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<li><a href="#introduction" id="toc-introduction" class="nav-link active" data-scroll-target="#introduction"><span class="header-section-number">7.1</span> Introduction</a></li>
<li><a href="#samples-and-labels-and-strings" id="toc-samples-and-labels-and-strings" class="nav-link" data-scroll-target="#samples-and-labels-and-strings">7.2 Samples and labels and strings</a></li>
<li><a href="#sec-types-values" id="toc-sec-types-values" class="nav-link" data-scroll-target="#sec-types-values">7.3 Types of values in R</a></li>
<li><a href="#sec-intro-to-strings" id="toc-sec-intro-to-strings" class="nav-link" data-scroll-target="#sec-intro-to-strings">7.4 String values</a></li>
<li><a href="#strings-in-s" id="toc-strings-in-s" class="nav-link" data-scroll-target="#strings-in-s">7.5 Strings in vectors</a></li>
<li><a href="#sec-repeating" id="toc-sec-repeating" class="nav-link" data-scroll-target="#sec-repeating">7.6 Repeating elements</a></li>
<li><a href="#resampling-with-and-without-replacement" id="toc-resampling-with-and-without-replacement" class="nav-link" data-scroll-target="#resampling-with-and-without-replacement">7.7 Resampling with and without replacement</a></li>
<li><a href="#conclusion" id="toc-conclusion" class="nav-link" data-scroll-target="#conclusion"><span class="header-section-number">7.8</span> Conclusion</a></li>
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<h1 class="title"><span id="sec-sampling-tools" class="quarto-section-identifier"><span class="chapter-number">7</span> <span class="chapter-title">Tools for samples and sampling</span></span></h1>
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<section id="introduction" class="level2" data-number="7.1">
<h2 data-number="7.1" class="anchored" data-anchor-id="introduction"><span class="header-section-number">7.1</span> Introduction</h2>
<p>Now you have some experience with R, probabilities and resampling, it is time to introduce some useful tools for our experiments and programs.</p>
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Note 7.1: Notebook: Sampling tools
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<p><a class="notebook-link" href="notebooks/sampling_tools.Rmd">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=sampling_tools.ipynb">Interact</a></p>
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<div class="nb-start" name="sampling_tools" title="Sampling tools">
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</section>
<section id="samples-and-labels-and-strings" class="level2" data-number="7.2">
<h2 data-number="7.2" class="anchored" data-anchor-id="samples-and-labels-and-strings">7.2 Samples and labels and strings</h2>
<p>Thus far we have used numbers such as 1 and 0 and 10 to represent the elements we are sampling from. For example, in <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a>, we were simulating the chance of a particular juror being black, given that 26% of the eligible jurors in the county were black. We used <em>integers</em> for that task, where we started with all the integers from 0 through 99, and asked R to select values at random from those integers. When R selected an integer from 0 through 25, we chose to label the resulting simulated juror as black — there are 26 integers in the range 0 through 25, so there is a 26% chance that any one integer will be in that range. If the integer was from 26 through 99, the simulated juror was white (there are 74 integers in the range 26 through 99).</p>
<p>Here is the process of simulating a single juror, adapted from <a href="resampling_with_code2.html#sec-random-zero-through-ninety-nine" class="quarto-xref"><span>Section 6.3.3</span></a>:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get 1 random number from 0 through 99</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="co"># replace=TRUE is redundant here (why?), but we leave it for consistency.</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>a <span class="ot"><-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">99</span>, <span class="dv">1</span>, <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>a</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 44</code></pre>
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<p>After that, we have to unpack our labeling of 0 through 25 as being “black” and 26 through 99 as being “white”. We might do that like this:</p>
<div class="cell" data-layout-align="center">
<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>this_juror_is_black <span class="ot"><-</span> a <span class="sc"><</span> <span class="dv">26</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>this_juror_is_black</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] FALSE</code></pre>
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<p>This all works as we want it to, but it’s just a little bit difficult to remember the coding (less than 26 means “black”, greater than 25 means “white”). We had to use that coding because we committed ourselves to using random numbers to simulate the outcomes.</p>
