From 90dc43356b36476b3a5f3d0e2d5653d59dd251a1 Mon Sep 17 00:00:00 2001 From: ethanweed Date: Fri, 19 Apr 2024 11:24:53 +0200 Subject: [PATCH] Update documentation --- 04.03-estimation.html | 176 +++++++++--------- ...7d00d4fac2a09b4cc118dacc05a178e1691bf4.png | Bin 0 -> 18903 bytes ...f9579bc1dc69fa8094b46ce17c7c54c7dfb18d.png | Bin 0 -> 19406 bytes ...760e613fa2ad204bbddd225e82ecebb0c74f80.png | Bin 0 -> 16337 bytes ...82aee5f2da77ba7eddaa9300894de289ffca45.png | Bin 0 -> 18383 bytes ...d008f0af87fc9fb061c97a17774331984debed.png | Bin 0 -> 44688 bytes ...65535bc4e6cbfe4f24ba359fb715b1c692cefa.png | Bin 0 -> 18647 bytes ...8ad8220c2643ced168592d281911a5dc3bc727.png | Bin 0 -> 36630 bytes ...259d66931a356fbbc68bbd1c34a440d3469423.png | Bin 0 -> 37468 bytes ...3f53643d5f7b69c55af74fd9274e8f3a6ae328.png | Bin 0 -> 43965 bytes ...8b6424822075e88c7efc96dd1bcdb3067489c2.png | Bin 0 -> 26722 bytes ...3cd92976a804eeed5d6eb780580ce85ba96162.png | Bin 0 -> 60896 bytes ...80ba51025cb178b6ac30ddd53401cbbcdb082e.png | Bin 0 -> 20383 bytes ...6795a9a894a6bb977e6333a42fef7f552f2b75.png | Bin 0 -> 13935 bytes _sources/04.03-estimation.ipynb | 2 +- searchindex.js | 2 +- 16 files changed, 90 insertions(+), 90 deletions(-) create mode 100644 _images/18a613c84d2a62121021c5bc777d00d4fac2a09b4cc118dacc05a178e1691bf4.png create mode 100644 _images/3b531d002a4096785d99b61772f9579bc1dc69fa8094b46ce17c7c54c7dfb18d.png create mode 100644 _images/40affcf665a1aa03d682e04a85760e613fa2ad204bbddd225e82ecebb0c74f80.png create mode 100644 _images/5294d389304b664d35c2d0850582aee5f2da77ba7eddaa9300894de289ffca45.png create mode 100644 _images/5d67480d4ae090f29ba2bbc2c7d008f0af87fc9fb061c97a17774331984debed.png create mode 100644 _images/5e65a846e0af730b84e022268d65535bc4e6cbfe4f24ba359fb715b1c692cefa.png create mode 100644 _images/7325fd9f87e21f7b45a565e6e58ad8220c2643ced168592d281911a5dc3bc727.png create mode 100644 _images/af7e2a301845c0184fd2c21753259d66931a356fbbc68bbd1c34a440d3469423.png create mode 100644 _images/c2dd62e6eb14fe4891579b86603f53643d5f7b69c55af74fd9274e8f3a6ae328.png create mode 100644 _images/c80cce23b577fe8f51c3f2345a8b6424822075e88c7efc96dd1bcdb3067489c2.png create mode 100644 _images/cf611046e2cde6a827614208b43cd92976a804eeed5d6eb780580ce85ba96162.png create mode 100644 _images/dc1b2baff44436df3147e0caf580ba51025cb178b6ac30ddd53401cbbcdb082e.png create mode 100644 _images/de5710abc1af1ae01e1c2751466795a9a894a6bb977e6333a42fef7f552f2b75.png diff --git a/04.03-estimation.html b/04.03-estimation.html index 778c3359..a9212544 100644 --- a/04.03-estimation.html +++ b/04.03-estimation.html @@ -540,7 +540,7 @@

