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<li><a href="#sec-logical-operators" id="toc-sec-logical-operators" class="nav-link active" data-scroll-target="#sec-logical-operators"><span class="header-section-number">23.1</span> Logical operators</a></li>
<li><a href="#comparisons-among-more-than-two-samples-of-counted-data" id="toc-comparisons-among-more-than-two-samples-of-counted-data" class="nav-link" data-scroll-target="#comparisons-among-more-than-two-samples-of-counted-data"><span class="header-section-number">23.2</span> Comparisons among more than two samples of counted data</a>
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<li><a href="#sec-fruitfly-four" id="toc-sec-fruitfly-four" class="nav-link" data-scroll-target="#sec-fruitfly-four"><span class="header-section-number">23.2.1</span> Example: Do Any of Four Treatments Affect Sex Ratio in Fruit Flies?</a></li>
<li><a href="#sec-four-rehabs" id="toc-sec-four-rehabs" class="nav-link" data-scroll-target="#sec-four-rehabs"><span class="header-section-number">23.2.2</span> Example: Do Four Psychological Treatments Differ in Effectiveness?</a></li>
<li><a href="#example-three-way-comparison" id="toc-example-three-way-comparison" class="nav-link" data-scroll-target="#example-three-way-comparison"><span class="header-section-number">23.2.3</span> Example: Three-way Comparison</a></li>
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<li><a href="#paired-comparisons-with-counted-data" id="toc-paired-comparisons-with-counted-data" class="nav-link" data-scroll-target="#paired-comparisons-with-counted-data"><span class="header-section-number">23.3</span> Paired Comparisons With Counted Data</a>
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<li><a href="#sec-pig-pairs" id="toc-sec-pig-pairs" class="nav-link" data-scroll-target="#sec-pig-pairs"><span class="header-section-number">23.3.1</span> Example: the pig rations again, but comparing pairs of pigs</a></li>
<li><a href="#example-merged-firms-compared-to-two-non-merged-groups" id="toc-example-merged-firms-compared-to-two-non-merged-groups" class="nav-link" data-scroll-target="#example-merged-firms-compared-to-two-non-merged-groups"><span class="header-section-number">23.3.2</span> Example: merged firms compared to two non-merged groups</a></li>
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<li><a href="#technical-note" id="toc-technical-note" class="nav-link" data-scroll-target="#technical-note"><span class="header-section-number">23.4</span> Technical note</a></li>
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<h1 class="title"><span class="chapter-number">23</span> <span class="chapter-title">The Statistics of Hypothesis-Testing with Counted Data, Part 2</span></h1>
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<p>Here’s the bad-news-good-news message again: The bad news is that the subject of inferential statistics is extremely difficult — not because it is complex but rather because it is subtle. The cause of the difficulty is that the world around us is difficult to understand, and spoon-fed mathematical simplifications which you manipulate mechanically simply mislead you into thinking you understand that about which you have not got a clue.</p>
<p>The good news is that you — and that means <em>you</em>, even if you say you are “no good at mathematics” — can understand these problems with a layperson’s hard thinking, even if you have no mathematical background beyond arithmetic and you think that you have no mathematical capability. That’s because the difficulty lies in such matters as pin-pointing the right question, and understanding how to interpret your results.</p>
<p>The problems in the previous chapter were tough enough. But this chapter considers problems with additional complications, such as when there are more than two groups, or paired comparisons for the same units of observation.</p>
<p>But first, we need another addition to our Python vocabulary.</p>
<section id="sec-logical-operators" class="level2" data-number="23.1">
<h2 data-number="23.1" class="anchored" data-anchor-id="sec-logical-operators"><span class="header-section-number">23.1</span> Logical operators</h2>
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Note 23.1: Notebook: Logical operators
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<p><a class="notebook-link" href="notebooks/logical_operators.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=logical_operators.ipynb">Interact</a></p>
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<div class="nb-start" name="logical_operators" title="Logical operators">
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<p>This section continues our programme of expanding the range of Python features that you can use to clear code. As we introduce each feature, we will use them in the following examples.</p>
<p>As motivation, we are about to do some simulations where we are interested in the number of some particular type of observations in each trial. For example, let’s do 10 coin tosses with <code>rnd.choice</code>:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="co"># For each element, heads of tails is equally likely.</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>coins <span class="op">=</span> rnd.choice([<span class="st">'heads'</span>, <span class="st">'tails'</span>], size<span class="op">=</span><span class="dv">10</span>)</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>coins</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>array(['tails', 'tails', 'tails', 'tails', 'tails', 'heads', 'tails',
'tails', 'tails', 'heads'], dtype='<U5')</code></pre>
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<p>Let us now say that we are interested to record if the trial had <em>either</em> 2 or fewer “heads” <em>or</em> two or fewer “tails”.</p>
<p>We could write it like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">2</span>:</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Trial is of interest'</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>Trial is of interest</code></pre>
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<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'tails'</span>) <span class="op"><=</span> <span class="dv">2</span>:</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Trial is of interest'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>It is a little repetitive to have to repeat the code identical code to print the same message for either of the two cases, and it would be even more repetitive if there were more lines of identical code to run for each of the two cases.</p>
<p>Python solves this problem with the <code>or</code> operator, like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">2</span> <span class="kw">or</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'tails'</span>) <span class="op"><=</span> <span class="dv">2</span>:</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Trial is of interest'</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>Trial is of interest</code></pre>
</div>
</div>
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Note 23.2: What is an operator?