<p>However, R can also store bits of text, called <em>strings</em>. Values that are bits of text can be very useful because the text values can be memorable labels for the entities we are sampling from, in our simulations.</p>
<p>Before we get to strings, let us consider the type of the values we have seen so far.</p>
</section>
<section id="sec-types-values" class="level2" data-number="7.3">
<h2 data-number="7.3" class="anchored" data-anchor-id="sec-types-values">7.3 Types of values in R</h2>
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<div class="r">
<p>So far, all the values we have seen in R are <em>numeric</em> — integers or floating point values. This is an integer value:</p>
<div class="cell" data-layout-align="center">
<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>v <span class="ot"><-</span> <span class="dv">10</span></span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>v</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 10</code></pre>
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<p>Here the variable <code>v</code> holds the value. We can see what type of value <code>v</code> holds by using the <code>class</code> function:</p>
<div class="cell" data-layout-align="center">
<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="fu">class</span>(v)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "numeric"</code></pre>
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<p>The value contained by the variable <code>v</code> is of <code>'numeric'</code> type (class). This is the type of value that can store both integer values (positive or negative whole numbers), or floating point values (values that can have digits after a decimal point. Here’s a floating point value.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>f <span class="ot"><-</span> <span class="fl">10.1</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="fu">class</span>(f)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "numeric"</code></pre>
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<p>Notice that R also see this as a <code>"numeric"</code> type of value. However, we are about to see that R values can be of other types, that are not numeric.</p>
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</section>
<section id="sec-intro-to-strings" class="level2" data-number="7.4">
<h2 data-number="7.4" class="anchored" data-anchor-id="sec-intro-to-strings">7.4 String values</h2>
<p>So far, all the values you have seen in R vectors have been numbers. Now we get on to values that are bits of text. These are called <em>strings</em>.</p>
<p>Here is a single R string value:</p>
<div class="cell" data-layout-align="center">
<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>s <span class="ot"><-</span> <span class="st">"Resampling"</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a>s</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "Resampling"</code></pre>
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<p>What is the <code>class</code> of the new bit-of-text value <code>s</code>?</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">class</span>(s)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "character"</code></pre>
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<p>The R <code>character</code> value is a bit of text, and therefore consists of a sequence of characters.</p>
<p>As vectors are containers for other things, such as numbers, strings are containers for characters.</p>
<div class="r">
<p>To get the length of a string, use the <code>nchar</code> function (Number of Characters):</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of characters in s</span></span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a><span class="fu">nchar</span>(s)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 10</code></pre>
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<div class="r">
<p>R has a <code>substring</code> function that allows you to select individual characters or sequences of characters from a string. The arguments to <code>substring</code> are: first — the string; second — the index of the first character you want to select; and third — the index of the last character you want to select. For example to select the second character in the string you would specify 2 as the starting index, and 2 as the ending index, like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Get the second character of the string</span></span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a>second_char <span class="ot"><-</span> <span class="fu">substring</span>(s, <span class="dv">2</span>, <span class="dv">2</span>)</span>
<span id="cb17-3"><a href="#cb17-3" aria-hidden="true" tabindex="-1"></a>second_char</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "e"</code></pre>
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</section>
<section id="strings-in-s" class="level2" data-number="7.5">
<h2 data-number="7.5" class="anchored" data-anchor-id="strings-in-s">7.5 Strings in vectors</h2>