11.1.5. Population parameters and sample
-_images/b59d951d489a29953f1f240211cf33bdf6e441bc8821b9726c0a9bcb25412816.png +_images/c80cce23b577fe8f51c3f2345a8b6424822075e88c7efc96dd1bcdb3067489c2.png

Now suppose I run an experiment. I select 100 people at random and administer an IQ test, giving me a simple random sample from the population. My sample would consist of a collection of numbers like this:

@@ -571,9 +571,9 @@

11.2. The law of large numbers -
10 samples. Mean:  103.53960547224054  Standard deviation:  16.186167782018202
-100 samples. Mean:  101.17191341063791  Standard deviation:  16.86103856998936
-10000 samples. Mean:  100.04894765095658  Standard deviation:  15.098657549401311
+
10 samples. Mean:  102.5083916244877  Standard deviation:  18.72448042232567
+100 samples. Mean:  97.95368145848558  Standard deviation:  17.12238658574764
+10000 samples. Mean:  99.93179094823645  Standard deviation:  14.95182846479696
 
@@ -607,8 +607,8 @@

11.3.1. Sampling distribution of the mea

-
Simulated data:  [106 125 106  77  98]
-Mean of simulated data:  102
+
Simulated data:  [ 93 118  89  88  79]
+Mean of simulated data:  93
 
@@ -623,8 +623,8 @@

11.3.1. Sampling distribution of the mea

-
Simulated data:  [ 89 107 121 102  56]
-Mean of simulated data:  95
+
Simulated data:  [ 90 109  91  91 103]
+Mean of simulated data:  96
 
@@ -683,93 +683,93 @@

11.3.1. Sampling distribution of the mea Replication 1 - 100 - 111 - 92 - 83 - 112 - 99.6 + 99 + 84 + 105 + 102 + 91 + 96.2 Replication 2 - 106 - 102 - 78 + 127 + 127 92 - 96 - 94.8 + 88 + 121 + 111.0 Replication 3 - 112 - 81 - 106 - 109 - 92 - 100.0 + 90 + 114 + 125 + 98 + 99 + 105.2 Replication 4 - 107 - 108 - 95 - 96 - 48 - 90.8 + 115 + 114 + 72 + 122 + 103 + 105.2 Replication 5 - 88 - 94 - 107 - 131 - 105 - 105.0 + 92 + 93 + 98 + 109 + 122 + 102.8 Replication 6 - 118 - 98 - 95 - 102 - 106 - 103.8 + 109 + 92 + 104 + 79 + 105 + 97.8 Replication 7 - 81 - 77 - 87 - 80 - 122 - 89.4 + 91 + 91 + 104 + 61 + 98 + 89.0 Replication 8 - 101 - 103 - 85 - 91 - 96 - 95.2 + 82 + 83 + 73 + 108 + 92 + 87.6 Replication 9 - 119 - 109 - 121 - 120 - 105 - 114.8 + 107 + 84 + 102 + 118 + 95 + 101.2 Replication 10 - 96 - 125 - 96 - 88 - 76 - 96.2 + 77 + 91 + 113 + 108 + 97 + 97.2 @@ -783,7 +783,7 @@

11.3.1. Sampling distribution of the mea

-
[99.6, 94.8, 100.0, 90.8, 105.0, 103.8, 89.4, 95.2, 114.8, 96.2]
+
[96.2, 111.0, 105.2, 105.2, 102.8, 97.8, 89.0, 87.6, 101.2, 97.2]
 
@@ -839,7 +839,7 @@

11.3.1. Sampling distribution of the mea

-_images/34967f35ee18db03222a750274dea854c2ca237d4d2d4c5947f979022246f78d.png +_images/18a613c84d2a62121021c5bc777d00d4fac2a09b4cc118dacc05a178e1691bf4.png

Sampling distributions are another important theoretical idea in statistics, and they’re crucial for understanding the behaviour of small samples. For instance, when I ran the very first “five IQ scores” experiment, the sample mean turned out to be 95. What the sampling distribution in fig-IQ_samp_dist tells us, though, is that the “five IQ scores” experiment is not very accurate. If I repeat the experiment, the sampling distribution tells me that I can expect to see a sample mean anywhere between 80 and 120.