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<p>Above, we called <code>or</code> an <em>operator</em>. An operator, for our purposes, is a special character , or a word, that sits between two values, and that tells Python how to combine these values.</p>
<p>For example <code>+</code> is an operator. When <code>+</code> sits between two numbers in code, Python interprets this to mean “take the two numbers on either side, and make a new number that is the result of adding the two numbers”:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="co"># + is an operator that, between two numbers, means "add the numbers".</span></span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a><span class="dv">1</span> <span class="op">+</span> <span class="dv">3</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>4</code></pre>
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<p><code>+</code>, <code>-</code>, <code>/</code> and <code>*</code> are all examples of operators that do <em>arithmetic</em> on the numbers to either side — they are <em>arithmetic</em> operators.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="co"># * is an operator that, between two numbers, means "multiply the numbers".</span></span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span> <span class="op">*</span> <span class="dv">4</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>8</code></pre>
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<p>We are about to use the operator <code>or</code>. <code>or</code> is a <em>logical</em> operator. It is a logical operator because it does not operate on <em>numbers</em> (as arithmetic operators do), but on <em>logical</em> (<em>Boolean</em>) values — values that can be either <code>True</code> or <code>False</code>.</p>
<p>For example, here we use <code>or</code>. to combine a <code>True</code> value (on the left) with a <code>False</code> value (on the right). It gives a result — <code>True</code>.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="va">True</span> <span class="kw">or</span> <span class="va">False</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>True</code></pre>
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<p><code>or</code> applies a very simple rule: if <em>either</em> the left-hand (LH) <em>or</em> the right-hand (RH) values are <code>True</code>, then <code>or</code> evaluates to <code>True</code>. Only if <em>neither</em> of the LH and RH values are <code>True</code>, does it return <code>False</code>.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Both LH and RH are True, return True.</span></span>
<span id="cb15-2"><a href="#cb15-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'True or True result:'</span>, <span class="va">True</span> <span class="kw">or</span> <span class="va">True</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>True or True result: True</code></pre>
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<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Only LH is True, return True.</span></span>
<span id="cb17-2"><a href="#cb17-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'True or False result:'</span>, <span class="va">True</span> <span class="kw">or</span> <span class="va">False</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>True or False result: True</code></pre>
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<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Only RH is True, return True.</span></span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'False or True result:'</span>, <span class="va">False</span> <span class="kw">or</span> <span class="va">True</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>False or True result: True</code></pre>
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<div class="sourceCode cell-code" id="cb21"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb21-1"><a href="#cb21-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Neither LH nor RH are True, return False.</span></span>
<span id="cb21-2"><a href="#cb21-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'False or False result:'</span>, <span class="va">False</span> <span class="kw">or</span> <span class="va">False</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>False or False result: False</code></pre>
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<p>Now let’s go back to the <code>if</code> statement above. The conditional part of the header line is:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb23"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb23-1"><a href="#cb23-1" aria-hidden="true" tabindex="-1"></a>np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">2</span> <span class="kw">or</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'tails'</span>) <span class="op"><=</span> <span class="dv">2</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>np.True_</code></pre>