<p>As we can store numbers as elements in vectors, we can also store strings as vector elements.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a>vector_of_strings <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Julian'</span>, <span class="st">'Lincoln'</span>, <span class="st">'Simon'</span>)</span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a>vector_of_strings</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "Julian" "Lincoln" "Simon" </code></pre>
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<p>As for any vector, you can select elements with <em>indexing</em>. When you select an element with a given position (index), you get the <em>string</em> at at that position:</p>
<p>Notice the output from this chunk:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Julian Lincoln Simon's second name</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a>middle_name <span class="ot"><-</span> vector_of_strings[<span class="dv">2</span>]</span>
<span id="cb21-3"><a href="#cb21-3" aria-hidden="true" tabindex="-1"></a>middle_name</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "Lincoln"</code></pre>
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<p>As for numbers, we can compare strings with, for example, the <code>==</code> operator, that asks whether the two strings are equal:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a>middle_name <span class="sc">==</span> <span class="st">'Lincoln'</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] TRUE</code></pre>
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</div>
</section>
<section id="sec-repeating" class="level2" data-number="7.6">
<h2 data-number="7.6" class="anchored" data-anchor-id="sec-repeating">7.6 Repeating elements</h2>
<p>Now let us go back to the problem of selecting black and white jurors.</p>
<p>We started with the strategy of using numbers 0 through 25 to mean “black” jurors, and 26 through 99 to mean “white” jurors. We selected values at random from 0 through 99, and then worked out whether the number meant a “black” juror (was less than 26) or a “white” juror (was greater than 25).</p>
<p>It would be good to use strings instead of numbers to identify the potential jurors. Then we would not have to remember our coding of 0 through 25 and 26 through 99.</p>
<p>If only there was a way to make a vector of 100 strings, where 26 of the strings were “black” and 74 were “white”. Then we could select randomly from that array, and it would be immediately obvious that we had a “black” or “white” juror.</p>
<p>Luckily we can do that, by using the <code>rep</code> function to construct the vector.</p>
<div class="r">
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<code>rep</code>
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<p><code>rep</code> is a useful function in R that makes a new output vector by repeating elements from the input vectors.</p>
<p>More specifically, <code>rep</code> accepts two arguments (inputs), and sends back a vector as output. The first input is a vector for which you want to repeat elements. The second input tells <code>rep</code> how many repeats to do, for each element.</p>
<p>This is most easy to see by examples.</p>
<p>Here we repeat all three elements in the first vector by a count (number) given in the second argument.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat each element in the first list (or array) 4 times.</span></span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a>to_repeat <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>) <span class="co"># First input argument.</span></span>
<span id="cb25-3"><a href="#cb25-3" aria-hidden="true" tabindex="-1"></a>n_repeats <span class="ot"><-</span> <span class="dv">4</span> <span class="co"># Second input argument.</span></span>
<span id="cb25-4"><a href="#cb25-4" aria-hidden="true" tabindex="-1"></a><span class="fu">rep</span>(to_repeat, n_repeats)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] 1 2 3 1 2 3 1 2 3 1 2 3</code></pre>
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<p>Of course we could also write this without the variables:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat each element in the first list (or array) 4 times.</span></span>
<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a><span class="fu">rep</span>(<span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">2</span>, <span class="dv">3</span>), <span class="dv">4</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] 1 2 3 1 2 3 1 2 3 1 2 3</code></pre>
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<p>The second argument (number of repeats) can also be a vector with one element for each element of the first vector. In that case, it gives the number of repeats for each element. We will use string elements for this example.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the first element 3 times, the second 2 times, and the third 4 times.</span></span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Second input is number of repeats for each element of the first argument.</span></span>
