{\beta}^T\boldsymbol{x_1})}{\sum \textrm{exp}(\beta^T\boldsymbol{x_i})}$$ +where summation is over all remembers of the risk set. + +Similar terms are derived for each time that a death occurred and are combined to form a likelihood (product of these probabilities over all event times $t_j$): +$$PL(\boldsymbol{\beta}) = \prod_{j=1}^{k} \frac{\exp(\boldsymbol{\beta}^T \boldsymbol{x_1})}{\sum_{i \in \text{risk set at } t_j} \exp(\boldsymbol{\beta}^T \boldsymbol{x_i})}$$ +Technically this is called *partial likelihood*, since the component terms are derived conditionally on the times that deaths occurred and the composition of the risk set at these times. This partial likelihood is used to estimate the regression coefficients $\boldsymbol{\beta}$ in the Cox model. Maximizing this partial likelihood gives the best estimates for $\boldsymbol{\beta}$, quantifying the effect of the covariates on the hazard. + +To estimate $\beta$, we maximize the partial likelihood function with respect to $\beta$. No closed-form solution is available, and so iterative algorithms are used. + +## Example with brain cancer data + +The `BrainCancer` data set from the `ISLR2` package contains survival times for patients with primary brain tumors undergoing treatment with radiation therapy. Variables included in the study are: + +- `sex`: male or female +- `diagnosis`: meningioma, LG glioma, HG glioma, or other +- `loc`: tumor location, infratentorial or supratentorial +- `gtv`: gross tumor volume $cm^3$ +- `ki`: Karnofsky index +- `stereo`: stereotatcic radiosurgery (SRS) or fractionated stereotactic (SRT) radiotherapy + +```{r} + +# preview data +data_brain <- BrainCancer + +data_brain %>% + str() + +# Cox model with multiple predictors +fit.cox_multi <- coxph(Surv(time, status) ~ sex + diagnosis + ki, data = data_brain) +summary(fit.cox_multi) + +``` + +### Coefficient interpration + +Above, we fitted Cox model using `sex` and `diagnosis` as predictors. The results, indicate, for instance that: + +- The estimated hazard ratio (HR) for a male patient is $e^{0.11} = 1.11$. This means that men have 1.11 times the hazard of dying than women while keeping other variables constant (sometimes referred to as adjusted hazard ration, AHR). +- However the associated Z value is small, resulting in large p-value of 0.75, which indicates that this difference is not statistically significant. +- We can also see that one-unit increase in the Karnofsky index is corresponds to a multiplier of $e^{-0.06} = 0.94$ in the chance of dying. The higher the Karnofsky index score, the lower the chance of dying at any given point in time, with this effect being highly significant $p$-value of 0.000158. +- Note that if you want to find HR associated with more than 1-unit dffererence, e.g. for patients with 10 units higher Karnofsky index, we would calculate HR as $e^{-0.06 \times 10} = 0.54$. This means that patients with 10 units higher Karnofsky index have 0.54 times the hazard of dying compared to those with lower Karnofsky index. + +Sometimes, in addition to reporting the numeric results of a Cox regression, **forest plots** are used to visualize HRs and their 95% confidence intervals. + +```{r} +#| label: forest-plot + +# make forest plot with ggforest() +ggforest(fit.cox_multi, data = data_brain) +``` + +### Global statistical signficance + +The output of the Cox regression contains also three methods for an overall test of whether the predictors in the model have a significant effect on the hazard (or risk) of the event occurring. Essentially, these check whether the model as a whole provides a better fit than a null model (a model without any covariates). + +**Likelihood Ratio Test (LRT)** compares the log-likelihoods of two models: + +- The **null model**, which includes only the baseline hazard (i.e., no covariates). +- The **full model**, which includes the covariates of interest. + +The test statistic is computed as: +$$ +2 \times (\text{log-likelihood of the full model} - \text{log-likelihood of the null model}) +$$ +This statistic follows a chi-squared distribution, and the associated p-value indicates whether the full model (with covariates) is significantly better at explaining the data than the null model. + +**Wald Test** + +The Wald test checks whether the estimated regression coefficients are significantly different from zero. For each predictor, the test examines whether its coefficient, $\beta_i$, is significantly different from 0, implying that the predictor has a significant effect on the hazard. The Wald test can also be used as a global test of significance. + +**Score Test (Log-Rank Test)** + +The Score test (also called the log-rank test in this context) evaluates the contribution of each predictor to the model's fit, considering the expected number of events. Like the likelihood ratio test, the Score test also compares the null and full models and uses a chi-squared distribution to compute significance. + +### Concordance + +In the context of a **Cox proportional hazards model**, **concordance** (also known as the **C-index** or **concordance index**) is a measure of how well the model predicts the order of