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<p>This will be <code>True</code> <em>either</em> when there there are two or fewer “heads”, <em>or</em> when there are two or fewer tails. Therefore, when we use this conditional in an <code>if</code> statement, we make the <em>body</em> of the <code>if</code> statement run only if either of the two conditions are <code>True</code>.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb25"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb25-1"><a href="#cb25-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">2</span> <span class="kw">or</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'tails'</span>) <span class="op"><=</span> <span class="dv">2</span>:</span>
<span id="cb25-2"><a href="#cb25-2" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Trial is of interest'</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>Trial is of interest</code></pre>
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<p>While we are here, Python has another very useful logical operator: <code>and</code>.</p>
<p><code>and</code> takes the LH and RH values, and returns <code>True</code> only if <em>both</em> values are <code>True</code>.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb27"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb27-1"><a href="#cb27-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Both LH and RH are True, return True.</span></span>
<span id="cb27-2"><a href="#cb27-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'True and True result:'</span>, <span class="va">True</span> <span class="kw">and</span> <span class="va">True</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>True and True result: True</code></pre>
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<div class="sourceCode cell-code" id="cb29"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb29-1"><a href="#cb29-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Only LH is True, return False.</span></span>
<span id="cb29-2"><a href="#cb29-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'True and False result:'</span>, <span class="va">True</span> <span class="kw">and</span> <span class="va">False</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>True and False result: False</code></pre>
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<div class="sourceCode cell-code" id="cb31"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb31-1"><a href="#cb31-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Only RH is True, return False.</span></span>
<span id="cb31-2"><a href="#cb31-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'False and True result:'</span>, <span class="va">False</span> <span class="kw">and</span> <span class="va">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>False and True result: False</code></pre>
</div>
<div class="sourceCode cell-code" id="cb33"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb33-1"><a href="#cb33-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Neither LH nor RH are True, return False.</span></span>
<span id="cb33-2"><a href="#cb33-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'False and False result:'</span>, <span class="va">False</span> <span class="kw">and</span> <span class="va">False</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>False and False result: False</code></pre>
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<p>We could, for example, ask whether the number of heads is >=3 <em>and</em> <=7 (is in the range 3 through 7).</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb35"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb35-1"><a href="#cb35-1" aria-hidden="true" tabindex="-1"></a><span class="cf">if</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op">>=</span> <span class="dv">3</span> <span class="kw">and</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">7</span>:</span>
<span id="cb35-2"><a href="#cb35-2" aria-hidden="true" tabindex="-1"></a> <span class="bu">print</span>(<span class="st">'Trial is of interest'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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Python interval comparison
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<p>In fact, Python has a special shortcut syntax called <em>interval comparison</em> for that last question — whether a number is within a range. It looks like this:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Asks whether thee number of heads is >= 3 *and* <= 7.</span></span>
<span id="cb36-2"><a href="#cb36-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span> <span class="op"><=</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">7</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>np.False_</code></pre>
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<p>Notice the value at one end of the range the left (here, the lower value), then the comparison operator, then the value to compare, then another comparison operator, followed by the value at the other end of the range on the right.</p>
<p>The interval comparison above is a shortcut for the more verbose version we would need when using <code>and</code>:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Also asks whether thee number of heads is >= 3 *and* <= 7.</span></span>