@@ -898,7 +898,7 @@

11.3.2. Sampling distributions exist for
-_images/dcb6bc54906231381f3944979f12aca8e15ef7c89923f6e1ba9a7dbb4c1fd779.png +_images/5294d389304b664d35c2d0850582aee5f2da77ba7eddaa9300894de289ffca45.png
@@ -964,7 +964,7 @@

11.3.2. Sampling distributions exist for
-_images/896164f9ef06dcf80e44e9ce42da9fda9d59eff50d78f402dcd18ba5a4c2723c.png +_images/c2dd62e6eb14fe4891579b86603f53643d5f7b69c55af74fd9274e8f3a6ae328.png

Okay, so that’s one part of the story. However, there’s something I’ve been glossing over so far. All my examples up to this point have been based on the “IQ scores” experiments, and because IQ scores are roughly normally distributed, I’ve assumed that the population distribution is normal. What if it isn’t normal? What happens to the sampling distribution of the mean? The remarkable thing is this: no matter what shape your population distribution is, as \(N\) increases the sampling distribution of the mean starts to look more like a normal distribution. To give you a sense of this, I ran some simulations using Python. To do this, I wrote a function called plotSamples that produces the “ramped” beta distribution shown in the first histogram below when \(N=1\). You can use the “click to show” button to take a look at the code, if you want to see how it works. The important thing for our purposes is that, as you can see by comparing the triangular shaped histogram to the bell curve plotted by the black line, the population distribution doesn’t look very much like a normal distribution at all. Then I provided the function with increasingly larger numbers of “participants” for each simulated experiment. As the size of \(N\) increases, the sampling distribution of the mean looks increasingly normal, and by the time we reach a sample size of \(N=8\) it’s almost perfectly normal. In other words, as long as your sample size isn’t tiny, the sampling distribution of the mean will be approximately normal no matter what your population distribution looks like!

@@ -1033,7 +1033,7 @@

11.3.2. Sampling distributions exist for
-_images/18daaf72a764216c35f5a6d05ca7d2c91b072dfb575f5bf5416ddfcfb700db12.png +_images/dc1b2baff44436df3147e0caf580ba51025cb178b6ac30ddd53401cbbcdb082e.png
@@ -1043,7 +1043,7 @@

11.3.2. Sampling distributions exist for

-_images/40e60e5572330779c322a9cf97d824e8d1c5fc6865627bdddd4a10bb7eb29b10.png +_images/3b531d002a4096785d99b61772f9579bc1dc69fa8094b46ce17c7c54c7dfb18d.png
@@ -1053,7 +1053,7 @@

11.3.2. Sampling distributions exist for

-_images/460918468a428625499180aa59a8157f85e56e654959763633953a837f72b65e.png +_images/5e65a846e0af730b84e022268d65535bc4e6cbfe4f24ba359fb715b1c692cefa.png
@@ -1063,7 +1063,7 @@

11.3.2. Sampling distributions exist for

-_images/e00c5386b61ef06697993150a21b871675965ba2cb64c425ecafa288f2efed29.png +_images/40affcf665a1aa03d682e04a85760e613fa2ad204bbddd225e82ecebb0c74f80.png

On the basis of these figures, it seems like we have evidence for all of the following claims about the sampling distribution of the mean:

@@ -1166,7 +1166,7 @@

11.4.2. Estimating the population standa
-_images/821ce1f34e583d8a2662e393bdd9766bbf1cade6bdc0c9e1f7629a20becd6394.png +_images/de5710abc1af1ae01e1c2751466795a9a894a6bb977e6333a42fef7f552f2b75.png