events. Specifically, it assesses the ability of the Cox model to correctly predict which of two individuals will experience an event first, based on their risk scores. + +The **concordance index** (C-index) ranges between 0.5 and 1: + +- A **C-index of 0.5** indicates that the model's predictions are no better than random chance. This means the model cannot distinguish between individuals who experience the event earlier and those who experience it later. +- A **C-index of 1** means perfect prediction, indicating that the model always correctly predicts the order of events. + +The C-index is similar to the **area under the ROC curve (AUC)** used in binary classification models, but it is adapted to survival analysis where the goal is to rank individuals by their risk of experiencing the event. + +In our example, a concordance of 0.782 means that the Cox model can correctly predict the ordering of survival times 78.2% of the time. In other words, if you randomly select two individuals and compare their predicted risk scores, the model will correctly predict which individual experiences the event earlier in 78.2% of cases. The standard error is 0.038 which provides a measure of uncertainty around the concordance estimate. A smaller standard error indicates more confidence in the estimate. + +### Predictions + +In Cox regression, we can estimate: + +- the survival probability at a specific time, $S(t|X=x)$ +- and the hazard ratio for an individual relative to a reference individual, $h(t|X=x) / $S(t|X=x_{ref})$, where $X_{ref}$ is a reference individual with known covariate values. + +For predicted survival, the choice of time matters. A predicted HR will not depend on time due to proportional hazards assumption. + +#### Survival +Let's predict the probability of survival at 40 days, for a new patient (man, with LG glioma, Karnofsky index of 80). + +```{r} +#| label: predict-survival +#| code-fold: false + +# new patient data +new_pat <- data.frame(time = 40, + sex = "Male", + diagnosis = "LG glioma", + ki = 80, + status = 0) + +predict(fit.cox_multi, newdata = new_pat, type = "survival", se.fit = T) +``` + +The probability of surviving through 40 days is 0.48. To get confidence interval we could use: + +```{r} +#| code-fold: false + + +unlist(summary(survfit(fit.cox_multi, + newdata = new_pat, + se.fit =T, conf.int = 0.95), + times=40)[c("surv", "std.err", "lower", "upper")]) +``` + +So the probability of surviving through 40 days is 0.48 with 95% CI [0.23, 1]. + +#### HR + +To predict the adjusted hazard ratio for a new patient, we use again the `predict()` function, this type with `type = "risk"`. We also need to specify the reference individual. Some options include: + +- `reference = "zero"`: the reference individual has continuous predictor values set to 0 and categorical predictors each at their reference level. This is not a good choice if a value of 0 is not plausible for continuous variables in the model, e.g. age. +- `reference = "sample"`: the reference individual has continuous predictors each their respective sample mean and categorical predictors at their reference level. +- `reference = "strata"`: this leads to the same results as in `reference = "sample" unless the model includes a `strata()`, in which case continuous variables will be set their within-stratum means. + +```{r} +#| label: predict-hr +#| code-fold: false + +# predict HR for our new patient +predict(fit.cox_multi, + newdata = new_pat, + type = "risk", + reference = "strata", + se.fit = T) +``` + +Our new patient 3.27 times the hazard of dying compared to a reference individual with level of each categorical predictor and the mean of each continuous predictor, which in our case are (Female for gender, Meningioma for diagnosis, and mean Karnofsky index of `r round(mean(data_brain$ki), 2)`). + +## Time-dependent covariates + +In all above example the predictors did not vary over time. Each individual had only one value for each predictor. Since survival data is the result of follow-up over time, it is possible to have predictors that do vary over time. For example, in a study of time to heart attack, researchers could record various other conditions’ occurrence over time, such as hypertension or angina. In a study of juvenile recidivism, researchers could record how education or employment status change over time. Cox regression is able to handle such time-varying predictors (also known as “time dependent covariates”). + +For time-varying covariates, the hazard function is extended to account for changes in the covariates over time: $$h(t | \boldsymbol{x}(t)) = h_0(t) \cdot \exp\left(\boldsymbol{\beta}^T \boldsymbol{x}(t)\right)$$ + +where: + +- $h(t | \boldsymbol{x}(t))$ is the hazard at time $t$, conditional on the covariate values at that time. +- $\boldsymbol{x}(t)$ represents the covariate vector at time $t$. In this case, one or more covariates can change over time. +- $\boldsymbol{\beta}$ is the vector of regression coefficients. +- $h_0(t)$ is the baseline hazard function, which remains time-dependent but does not depend on the covariates. + +In practice, time-varying covariates are handled by breaking the follow-up