<span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span> <span class="op"><=</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="kw">and</span> np.<span class="bu">sum</span>(coins <span class="op">==</span> <span class="st">'heads'</span>) <span class="op"><=</span> <span class="dv">7</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>np.False_</code></pre>
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End of notebook: Logical operators
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<p><code>logical_operators</code> starts at <a href="#nte-logical_operators" class="quarto-xref">Note <span>23.1</span></a>.</p>
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</section>
<section id="comparisons-among-more-than-two-samples-of-counted-data" class="level2" data-number="23.2">
<h2 data-number="23.2" class="anchored" data-anchor-id="comparisons-among-more-than-two-samples-of-counted-data"><span class="header-section-number">23.2</span> Comparisons among more than two samples of counted data</h2>
<section id="sec-fruitfly-four" class="level3" data-number="23.2.1">
<h3 data-number="23.2.1" class="anchored" data-anchor-id="sec-fruitfly-four"><span class="header-section-number">23.2.1</span> Example: Do Any of Four Treatments Affect Sex Ratio in Fruit Flies?</h3>
<p>This is an example of the general problem — <strong>when the benchmark universe proportion is known, is the proportion of the binomial population affected by any of the treatments?</strong></p>
<p>Suppose that, instead of experimenting with just one type of radiation treatment on the flies (as in <a href="testing_counts_1.html#sec-fruitfly" class="quarto-xref"><span>Section 21.2.1</span></a>), you try <em>four</em> different treatments, which we shall label A, B, C, and D. Treatment A produces fourteen males and six females, but treatments B, C, and D produce ten, eleven, and ten males, respectively. It is immediately obvious that there is no reason to think that treatment B, C, or D affects the sex ratio. But what about treatment A?</p>
<p>A frequent and dangerous mistake made by young scientists is to scrounge around in the data for the most extreme result, and then treat it as if it were the only result. In the context of this example, it would be fallacious to think that the probability of the fourteen-males-to-six females split observed for treatment A is the same as the probability that we figured for a <em>single</em> experiment in the example <a href="testing_counts_1.html#sec-fruitfly" class="quarto-xref"><span>Section 21.2.1</span></a>. Instead, we must consider that our benchmark universe is composed of <em>four sets</em> of twenty trials, each trial having a 50-50 probability of being male. We can consider that our previous trials 1-4 in <a href="testing_counts_1.html#sec-fruitfly" class="quarto-xref"><span>Section 21.2.1</span></a> constitute a <em>single</em> new trial, and each subsequent set of four previous trials constitute another new trial. We then ask how likely a new trial of our sets of twenty flips is to produce <em>one</em> set with fourteen or more of one or the other sex.</p>
<p>Let us make the procedure explicit, starting at the procedure from <a href="testing_counts_1.html#sec-fruitfly" class="quarto-xref"><span>Section 21.2.1</span></a>. Again, we will check for 14 or more males, <em>or</em> 6 or fewer males (meaning, 14 or more females).</p>
<ul>
<li><strong>Step 1.</strong> Let tails = male, heads = female.</li>
<li><strong>Step 2.</strong> Flip twenty coins and count the number of tails (males). Call this the count for group A. Repeat three more times to get counts for groups B, C and D.</li>
<li><strong>Step 5.</strong> If <em>any</em> of the group counts for A, or B or C or D is 14 ore more then record “Yes”, or if <em>any</em> of the group counts A, B, C, D are 6 or less (meaning >= 14 females), record “yes”. If neither is true, record “No”.</li>
<li><strong>Step 4.</strong> Repeat steps 2 and 3 perhaps 100 times.</li>
<li><strong>Step 5.</strong> Calculate the proportion of “yes” results in the 100 trials. This proportion estimates the probability that a fruit fly population with a proportion of 50 percent males will produce as many as 14 males, or as many as 14 females, in at least one of four groups of 20 flies.</li>
</ul>
<p>We begin the trials with data as in <a href="#tbl-fruitfly-4-groups" class="quarto-xref">Table <span>23.1</span></a>. In two of the six simulation trials, one or more one samples (groups) shows 14 or more males. Without even concerning ourselves about whether we should be looking at males or females, or just males, or needing to do more trials, we can see that it would be <em>very common indeed</em> to have one of four treatments show fourteen or more of one sex just by chance. This discovery clearly indicates that a result that would be fairly unusual (five in 25) for a single sample alone is commonplace in one of four observed samples.</p>