This intuition feels right, but it would be nice to demonstrate this somehow. There are in fact mathematical proofs that confirm this intuition, but unless you have the right mathematical background they don’t help very much. Instead, what I’ll do is use Python to simulate the results of some experiments. With that in mind, let’s return to our IQ studies. Suppose the true population mean IQ is 100 and the standard deviation is 15. I can use the rnorm() function to generate the the results of an experiment in which I measure \(N=2\) IQ scores, and calculate the sample standard deviation. If I do this over and over again, and plot a histogram of these sample standard deviations, what I have is the sampling distribution of the standard deviation. I’ve plotted this distribution in fig-sampdistsd. Even though the true population standard deviation is 15, the average of the sample standard deviations is only 8.5. Notice that this is a very different result to what we found in fig-IQ-clm when we plotted the sampling distribution of the mean. If you look at that sampling distribution, what you see is that the population mean is 100, and the average of the sample means is also 100.

@@ -1248,7 +1248,7 @@

11.4.2. Estimating the population standa
-_images/4531874074920abf560164e981a1b1adfaec95dfa97869cca8cd78e571562065.png +_images/7325fd9f87e21f7b45a565e6e58ad8220c2643ced168592d281911a5dc3bc727.png

The fix to this systematic bias turns out to be very simple. Here’s how it works. Before tackling the standard deviation, let’s look at the variance. If you recall from the section on measures of variability, the sample variance is defined to be the average of the squared deviations from the sample mean. That is:

@@ -1471,9 +1471,9 @@

11.5.2. Interpreting a confidence interv
-
/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_63592/4220361125.py:39: DeprecationWarning: Use of keyword argument 'alpha' for method 'interval' is deprecated and wil be removed in SciPy 1.11.0. Use first positional argument or keyword argument 'confidence' instead.
+
/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_7964/4220361125.py:39: DeprecationWarning: Use of keyword argument 'alpha' for method 'interval' is deprecated and wil be removed in SciPy 1.11.0. Use first positional argument or keyword argument 'confidence' instead.
   ci = t.interval(alpha=.95, df=len(simdata)-1, loc=np.mean(simdata), scale=sem(simdata))
-/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_63592/4220361125.py:39: DeprecationWarning: Use of keyword argument 'alpha' for method 'interval' is deprecated and wil be removed in SciPy 1.11.0. Use first positional argument or keyword argument 'confidence' instead.
+/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_7964/4220361125.py:39: DeprecationWarning: Use of keyword argument 'alpha' for method 'interval' is deprecated and wil be removed in SciPy 1.11.0. Use first positional argument or keyword argument 'confidence' instead.
   ci = t.interval(alpha=.95, df=len(simdata)-1, loc=np.mean(simdata), scale=sem(simdata))
 
@@ -1485,7 +1485,7 @@

11.5.2. Interpreting a confidence interv

11.5.3. Calculating confidence intervals in Python#

-

To produce the confidence intervals for the plots of simulated IQ data above, I used the t, sem, and meanfunctions available in the scipy.statspackage. Another option is to use the tconfint_mean function from the statsmodels package. As you can see, both methods give nearly identical results. Method 1 is good insofar is at requires you to explicitly specify the desired confidence interval, the degrees of freedom, and the standard error of the mean. Method takes care of all of this for us, which makes it easier, but a bit more of a black box.

+

To produce the confidence intervals for the plots of simulated IQ data above, I used the t, sem, and meanfunctions available in the scipy.statspackage. Another option is to use the tconfint_mean function from the statsmodels package. As you can see, both methods give nearly identical results. Method 1 is good insofar is at requires you to explicitly specify the desired confidence interval, the degrees of freedom, and the standard error of the mean. Method 2 takes care of all of this for us, which makes it easier, but is a bit more of a black box.

