Merge pull request #249 from NBISweden/evaupdate

Update probability session (move an exercise)

session-inference/lectures/inferenceI.qmd

@@ -26,14 +26,21 @@ options(digits=2)
 
 ## Introduction to hypothesis tests 
 
-::: {.r-fit-text}
-
 **Statistical inference** is to draw conclusions regarding properties of a population based on observations of a random sample from the population.
 
+:::{.columns}
+:::{.column width="7%"}
+![](../figures/Lampa.jpg)
+:::
+:::{.column width="92%"}
 A **hypothesis test** is a type of inference about evaluating if a hypothesis about a population is supported by the observations of a random sample (i.e by the data available).
+:::
+::::
 
 Typically, the hypotheses that are tested are assumptions about properties of a population, such as proportion, mean, mean difference, variance etc.
-:::
+
+
+![](../figures/hypotestest.jpg){width=50% fig-align="center"}
 
 ## The null and alternative hypothesis 
 
@@ -92,8 +99,23 @@ df1 |> ggplot(aes(x=group, y=x, color=group)) + geom_boxplot() + theme_bw() + xl
 ::: {.smaller style="font-size: 35px" .incremental}
 1.  Define $H_0$ and $H_1$
 2.  Select an appropriate significance level, $\alpha$
+
+::: {.notes}
+The significance level is the acceptable risk of false alarm, i.e. to say *"I have a hit"*, *"I found a difference"*, when the the null hypothesis (*"there is no difference"*) is true.
+:::
+
 3.  Select appropriate test statistic, $T$, and compute the observed value, $t_{obs}$
+
+::: {.notes}
+For example the difference in means, the proportion of successes, the correlation coefficient etc.
+:::
+
 4.  Assume that the $H_0$ is true and compute the sampling distribution of $T$.
+
+::: {.notes}
+We will get back to the null distribution in the next slide
+:::
+
 5.  Compare the observed value, $t_{obs}$, with the computed sampling distribution under $H_0$ and compute a p-value. The p-value is the probability of observing a value at least as extreme as the observed value, if $H_0$ is true.
 6.  Based on the p-value either accept or reject $H_0$.
 :::
@@ -218,8 +240,6 @@ ggplot(df, aes(x=x, y=f, color=h, fill=h)) + geom_line()  + geom_area(data=df %>
 
 You suspect that a dice is loaded, i.e. showing 'six' more often than expected of a fair dice. To test this you throw the dice 10 times and count the total number of sixes. You got 5 sixes. Is there reason to believe that the dice is loaded?
 
-*Live coding!*
-
 1.  Define $H_0$ and $H_1$
 2.  Select an appropriate significance level, $\alpha$
 3.  Select appropriate test statistic, $T$, and $t_{obs}$
@@ -658,7 +678,6 @@ If high-fat diet has no effect, i.e. if $H_0$ was true, the result would be as i
 ::: {.column width="50%"}
 The 24 mice were initially from the same population, depending on how the mice are randomly assigned to high-fat and normal group, the mean weights would differ, even if the two groups were treated the same.
 
-Random reassignment to two groups can be accomplished using permutation.
 :::
 ::: {.column width="50%"}
 ```{r results="asis"}
@@ -670,6 +689,45 @@ for (i in o) {
 :::
 ::::
 
+## Simulation example
+
+**4. Null distribution**
+
+If high-fat diet has no effect, i.e. if $H_0$ was true, the result would be as if all mice were given the same diet.
+
+::: {.columns}
+::: {.column width="50%"}
+The 24 mice were initially from the same population, depending on how the mice are randomly assigned to high-fat and normal group, the mean weights would differ, even if the two groups were treated the same.
+:::
+::: {.column width="50%"}
+:::{.columns}
+:::{.column width="40%"}
+```{r results="asis"}
+sc <- 2
+o2 <- sample(o)
+for (i in o2[1:12]) {
+  cat(sprintf('![](../figures/Mus.jpg){width=%i data-id="%s"}', round(c(xHF, xN)[i]*sc), name[i]))
+}
+```
+:::
+:::{.column width="40%"}
+```{r results="asis"}
+for (i in o2[13:24]) {
+  cat(sprintf('![](../figures/Mus.jpg){width=%i data-id="%s"}', round(c(xHF, xN)[i]*sc), name[i]))
+}
+```
+:::
+::::
+
+:::
+::::
+
+## Simulation example
+
+**4. Null distribution**
+
+Random reassignment to two groups can be accomplished using permutation.
+
 Assume $H_0$ is true, i.e. assume all mice are equivalent and
 
 1.  Randomly reassign 12 of the 24 mice to 'high-fat' and the remaining 12 to 'control'.

session-probability/docs/prob_02discrv.html

@@ -426,7 +426,7 @@ <h2 data-number="2.2" class="anchored" data-anchor-id="simulate-distributions"><
 <div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="do">## Another coin toss</span></span>
 <span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sample</span>(<span class="fu">c</span>(<span class="st">"H"</span>, <span class="st">"T"</span>), <span class="at">size=</span><span class="dv">1</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>[1] "T"</code></pre>
+<pre><code>[1] "H"</code></pre>
 </div>
 </div>
 <p>Every time you run <code>sample</code> a new coin toss is simulated.</p>
@@ -436,7 +436,7 @@ <h2 data-number="2.2" class="anchored" data-anchor-id="simulate-distributions"><
 <div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="do">## 20 independent coin tosses</span></span>
 <span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>(coins <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="fu">c</span>(<span class="st">"H"</span>, <span class="st">"T"</span>), <span class="at">size=</span><span class="dv">20</span>, <span class="at">replace=</span><span class="cn">TRUE</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code> [1] "H" "T" "H" "H" "T" "T" "T" "H" "T" "T" "H" "H" "T" "T" "H" "T" "H" "H" "H"
+<pre><code> [1] "H" "T" "H" "H" "H" "T" "H" "H" "T" "H" "H" "T" "T" "T" "H" "T" "H" "T" "H"
 [20] "H"</code></pre>
 </div>
 </div>
@@ -445,7 +445,7 @@ <h2 data-number="2.2" class="anchored" data-anchor-id="simulate-distributions"><
 <div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="do">## How many heads?</span></span>
 <span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(coins <span class="sc">==</span> <span class="st">"H"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>[1] 11</code></pre>
+<pre><code>[1] 12</code></pre>
 </div>
 </div>
 <p>We can repeat this experiment (toss 20 coins and count the number of heads) several times to estimate the distribution of number of heads in 20 coin tosses.</p>
@@ -470,11 +470,11 @@ <h2 data-number="2.2" class="anchored" data-anchor-id="simulate-distributions"><
 <div class="cell">
 <div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sum</span>(Nheads <span class="sc">&gt;=</span> <span class="dv">15</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>[1] 202</code></pre>
+<pre><code>[1] 217</code></pre>
 </div>
 </div>
 <p>From this we conclude that</p>
-<p><span class="math inline">\(P(Y \geq 15) =\)</span> 202/10000 = 0.0202</p>
+<p><span class="math inline">\(P(Y \geq 15) =\)</span> 217/10000 = 0.0217</p>
 </div>
 <p>Resampling can also be used to compute other properties of a random variable, such as the expected value.</p>
 <p>The <strong>law of large numbers</strong> states that if the same experiment is performed many times the average of the result will be close to the expected value.</p>