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<div id="tbl-fruitfly-4-groups" class="lightable-paper lightable-striped lightable-hover quarto-float quarto-figure quarto-figure-center anchored" style="font-family: "Arial Narrow", arial, helvetica, sans-serif; width: auto !important; margin-left: auto; margin-right: auto;" data-quarto-postprocess="true">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-fruitfly-4-groups-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 23.1: Results from 6 random trials for Fruitfly 4-group problem
</figcaption>
<div aria-describedby="tbl-fruitfly-4-groups-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="lightable-paper lightable-striped lightable-hover caption-top table table-sm table-striped small" data-quarto-postprocess="true">
<thead>
<tr class="header">
<th style="text-align: right;" data-quarto-table-cell-role="th">Trial no</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">Count for group A</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">Count for group B</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">Count for group C</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">Count for group D</th>
<th style="text-align: left;" data-quarto-table-cell-role="th">Any >=14 or any <=6</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">1</td>
<td style="text-align: right;">12</td>
<td style="text-align: right;">12</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">11</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">2</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">10</td>
<td style="text-align: right;">11</td>
<td style="text-align: right;">6</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="odd">
<td style="text-align: right;">3</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">10</td>
<td style="text-align: right;">11</td>
<td style="text-align: right;">12</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">4</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">15</td>
<td style="text-align: right;">13</td>
<td style="text-align: right;">9</td>
<td style="text-align: left;">Yes</td>
</tr>
<tr class="odd">
<td style="text-align: right;">5</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">10</td>
<td style="text-align: right;">10</td>
<td style="text-align: right;">9</td>
<td style="text-align: left;">No</td>
</tr>
<tr class="even">
<td style="text-align: right;">6</td>
<td style="text-align: right;">12</td>
<td style="text-align: right;">11</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">16</td>
<td style="text-align: left;">Yes</td>
</tr>
</tbody>
</table>
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</figure>
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Note 23.3: Notebook: Fruit fly simulation of four groups
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<p><a class="notebook-link" href="notebooks/fruit_fly4.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=fruit_fly4.ipynb">Interact</a></p>
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<div class="nb-start" name="fruit_fly4" title="Fruit fly simulation of four groups">
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<p>A key point of the notebook here is that each trial consists of <em>four groups</em> of 20 randomly generated hypothetical fruit flies. And if we consider 10,000 trials, we will be examining 40,000 sets of 20 fruit flies.</p>
<p>In each trial we generate 4 random samples (groups) of 20 fruit flies, and for each, we count the number of males (“males”s) and then check whether that group has more than 13 of either sex (actually, more than 13 “males”s or less than 7 “males”). If it does, then we change a variable called <code>unusual</code> to 1, which informs us that for this sample, at least 1 group of 20 fruit flies had results as unusual as the results from the fruit flies exposed to the four treatments.</p>
<p>After the 10,000 runs are made, we count the number of trials where one sample had a group of fruit flies with 14 or more of either sex, and show the results.</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb40-2"><a href="#cb40-2" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb40-3"><a href="#cb40-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb40-4"><a href="#cb40-4" aria-hidden="true" tabindex="-1"></a>n_iters <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb40-5"><a href="#cb40-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb40-6"><a href="#cb40-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Make array to store results for each trial.</span></span>
<span id="cb40-7"><a href="#cb40-7" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_iters)</span>
<span id="cb40-8"><a href="#cb40-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb40-9"><a href="#cb40-9" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_iters):</span>
<span id="cb40-10"><a href="#cb40-10" aria-hidden="true" tabindex="-1"></a> <span class="co"># unusual indicates whether we have obtained any trial group with more</span></span>
<span id="cb40-11"><a href="#cb40-11" aria-hidden="true" tabindex="-1"></a> <span class="co"># than 13 of either sex. We start at 0 (= no).</span></span>