<span id="cb29-3"><a href="#cb29-3" aria-hidden="true" tabindex="-1"></a><span class="fu">rep</span>(<span class="fu">c</span>(<span class="st">'one'</span>, <span class="st">'two'</span>, <span class="st">'three'</span>), <span class="fu">c</span>(<span class="dv">3</span>, <span class="dv">2</span>, <span class="dv">4</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] "one" "one" "one" "two" "two" "three" "three" "three" "three"</code></pre>
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<p>Here we use <code>rep</code> to make up our simulated jury pool.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The values that we will repeat to fill up the larger array.</span></span>
<span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a>juror_types <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'black'</span>, <span class="st">'white'</span>)</span>
<span id="cb31-3"><a href="#cb31-3" aria-hidden="true" tabindex="-1"></a><span class="co"># The number of times we want to repeat "black" and "white".</span></span>
<span id="cb31-4"><a href="#cb31-4" aria-hidden="true" tabindex="-1"></a>repeat_nos <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">26</span>, <span class="dv">74</span>)</span>
<span id="cb31-5"><a href="#cb31-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat "black" 26 times and "white" 74 times.</span></span>
<span id="cb31-6"><a href="#cb31-6" aria-hidden="true" tabindex="-1"></a>jury_pool <span class="ot"><-</span> <span class="fu">rep</span>(juror_types, repeat_nos)</span>
<span id="cb31-7"><a href="#cb31-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb31-8"><a href="#cb31-8" aria-hidden="true" tabindex="-1"></a>jury_pool</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] "black" "black" "black" "black" "black" "black" "black" "black" "black"
[10] "black" "black" "black" "black" "black" "black" "black" "black" "black"
[19] "black" "black" "black" "black" "black" "black" "black" "black" "white"
[28] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[37] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[46] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[55] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[64] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[73] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[82] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[91] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[100] "white"</code></pre>
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<p>We can use this vector of repeats of strings, to sample from. The result is easier to grasp, because we are using the string labels, instead of numbers:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Select one juror at random from the black / white pool.</span></span>
<span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a><span class="co"># replace=TRUE is redundant here, but we leave it for consistency.</span></span>
<span id="cb33-3"><a href="#cb33-3" aria-hidden="true" tabindex="-1"></a>one_juror <span class="ot"><-</span> <span class="fu">sample</span>(jury_pool, <span class="dv">1</span>, <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb33-4"><a href="#cb33-4" aria-hidden="true" tabindex="-1"></a>one_juror</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] "black"</code></pre>
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<p>We can select our full jury of 12 jurors, and see the results in a more obvious form:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Select one juror at random from the black / white pool.</span></span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a>one_jury <span class="ot"><-</span> <span class="fu">sample</span>(jury_pool, <span class="dv">12</span>, <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb35-3"><a href="#cb35-3" aria-hidden="true" tabindex="-1"></a>one_jury</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] "white" "white" "white" "white" "white" "white" "white" "black" "black"
[10] "white" "white" "black"</code></pre>
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Using the <code>size</code> argument to <code>sample</code>
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<p>In the code above, we have specified the <em>size</em> of the sample we want (12) with the second argument to <code>sample</code>. As you saw in <a href="resampling_with_code.html#sec-named-arguments" class="quarto-xref"><span>Section 5.8</span></a>, we can also give names to the function arguments, in this case, to make it clearer what we mean by “12” in the code above. In fact, from now on, that is what we will do; we will specify the <em>size</em> of our sample by using the <em>name</em> for the function argument to <code>sample</code> — <code>size</code> — like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb37"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb37-1"><a href="#cb37-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Select one juror at random from the black / white pool.</span></span>
<span id="cb37-2"><a href="#cb37-2" aria-hidden="true" tabindex="-1"></a><span class="co"># Specify the sample size using the "size" named argument.</span></span>
