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<li><a href="#preview" id="toc-preview" class="nav-link active" data-scroll-target="#preview"><span class="header-section-number">29.1</span> Preview</a></li>
<li><a href="#introduction-to-correlation-and-causation" id="toc-introduction-to-correlation-and-causation" class="nav-link" data-scroll-target="#introduction-to-correlation-and-causation"><span class="header-section-number">29.2</span> Introduction to correlation and causation</a></li>
<li><a href="#a-note-on-association-compared-to-testing-a-hypothesis" id="toc-a-note-on-association-compared-to-testing-a-hypothesis" class="nav-link" data-scroll-target="#a-note-on-association-compared-to-testing-a-hypothesis"><span class="header-section-number">29.3</span> A Note on Association Compared to Testing a Hypothesis</a>
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<li><a href="#sec-athletic-iq-ranks" id="toc-sec-athletic-iq-ranks" class="nav-link" data-scroll-target="#sec-athletic-iq-ranks"><span class="header-section-number">29.3.1</span> Example: Is Athletic Ability Directly Related to Intelligence?</a></li>
<li><a href="#example-athletic-ability-and-i.q.-a-third-way" id="toc-example-athletic-ability-and-i.q.-a-third-way" class="nav-link" data-scroll-target="#example-athletic-ability-and-i.q.-a-third-way"><span class="header-section-number">29.3.2</span> Example: Athletic ability and I.Q. — a third way</a></li>
</ul></li>
<li><a href="#sec-sum-products" id="toc-sec-sum-products" class="nav-link" data-scroll-target="#sec-sum-products"><span class="header-section-number">29.4</span> Correlation with sum of products</a>
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<li><a href="#sec-ath-iq-sop" id="toc-sec-ath-iq-sop" class="nav-link" data-scroll-target="#sec-ath-iq-sop"><span class="header-section-number">29.4.1</span> Example: sum of products correlation of athletic and IQ scores</a></li>
<li><a href="#example-correlation-between-adherence-to-medication-regime-and-change-in-cholesterol" id="toc-example-correlation-between-adherence-to-medication-regime-and-change-in-cholesterol" class="nav-link" data-scroll-target="#example-correlation-between-adherence-to-medication-regime-and-change-in-cholesterol"><span class="header-section-number">29.4.2</span> Example: Correlation Between Adherence to Medication Regime and Change in Cholesterol</a></li>
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<li><a href="#sec-correlation-coefficient" id="toc-sec-correlation-coefficient" class="nav-link" data-scroll-target="#sec-correlation-coefficient"><span class="header-section-number">29.5</span> The correlation coefficient</a>
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<li><a href="#correlations-are-symmetrical" id="toc-correlations-are-symmetrical" class="nav-link" data-scroll-target="#correlations-are-symmetrical"><span class="header-section-number">29.5.1</span> Correlations are symmetrical</a></li>
<li><a href="#the-correlation-coefficient-in" id="toc-the-correlation-coefficient-in" class="nav-link" data-scroll-target="#the-correlation-coefficient-in"><span class="header-section-number">29.5.2</span> The correlation coefficient in Python</a></li>
<li><a href="#test-linear-association-with-the-correlation-coefficient" id="toc-test-linear-association-with-the-correlation-coefficient" class="nav-link" data-scroll-target="#test-linear-association-with-the-correlation-coefficient"><span class="header-section-number">29.5.3</span> Test linear association with the correlation coefficient</a></li>
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<li><a href="#sec-counted-association" id="toc-sec-counted-association" class="nav-link" data-scroll-target="#sec-counted-association"><span class="header-section-number">29.6</span> Testing for a relationship between counted-data variables</a>
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<li><a href="#example-drinking-beer-and-being-in-favor-of-selling-beer" id="toc-example-drinking-beer-and-being-in-favor-of-selling-beer" class="nav-link" data-scroll-target="#example-drinking-beer-and-being-in-favor-of-selling-beer"><span class="header-section-number">29.6.1</span> Example: Drinking Beer And Being In Favor of Selling Beer</a></li>
<li><a href="#example-do-athletes-really-have-slumps" id="toc-example-do-athletes-really-have-slumps" class="nav-link" data-scroll-target="#example-do-athletes-really-have-slumps"><span class="header-section-number">29.6.2</span> Example: do athletes really have “slumps”?</a></li>
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<li><a href="#exercises" id="toc-exercises" class="nav-link" data-scroll-target="#exercises"><span class="header-section-number">29.7</span> Exercises</a>