# Sample data:
@@ -1511,12 +1511,12 @@ 

11.5.3. Calculating confidence intervals

-
Method 1:  (9.514149929408497, 15.485850070591503)
-Method 2:  (9.514149929408497, 15.485850070591503)
+
/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_7964/1339756836.py:7: DeprecationWarning: Use of keyword argument 'alpha' for method 'interval' is deprecated and wil be removed in SciPy 1.11.0. Use first positional argument or keyword argument 'confidence' instead.
+  ci_1 = t.interval(alpha=0.95,
 
-
/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_63592/1339756836.py:7: DeprecationWarning: Use of keyword argument 'alpha' for method 'interval' is deprecated and wil be removed in SciPy 1.11.0. Use first positional argument or keyword argument 'confidence' instead.
-  ci_1 = t.interval(alpha=0.95,
+
Method 1:  (9.514149929408497, 15.485850070591503)
+Method 2:  (9.514149929408497, 15.485850070591503)
 
@@ -1641,14 +1641,14 @@

11.5.4. Plotting confidence intervals in

-
/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_63592/3591514384.py:9: FutureWarning: 
+
/var/folders/6m/mwt0c30539b1zrk5vf14dpr44pd98r/T/ipykernel_7964/3591514384.py:9: FutureWarning: 
 
 The `ci` parameter is deprecated. Use `errorbar=('ci', 40)` for the same effect.
 
   sns.pointplot(x="time", y="total_bill", hue="smoker",  ci = 40, data=tips, ax=axes[1])
 
-_images/836d599b130ba5b08f6e8e9db87c138ea6de81f3fd0d08f3f3205c455851526d.png +_images/af7e2a301845c0184fd2c21753259d66931a356fbbc68bbd1c34a440d3469423.png

For regression plots, seaborn computes a confidence interval for regression line by default. This can be turned off with ci=None, but I think it is good practice to include it, because it gives a nice visual indication of the strength of the model.

@@ -1673,7 +1673,7 @@

11.5.4. Plotting confidence intervals in

-_images/7f2959939539671e8609a07c12cd7f8cbdb2c3981711cee476c39dfff6d9e21b.png +_images/cf611046e2cde6a827614208b43cd92976a804eeed5d6eb780580ce85ba96162.png

For regression plots with discrete variables on the x-axis, seaborn has options for either showing all datapoints, or showing only the mean with error-bars indicating the confidence interval. There are many more details and options to be found in the seaborn documentation. For more complex or custom figures, like the one in fig-cirep showing confidence intervals for simulated IQ data, you will need to dive into matplotlib, which allows much more customization than is available simply using seaborn.

@@ -1700,7 +1700,7 @@

11.5.4. Plotting confidence intervals in

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Why do we learn statistics?", "2. A brief introduction to research design", "3. Getting Started with Python", "4. More Python Concepts", "5. Descriptive statistics", "6. Drawing Graphs", "7. Data Wrangling", "8. Basic Programming", "9. Statistical theory", "10. Introduction to Probability", "11. Estimating unknown quantities from a sample", "12. Hypothesis Testing", "13. Categorical data analysis", "14. Comparing Two Means", "15. Comparing several means (one-way ANOVA)", "16. Linear regression", "17. Factorial ANOVA", "18. Bayesian Statistics", "19. Epilogue", "20. 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Why do we learn statistics?", "2. A brief introduction to research design", "3. Getting Started with Python", "4. More Python Concepts", "5. Descriptive statistics", "6. Drawing Graphs", "7. Data Wrangling", "8. Basic Programming", "9. Statistical theory", "10. Introduction to Probability", "11. Estimating unknown quantities from a sample", "12. Hypothesis Testing", "13. Categorical data analysis", "14. Comparing Two Means", "15. Comparing several means (one-way ANOVA)", "16. Linear regression", "17. Factorial ANOVA", "18. Bayesian Statistics", "19. Epilogue", "20. 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