<span id="cb40-12"><a href="#cb40-12" aria-hidden="true" tabindex="-1"></a> unusual <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb40-13"><a href="#cb40-13" aria-hidden="true" tabindex="-1"></a> <span class="co"># Repeat the following steps 4 times to constitute 4 trial groups</span></span>
<span id="cb40-14"><a href="#cb40-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># (representing treatments A, B, C, and D) of 20 flies each.</span></span>
<span id="cb40-15"><a href="#cb40-15" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> j <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">4</span>):</span>
<span id="cb40-16"><a href="#cb40-16" aria-hidden="true" tabindex="-1"></a> flies <span class="op">=</span> rnd.choice([<span class="st">'male'</span>, <span class="st">'female'</span>], size<span class="op">=</span><span class="dv">20</span>)</span>
<span id="cb40-17"><a href="#cb40-17" aria-hidden="true" tabindex="-1"></a> n_males <span class="op">=</span> np.<span class="bu">sum</span>(flies <span class="op">==</span> <span class="st">'male'</span>)</span>
<span id="cb40-18"><a href="#cb40-18" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> n_males <span class="op">>=</span> <span class="dv">14</span> <span class="kw">or</span> n_males <span class="op"><=</span> <span class="dv">6</span>:</span>
<span id="cb40-19"><a href="#cb40-19" aria-hidden="true" tabindex="-1"></a> unusual <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb40-20"><a href="#cb40-20" aria-hidden="true" tabindex="-1"></a> <span class="co"># unusual now tells us whether we got a result as extreme as that</span></span>
<span id="cb40-21"><a href="#cb40-21" aria-hidden="true" tabindex="-1"></a> <span class="co"># observed (unusual == 1 if we did, unusual == 0 if not). We must</span></span>
<span id="cb40-22"><a href="#cb40-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># keep track of this result in the results variable, for each experiment.</span></span>
<span id="cb40-23"><a href="#cb40-23" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> unusual</span>
<span id="cb40-24"><a href="#cb40-24" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb40-25"><a href="#cb40-25" aria-hidden="true" tabindex="-1"></a><span class="co"># The number of trials for which at least one of the four tests</span></span>
<span id="cb40-26"><a href="#cb40-26" aria-hidden="true" tabindex="-1"></a><span class="co"># exceeded 13 males or 13 females.</span></span>
<span id="cb40-27"><a href="#cb40-27" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(results)</span>
<span id="cb40-28"><a href="#cb40-28" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n_iters</span>
<span id="cb40-29"><a href="#cb40-29" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb40-30"><a href="#cb40-30" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="st">'Proportion of trials with one or more group >=14 or <=6 :'</span>, kk)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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<pre><code>Proportion of trials with one or more group >=14 or <=6 : 0.3808</code></pre>
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End of notebook: Fruit fly simulation of four groups
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<p><code>fruit_fly4</code> starts at <a href="#nte-fruit_fly4" class="quarto-xref">Note <span>23.3</span></a>.</p>
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<p>In one set of 10,000 trials, there were more than 13 males or more than 13 females 38 percent of the time — clearly not an unusual occurrence.</p>
</section>
<section id="sec-four-rehabs" class="level3" data-number="23.2.2">
<h3 data-number="23.2.2" class="anchored" data-anchor-id="sec-four-rehabs"><span class="header-section-number">23.2.2</span> Example: Do Four Psychological Treatments Differ in Effectiveness?</h3>
<p><strong>Do Several Two-Outcome Samples Differ Among Themselves in Their Proportions?</strong></p>
<p>Consider four different psychological treatments designed to rehabilitate young offenders. Instead of a numerical test score, there is only a “yes” or a “no” answer as to whether the young person has kept their record clean or has gotten into trouble again. Call a clean record “success”. Label the treatments P, R, S, and T, each of which is administered to a separate group of twenty young offenders. The number of “success” outcomes per group has been: P, 17; R, 10; S, 10; T, 7. Is it improbable that all four groups come from the same universe?</p>
<p>This problem is like the placebo vs. cancer-cure problem, but now there are more than two samples. It is also like the four-sample irradiated-fruit flies example (<a href="#sec-fruitfly-four" class="quarto-xref"><span>Section 23.2.1</span></a>), except that now we are <em>not</em> asking whether any or some of the samples differ from a <em>given universe</em> (50-50 sex ratio in that case). Rather, we are now asking whether there are differences <em>among</em> the samples themselves. Please keep in mind that we are still dealing with two-outcome (success-or-failure, yes-or-no, well-or-sick) problems. The outcomes fall into categories, to which we give labels (“success” or “failure”, “well” or “sick”). Later we shall take up problems that are similar except that the outcomes are “quantitative” — in that the outcomes are numbers rather than labels.</p>