<span id="cb37-3"><a href="#cb37-3" aria-hidden="true" tabindex="-1"></a>one_jury <span class="ot"><-</span> <span class="fu">sample</span>(jury_pool, <span class="at">size=</span><span class="dv">12</span>, <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb37-4"><a href="#cb37-4" aria-hidden="true" tabindex="-1"></a>one_jury</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] "white" "white" "white" "white" "white" "black" "white" "white" "white"
[10] "white" "white" "white"</code></pre>
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<p>We can use <code>==</code> on the vector to get <code>TRUE</code> values where the juror was “black” and <code>FALSE</code> values otherwise:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb39"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb39-1"><a href="#cb39-1" aria-hidden="true" tabindex="-1"></a>are_black <span class="ot"><-</span> one_jury <span class="sc">==</span> <span class="st">'black'</span></span>
<span id="cb39-2"><a href="#cb39-2" aria-hidden="true" tabindex="-1"></a>are_black</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code> [1] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE</code></pre>
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<p>Finally, we can <code>sum</code> to find the number of black jurors (<a href="resampling_with_code.html#sec-count-with-sum" class="quarto-xref"><span>Section 5.15</span></a>):</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb41"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of black jurors in this simulated jury.</span></span>
<span id="cb41-2"><a href="#cb41-2" aria-hidden="true" tabindex="-1"></a>n_black <span class="ot"><-</span> <span class="fu">sum</span>(are_black)</span>
<span id="cb41-3"><a href="#cb41-3" aria-hidden="true" tabindex="-1"></a>n_black</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] 1</code></pre>
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<p>Putting that all together, this is our new procedure to select one jury and count the number of black jurors:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb43"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a>one_jury <span class="ot"><-</span> <span class="fu">sample</span>(jury_pool, <span class="at">size=</span><span class="dv">12</span>, <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb43-2"><a href="#cb43-2" aria-hidden="true" tabindex="-1"></a>are_black <span class="ot"><-</span> one_jury <span class="sc">==</span> <span class="st">'black'</span></span>
<span id="cb43-3"><a href="#cb43-3" aria-hidden="true" tabindex="-1"></a>n_black <span class="ot"><-</span> <span class="fu">sum</span>(are_black)</span>
<span id="cb43-4"><a href="#cb43-4" aria-hidden="true" tabindex="-1"></a>n_black</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] 4</code></pre>
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<p>Or we can be even more compact by putting several statements together into one line:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb45"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a><span class="co"># The same as above, but on one line.</span></span>
<span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a>n_black <span class="ot">=</span> <span class="fu">sum</span>(<span class="fu">sample</span>(jury_pool, <span class="at">size=</span><span class="dv">12</span>, <span class="at">replace=</span><span class="cn">TRUE</span>) <span class="sc">==</span> <span class="st">'black'</span>)</span>
<span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a>n_black</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] 4</code></pre>
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</section>
<section id="resampling-with-and-without-replacement" class="level2" data-number="7.7">
<h2 data-number="7.7" class="anchored" data-anchor-id="resampling-with-and-without-replacement">7.7 Resampling with and without replacement</h2>
<p>Now let us return to the details of Robert Swain’s case, that you first saw in <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a>.</p>
<p>We looked at the composition of Robert Swain’s 12-person jury — but in fact, by law, that does not have to be representative of the eligible jurors. The 12-person jury is drawn from a jury <em>panel</em>, of 100 people, and this should, in turn, be drawn from the population of all eligible jurors in the county, consisting, at the time, of “all male citizens in the community over 21 who are reputed to be honest, intelligent men and are esteemed for their integrity, good character and sound judgment.” So, unless there was some bias against black jurors, we might expect the 100-person jury panel to be a plausibly random sample of the eligible jurors, of whom 26% were black. See <a href="https://supreme.justia.com/cases/federal/us/380/202">the Supreme Court case judgement</a> for details.</p>
<p>In fact, in Robert Swain’s trial, there were 8 black members in the 100-person jury panel. We will leave it to you to adapt the simulation from <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a> to ask the question — is 8% surprising as a random sample from a population with 26% black people?</p>