time into intervals where the covariate values are assumed to remain constant, and then updating the covariate values at each interval. Suppose a covariate (e.g., blood pressure, treatment status, or exposure) varies over time. For an individual, we divide their follow-up time into periods where the covariate value is constant, and then treat each period as a separate "observation" in the Cox model. The Cox model then uses these updated covariate values when estimating the hazard at each time point. + +The partial likelihood function in the Cox model is modified to incorporate the updated covariate values $\boldsymbol{x}(t)$. For each event time $t_j$, the likelihood is based on the covariate values that are valid at that specific time. + +The partial likelihood becomes: + +$$PL(\boldsymbol{\beta}) = \prod_{j=1}^{k} \frac{\exp\left(\boldsymbol{\beta}^T \boldsymbol{x}_1(t_j)\right)}{\sum_{i \in \text{risk set at } t_j} \exp\left(\boldsymbol{\beta}^T \boldsymbol{x}_i(t_j)\right)}$$ + +where: + +- $\boldsymbol{x}_1(t_j)$ is the covariate vector for the subject who experienced the event at time $t_j$, +- and $\boldsymbol{x}_i(t_j)$ is the covariate vector for each subject in the risk set at time $t_j$. + +A dataset with time-varying predictors will have multiple rows per individual, with different rows having different values for the time-varying predictors, reflecting how they change over time. Additionally, rather than having a single event time variable, each row will have two time variables indicating the beginning and end of the time interval represented by that row of data. + +For instance, in the study of heroin usage, we can see a male participant who started using non-prescribed pharmaceutical opioids (NPPOs) at age 19 (time independent covariates). He did not meet the criteria for lifetime opioid dependence at baseline (wave 1), but did at the next interview (wave2), and he first reported using heroin at his 6th interview (both time varying variables). + +```{r} +#| label: laod-data-opioid +data_opioid <- readRDS("data/data-opioid.rds") + +data_opioid %>% + filter(RANDID == 10) %>% + select(RANDID, wave, START, STOP, heroin, age_at_init, sex, dep_lifetime) +``` + + +## Multiple events: competing risks + +In survival analysis, **competing risks** occur when an individual is at risk of more than one mutually exclusive event, and the occurrence of one event **precludes** the occurrence of the others. Traditional survival models, such as the **Cox proportional hazards model**, typically focus on a single type of event (e.g., death or failure), but in many real-world scenarios, individuals may face different types of events that compete with each other. + +A **competing risk** refers to events that prevent the event of primary interest from happening. For example: +- In a study of heart disease, **death due to cancer** is a competing risk if the primary event of interest is **death due to heart disease**. +- In a study of cancer treatment, **death from any cause** is a competing risk if the event of interest is **relapse**. + +In these situations, standard survival analysis methods may overestimate the probability of the primary event because they do not account for the possibility of competing events. + +**The Cumulative Incidence Function (CIF)** is commonly used in competing risks analysis. It represents the probability of experiencing a specific event (e.g., death from heart disease) by a certain time, accounting for the presence of competing risks (e.g., death from cancer). + +CIF is a product of two estimates. + +1. The estimate of hazard at ordered failure time $t_j$ for the event of interest: $$\hat{h_c(t_j)}=\frac{d_{cj}}{n_j}$$ +where: + +- $m_{cj}$ denotes the number of events for risk $c$ at time $t_j$ +- and $n_j$ is the number of subjects at risk at time $t_j$ + +2. The estimate of overall probability of surviving previous time: $$\hat{S}(t_{j-1})$$ We consider the overall survival as a subject must have survived all other competing events in order to fail from event type $c$ at time $t_j$. + +The estimated incidence probability of failing from even type $c$ at time $t_j$ is then: $$\hat{I_c}(t_j) = \hat{S}(t_{j-1}) \times \hat{h_c(t_j)}$$ The probability of failing form even type $c$ at time $t_j$ is a product of surviving the previous time periods and the cause specific hazard at time $t_j$. + +The CIF for event type $c$ at time $t_j$ is then the cumulative sum up to time $t_j$, i.e. from $f' = 1$ to $f'=f$, of these incidence probabilities over all event type $c$ failure times, which is expressed: $$CIF_c(t_j) = \sum_{f'=1}^{f}\hat{I_c}(t_j) = \sum_{f'=1}^{f}\hat{S}(t_{f'-1})\times\hat{h_c}(t_{f'})$$ + +In 1999 @Gray1998 proposed a non-parametric test to compare two or more CIFs. The test is analogous to the log-rank test comparing KM curves, using a modified Chi-squared test statistic. This test does not require the independent censoring assumption. + +In Fine and Gray @Fine1999 proposed a proportional hazards model aims at modeling the CIF with covariates, by treating the CIF curve as a subdistribution function. 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