<p>If all four groups were drawn from the same universe, that universe has an estimated success rate of 17/20 + 10/20 + 10/20 + 7/20 = 44/80 = 55/100 = 55%, because the observed data <em>taken as a whole</em> constitute our best guess as to the nature of the universe from which they come — again, <em>if</em> they all come from the same universe. (Please think this matter over a bit, because it is important and subtle. It may help you to notice the absence of any <em>other</em> information about the universe from which they have all come, if they have come from the same universe.)</p>
<p>Therefore, select twenty two-digit numbers for each group from the random-number table, marking “success” for each number in the range 1 through 55 and “failure” for each number 56 through 100. Conduct a number of such trials. Then count the proportion of times that the difference between the highest and lowest groups is larger than the widest observed difference, the difference between P and T (17-7 = 10). In <a href="#tbl-offenders" class="quarto-xref">Table <span>23.2</span></a>, none of the first six trials shows anywhere near as large a difference as the observed range of 10, suggesting that it would be rare for four treatments that are “really” similar to show so great a difference. There is thus reason to believe that P and T differ in their effects.</p>
<div id="tbl-offenders" class="quarto-float quarto-figure quarto-figure-center anchored">
<figure class="quarto-float quarto-float-tbl figure">
<figcaption class="quarto-float-caption-top quarto-float-caption quarto-float-tbl" id="tbl-offenders-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 23.2: Results of Six Random Trials for Problem “offenders”
</figcaption>
<div aria-describedby="tbl-offenders-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="caption-top table">
<thead>
<tr class="header">
<th>Trial</th>
<th>P</th>
<th>R</th>
<th>S</th>
<th>T</th>
<th>Largest Minus Smallest</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1</td>
<td>11</td>
<td>9</td>
<td>8</td>
<td>12</td>
<td>4</td>
</tr>
<tr class="even">
<td>2</td>
<td>10</td>
<td>10</td>
<td>12</td>
<td>12</td>
<td>2</td>
</tr>
<tr class="odd">
<td>3</td>
<td>9</td>
<td>12</td>
<td>8</td>
<td>12</td>
<td>4</td>
</tr>
<tr class="even">
<td>4</td>
<td>9</td>
<td>11</td>
<td>12</td>
<td>10</td>
<td>3</td>
</tr>
<tr class="odd">
<td>5</td>
<td>10</td>
<td>10</td>
<td>11</td>
<td>12</td>
<td>1</td>
</tr>
<tr class="even">
<td>6</td>
<td>11</td>
<td>11</td>
<td>9</td>
<td>11</td>
<td>2</td>
</tr>
</tbody>
</table>
</div>
</figure>
</div>
<p>The strategy of the Python solution to “offenders” is similar to the strategy for previous problems in this chapter. The benchmark (null) hypothesis is that the treatments do not differ in their effects observed, and we estimate the probability that the observed results would occur by chance using the benchmark universe. The only new twist is that we must instruct the computer to find the groups with the highest and the lowest numbers of success rates.</p>
<p>Using Python, we generate four “treatments,” each represented by 20 labels. We draw these 20 labels from the choices “success” or “failure”, with a 55% chance of getting “success” and a 45% chance of “failure”. Follow along in the program for the rest of the procedure:</p>
<div id="nte-offenders" class="callout callout-style-default callout-note callout-titled">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
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<div class="callout-title-container flex-fill">
Note 23.4: Notebook: Simulation of offender strategies
</div>
</div>
<div class="callout-body-container callout-body">
<div class="nb-links">
<p><a class="notebook-link" href="notebooks/offenders.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=offenders.ipynb">Interact</a></p>
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</div>
</div>
<div class="nb-start" name="offenders" title="Simulation of offender strategies">
</div>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb42"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb42-1"><a href="#cb42-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb42-2"><a href="#cb42-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb42-3"><a href="#cb42-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-4"><a href="#cb42-4" aria-hidden="true" tabindex="-1"></a><span class="co"># set up the random number generator</span></span>
<span id="cb42-5"><a href="#cb42-5" aria-hidden="true" tabindex="-1"></a>rnd <span class="op">=</span> np.random.default_rng()</span>
<span id="cb42-6"><a href="#cb42-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-7"><a href="#cb42-7" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the number of trials</span></span>
<span id="cb42-8"><a href="#cb42-8" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10_000</span></span>