<p>But we have a different question: given that 8 out of 100 of the jury panel were black, is it surprising that none of the 12-person jury were black? As usual, we can answer that question with simulation.</p>
<p>Let’s think about what a single simulated jury selection would look like.</p>
<p>First we compile a representation of the actual jury panel, using the tools we have used above.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb47"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a>juror_types <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">'black'</span>, <span class="st">'white'</span>)</span>
<span id="cb47-2"><a href="#cb47-2" aria-hidden="true" tabindex="-1"></a><span class="co"># in fact there were 8 black jurors and 92 white jurors.</span></span>
<span id="cb47-3"><a href="#cb47-3" aria-hidden="true" tabindex="-1"></a>panel_nos <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">8</span>, <span class="dv">92</span>)</span>
<span id="cb47-4"><a href="#cb47-4" aria-hidden="true" tabindex="-1"></a>jury_panel <span class="ot"><-</span> <span class="fu">rep</span>(juror_types, panel_nos)</span>
<span id="cb47-5"><a href="#cb47-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result</span></span>
<span id="cb47-6"><a href="#cb47-6" aria-hidden="true" tabindex="-1"></a>jury_panel</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] "black" "black" "black" "black" "black" "black" "black" "black" "white"
[10] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[19] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[28] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[37] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[46] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[55] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[64] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[73] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[82] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[91] "white" "white" "white" "white" "white" "white" "white" "white" "white"
[100] "white"</code></pre>
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</div>
<p>Now consider taking a 12-person jury at random from this panel. We select the first juror at random, so that juror has an 8 out of 100 chance of being black. But when we select the second jury member, the situation has changed slightly. We can’t select the first juror again, so our panel is now 99 people. If our first juror was black, then the chances of selecting another black juror next are not 8 out of 100, but 7 out of 99 — a smaller chance. The problem is, as we shall see in more detail later, the chances of getting a black juror as the second, and third and fourth members of the jury depend on whether we selected a black juror as the first and second and third jury members. At its most extreme, imagine we had already selected eight jurors, and by some strange chance, all eight were black. Now our chances of selecting a black juror as the ninth juror are zero — there are no black jurors left to select from the panel.</p>
<p>In this case we are selecting jurors from the panel <em>without replacement</em>, meaning, that once we have selected a particular juror, we cannot select them again, and we do not put them back into the panel when we select our next juror.</p>
<p>This is the probability equivalent of the situation when you are dealing a hand of cards. Let’s say someone is dealing you, and you only, a hand of five cards. You get an ace as your first card. Your chances of getting an ace as your first card were just the number of aces in the deck divided by the number of cards — four in 52 – <span class="math inline">\(\frac{4}{52}\)</span>. But for your second card, the probability has changed, because there is one less ace remaining in the pack, and one less card, so your chances of getting an ace as your second card are now <span class="math inline">\(\frac{3}{51}\)</span>. This is sampling without replacement — in a normal game, you can’t get the same card twice. Of course, you could imagine getting a hand where you sampled <em>with replacement</em>. In that case, you’d get a card, you’d write down what it was, and you’d give the card back to the dealer, who would <em>replace</em> the card in the deck, shuffle again, and give you another card.</p>
<p>As you can see, the chances change if you are sampling with or without replacement, and the kind of sampling you do, will dictate how you model your chances in your simulations.</p>
<p>Because this distinction is so common, and so important, the machinery you have already seen in <code>sample</code> has simple ways for you to select your sampling type. You have already seen sampling <em>with replacement</em>, and it looks like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb49"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Take a sample of 12 jurors from the panel *with replacement*</span></span>
<span id="cb49-2"><a href="#cb49-2" aria-hidden="true" tabindex="-1"></a>strange_jury <span class="ot"><-</span> <span class="fu">sample</span>(jury_panel, <span class="at">size=</span><span class="dv">12</span>, <span class="at">replace=</span><span class="cn">TRUE</span>)</span>
<span id="cb49-3"><a href="#cb49-3" aria-hidden="true" tabindex="-1"></a>strange_jury</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] "white" "white" "white" "white" "black" "white" "white" "white" "white"
[10] "white" "white" "white"</code></pre>
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<p>This is a strange jury, because it can select any member of the jury pool more than once. Perhaps that juror would have to fill two (or more!) seats, or run quickly between them. But of course, that is not how juries are selected. They are selected <em>without replacement</em>:</p>
<div class="r">
<p>Thus far, we have always done sampling <em>with replacement</em>, and, in order to do that with <code>sample</code>, we pass the argument <code>replace=TRUE</code>. We do that because the default for <code>sample</code> is <code>replace=FALSE</code>, that is, by default, <code>sample</code> does sampling without replacement. If you want to do sampling without replacement, you can just omit the <code>replace=TRUE</code> argument to <code>sample</code>, or you can specify <code>replace=FALSE</code> explicitly, perhaps to remind yourself that this is sampling without replacement. Whether you omit the <code>replace</code> argument, or specify <code>replace=FALSE</code>, the behavior is the same.</p>
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb51"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb51-1"><a href="#cb51-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Take a sample of 12 jurors from the panel *with replacement*</span></span>
<span id="cb51-2"><a href="#cb51-2" aria-hidden="true" tabindex="-1"></a><span class="co"># replace=FALSE is the default for sample.</span></span>
<span id="cb51-3"><a href="#cb51-3" aria-hidden="true" tabindex="-1"></a>ok_jury <span class="ot"><-</span> <span class="fu">sample</span>(jury_panel, <span class="at">size=</span><span class="dv">12</span>)</span>
<span id="cb51-4"><a href="#cb51-4" aria-hidden="true" tabindex="-1"></a>ok_jury</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] "white" "white" "black" "white" "black" "white" "white" "white" "black"
[10] "white" "white" "white"</code></pre>
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<p>You have already seen comment lines. These are lines beginning with <code>#</code>, to signal to R that the rest of the line is text for humans to read, but R to ignore.</p>
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<div class="sourceCode cell-code" id="cb53"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb53-1"><a href="#cb53-1" aria-hidden="true" tabindex="-1"></a><span class="co"># This is a comment. R ignores this line.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<p>You can also put comments at the <em>end of code lines</em>, by finishing the code part of the line, and then putting a <code>#</code>, followed by more text. Again, R will ignore everything after the <code>#</code> as a text for humans, but not for R.</p>
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<div class="sourceCode cell-code" id="cb54"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(<span class="st">'Hello'</span>) <span class="co"># This is a comment at the end of the line.</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Hello</code></pre>
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<p>To finish the procedure for simulating a single jury selection, we count the number of black jurors:</p>
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<div class="sourceCode cell-code" id="cb56"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb56-1"><a href="#cb56-1" aria-hidden="true" tabindex="-1"></a>n_black <span class="ot"><-</span> <span class="fu">sum</span>(ok_jury <span class="sc">==</span> <span class="st">'black'</span>) <span class="co"># How many black jurors?</span></span>
<span id="cb56-2"><a href="#cb56-2" aria-hidden="true" tabindex="-1"></a>n_black</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>[1] 3</code></pre>
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<p>Now we have the procedure for one simulated trial, here is the procedure for 10,000 simulated trials.</p>
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<div class="sourceCode cell-code" id="cb58"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb58-1"><a href="#cb58-1" aria-hidden="true" tabindex="-1"></a>counts <span class="ot"><-</span> <span class="fu">numeric</span>(<span class="dv">10000</span>)</span>
<span id="cb58-2"><a href="#cb58-2" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="dv">10000</span>) {</span>
<span id="cb58-3"><a href="#cb58-3" aria-hidden="true" tabindex="-1"></a> <span class="co"># Single trial procedure</span></span>
<span id="cb58-4"><a href="#cb58-4" aria-hidden="true" tabindex="-1"></a> jury <span class="ot"><-</span> <span class="fu">sample</span>(jury_panel, <span class="at">size=</span><span class="dv">12</span>) <span class="co"># replace=FALSE is the default.</span></span>