<span id="cb42-9"><a href="#cb42-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-10"><a href="#cb42-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Set the sample size for each trial</span></span>
<span id="cb42-11"><a href="#cb42-11" aria-hidden="true" tabindex="-1"></a>sample_size <span class="op">=</span> <span class="dv">20</span></span>
<span id="cb42-12"><a href="#cb42-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-13"><a href="#cb42-13" aria-hidden="true" tabindex="-1"></a><span class="co"># An empty array to store the trial results.</span></span>
<span id="cb42-14"><a href="#cb42-14" aria-hidden="true" tabindex="-1"></a>scores <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb42-15"><a href="#cb42-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-16"><a href="#cb42-16" aria-hidden="true" tabindex="-1"></a><span class="co"># Do 10000 trials</span></span>
<span id="cb42-17"><a href="#cb42-17" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n_trials):</span>
<span id="cb42-18"><a href="#cb42-18" aria-hidden="true" tabindex="-1"></a> <span class="co"># The first treatment group</span></span>
<span id="cb42-19"><a href="#cb42-19" aria-hidden="true" tabindex="-1"></a> a <span class="op">=</span> rnd.choice([<span class="st">'success'</span>, <span class="st">'failure'</span>], p<span class="op">=</span>[<span class="fl">0.55</span>, <span class="fl">0.45</span>], size<span class="op">=</span><span class="dv">20</span>)</span>
<span id="cb42-20"><a href="#cb42-20" aria-hidden="true" tabindex="-1"></a> <span class="co"># Count successes.</span></span>
<span id="cb42-21"><a href="#cb42-21" aria-hidden="true" tabindex="-1"></a> a_count <span class="op">=</span> np.<span class="bu">sum</span>(a <span class="op">==</span> <span class="st">'success'</span>)</span>
<span id="cb42-22"><a href="#cb42-22" aria-hidden="true" tabindex="-1"></a> <span class="co"># Second, third and fourth treatment groups, and successes.</span></span>
<span id="cb42-23"><a href="#cb42-23" aria-hidden="true" tabindex="-1"></a> b <span class="op">=</span> rnd.choice([<span class="st">'success'</span>, <span class="st">'failure'</span>], p<span class="op">=</span>[<span class="fl">0.55</span>, <span class="fl">0.45</span>], size<span class="op">=</span><span class="dv">20</span>)</span>
<span id="cb42-24"><a href="#cb42-24" aria-hidden="true" tabindex="-1"></a> b_count <span class="op">=</span> np.<span class="bu">sum</span>(b <span class="op">==</span> <span class="st">'success'</span>)</span>
<span id="cb42-25"><a href="#cb42-25" aria-hidden="true" tabindex="-1"></a> c <span class="op">=</span> rnd.choice([<span class="st">'success'</span>, <span class="st">'failure'</span>], p<span class="op">=</span>[<span class="fl">0.55</span>, <span class="fl">0.45</span>], size<span class="op">=</span><span class="dv">20</span>)</span>
<span id="cb42-26"><a href="#cb42-26" aria-hidden="true" tabindex="-1"></a> c_count <span class="op">=</span> np.<span class="bu">sum</span>(c <span class="op">==</span> <span class="st">'success'</span>)</span>
<span id="cb42-27"><a href="#cb42-27" aria-hidden="true" tabindex="-1"></a> d <span class="op">=</span> rnd.choice([<span class="st">'success'</span>, <span class="st">'failure'</span>], p<span class="op">=</span>[<span class="fl">0.55</span>, <span class="fl">0.45</span>], size<span class="op">=</span><span class="dv">20</span>)</span>
<span id="cb42-28"><a href="#cb42-28" aria-hidden="true" tabindex="-1"></a> d_count <span class="op">=</span> np.<span class="bu">sum</span>(d <span class="op">==</span> <span class="st">'success'</span>)</span>
<span id="cb42-29"><a href="#cb42-29" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-30"><a href="#cb42-30" aria-hidden="true" tabindex="-1"></a> <span class="co"># Now find all the pairwise differences in successes among the groups.</span></span>
<span id="cb42-31"><a href="#cb42-31" aria-hidden="true" tabindex="-1"></a> a_minus_b <span class="op">=</span> a_count <span class="op">-</span> b_count</span>
<span id="cb42-32"><a href="#cb42-32" aria-hidden="true" tabindex="-1"></a> a_minus_c <span class="op">=</span> a_count <span class="op">-</span> c_count</span>
<span id="cb42-33"><a href="#cb42-33" aria-hidden="true" tabindex="-1"></a> a_minus_d <span class="op">=</span> a_count <span class="op">-</span> d_count</span>
<span id="cb42-34"><a href="#cb42-34" aria-hidden="true" tabindex="-1"></a> b_minus_c <span class="op">=</span> b_count <span class="op">-</span> c_count</span>
<span id="cb42-35"><a href="#cb42-35" aria-hidden="true" tabindex="-1"></a> b_minus_d <span class="op">=</span> b_count <span class="op">-</span> d_count</span>
<span id="cb42-36"><a href="#cb42-36" aria-hidden="true" tabindex="-1"></a> c_minus_d <span class="op">=</span> c_count <span class="op">-</span> d_count</span>
<span id="cb42-37"><a href="#cb42-37" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb42-38"><a href="#cb42-38" aria-hidden="true" tabindex="-1"></a> <span class="co"># Concatenate, or join, all the differences in a single array "diffs".</span></span>