<span id="cb58-5"><a href="#cb58-5" aria-hidden="true" tabindex="-1"></a> n_black <span class="ot"><-</span> <span class="fu">sum</span>(jury <span class="sc">==</span> <span class="st">'black'</span>) <span class="co"># How many black jurors?</span></span>
<span id="cb58-6"><a href="#cb58-6" aria-hidden="true" tabindex="-1"></a> <span class="co"># Store the result</span></span>
<span id="cb58-7"><a href="#cb58-7" aria-hidden="true" tabindex="-1"></a> counts[i] <span class="ot"><-</span> n_black</span>
<span id="cb58-8"><a href="#cb58-8" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb58-9"><a href="#cb58-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Number of juries with 0 black jurors.</span></span>
<span id="cb58-10"><a href="#cb58-10" aria-hidden="true" tabindex="-1"></a>zero_black <span class="ot"><-</span> <span class="fu">sum</span>(counts <span class="sc">==</span> <span class="dv">0</span>)</span>
<span id="cb58-11"><a href="#cb58-11" aria-hidden="true" tabindex="-1"></a><span class="co"># Proportion</span></span>
<span id="cb58-12"><a href="#cb58-12" aria-hidden="true" tabindex="-1"></a>p_zero_black <span class="ot"><-</span> zero_black <span class="sc">/</span> <span class="dv">10000</span></span>
<span id="cb58-13"><a href="#cb58-13" aria-hidden="true" tabindex="-1"></a><span class="fu">message</span>(p_zero_black)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>0.3375</code></pre>
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<p>We have found that, when there are only 8% black jurors in the jury panel, having no black jurors in the final jury happens about 34% of the time, even in this case, where the jury is selected completely at random from the jury panel.</p>
<p>We should look for the main source of bias in the initial selection of the jury panel, not in the selection of the jury from the panel.</p>
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End of notebook: Sampling tools
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<p><code>sampling_tools</code> starts at <a href="#nte-sampling_tools" class="quarto-xref">Note <span>7.1</span></a>.</p>
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With or without replacement for the original jury selection
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<p>You may have noticed in <a href="resampling_with_code2.html" class="quarto-xref"><span>Chapter 6</span></a> that we were sampling Robert Swain’s jury from the eligible pool of jurors, <em>with replacement</em>. You might reasonably ask whether we should have selected from the eligible jurors <em>without replacement</em>, given that the same juror cannot serve more than once in the same jury, and therefore, the same argument applies there as here.</p>
<p>The trick there was that we were selecting from a very large pool of many thousand eligible jurors, of whom 26% were black. Let’s say there were 10,000 eligible jurors, of whom 2,600 were black. When selecting the first juror, there is exactly a 2,600 in 10,000 chance of getting a black juror — 26%. If we do get a black juror first, then the chance that the second juror will be black has changed slightly, 2,599 in 9,999. But these changes are very small; even if we select eleven black jurors out of eleven, when we come to the twelfth juror, we still have a 2,589 out of 9,989 chance of getting another black juror, and that works out at a 25.92% chance — hardly changed from the original 26%. So yes, you’d be right, we really should have compiled our population of 2,600 black jurors and 7,400 white jurors, and then sampled without replacement from that population, but as the resulting sample probabilities will be very similar to the simpler sampling with replacement, we chose to try and slide that one quietly past you, in the hope you would forgive us when you realized.</p>
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<section id="conclusion" class="level2" data-number="7.8">
<h2 data-number="7.8" class="anchored" data-anchor-id="conclusion"><span class="header-section-number">7.8</span> Conclusion</h2>
<p>This chapter introduced you to the idea of strings — values in R that store bits of text. Strings are very useful as labels for the entities we are sampling from, when we do our simulations. Strings are particularly useful when we use them with vectors, and one way we often do that is to build up vectors of strings to sample from, using the <span class="r"><code>rep</code></span> function.</p>
<p>There is a fundamental distinction between two different types of sampling — sampling <em>with replacement</em>, where we draw an element from a larger pool, then put that element back before drawing again, and sampling <em>without replacement</em>, where we remove the element from the remaining pool when we draw it into the sample. As we will see later, it is often a judgment call which of these two types of sampling is a more reasonable model of the world you are trying to simulate.</p>
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