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<li><a href="#sec-exr-voter-participation" id="toc-sec-exr-voter-participation" class="nav-link" data-scroll-target="#sec-exr-voter-participation"><span class="header-section-number">29.7.1</span> Exercise: voter participation</a></li>
<li><a href="#sec-exr-runs-strikeouts" id="toc-sec-exr-runs-strikeouts" class="nav-link" data-scroll-target="#sec-exr-runs-strikeouts"><span class="header-section-number">29.7.2</span> Exercise: association of runs and strikeouts</a></li>
<li><a href="#sec-exr-runs-strikeouts-r" id="toc-sec-exr-runs-strikeouts-r" class="nav-link" data-scroll-target="#sec-exr-runs-strikeouts-r"><span class="header-section-number">29.7.3</span> Exercise: runs, strikeouts, correlation coefficient</a></li>
<li><a href="#sec-exr-money-exchange" id="toc-sec-exr-money-exchange" class="nav-link" data-scroll-target="#sec-exr-money-exchange"><span class="header-section-number">29.7.4</span> Exercise: money and exchange rate</a></li>
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<h1 class="title"><span id="sec-correlation-causation" class="quarto-section-identifier"><span class="chapter-number">29</span> <span class="chapter-title">Correlation and Causation</span></span></h1>
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<section id="preview" class="level2" data-number="29.1">
<h2 data-number="29.1" class="anchored" data-anchor-id="preview"><span class="header-section-number">29.1</span> Preview</h2>
<p>The correlation (speaking in a loose way for now) between two variables measures the strength of the relationship between them. A positive “linear” correlation between two variables <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> implies that high values of <span class="math inline">\(x\)</span> are associated with high values of <span class="math inline">\(y\)</span>, and that low values of <span class="math inline">\(x\)</span> are associated with low values of <span class="math inline">\(y\)</span>. A negative correlation implies the opposite; high values of <span class="math inline">\(x\)</span> are associated with <em>low</em> values of <span class="math inline">\(y\)</span>. By definition a “correlation coefficient” (<a href="#sec-correlation-coefficient" class="quarto-xref"><span>Section 29.5</span></a>) close to zero indicates little or no linear relationship between two variables; correlation coefficients close to 1 and -1 denote a strong positive or negative relationship. We will start by using a simpler measure of correlation than the correlation coefficient, however.</p>
<p>One way to measure correlation with the resampling method is to rank both variables from highest to lowest, and investigate how often in randomly-generated samples the rankings of the two variables are as close to each other as the rankings in the observed variables. A better approach, because it uses more of the quantitative information contained in the data — though it requires more computation — is to multiply the values for the corresponding pairs of values for the two variables, and compare the sum of the resulting products to the analogous sum for randomly-generated pairs of the observed variable values (<a href="#sec-sum-products" class="quarto-xref"><span>Section 29.4</span></a>). The last section of the chapter (<a href="#sec-counted-association" class="quarto-xref"><span>Section 29.6</span></a>) shows how the strength of a relationship can be determined when the data are counted, rather than measured. First comes some discussion of the philosophical issues involved in correlation and causation.</p>
</section>
<section id="introduction-to-correlation-and-causation" class="level2" data-number="29.2">
<h2 data-number="29.2" class="anchored" data-anchor-id="introduction-to-correlation-and-causation"><span class="header-section-number">29.2</span> Introduction to correlation and causation</h2>
<p>The questions in examples <a href="probability_theory_3.html#sec-birthday-problem" class="quarto-xref"><span>Section 12.1</span></a> to <a href="probability_theory_4_finite.html#sec-fifteen-bridge" class="quarto-xref"><span>Section 13.3.3</span></a> have been stated in the following form: Does the independent variable (say, irradiation; or type of pig ration) have an effect upon the dependent variable (say, sex of fruit flies; or weight gain of pigs)? This is another way to state the following question: Is there a <em>causal relationship</em> between the independent variable(s) and the dependent variable? (“Independent” or “control” is the name we give to the variable(s) the researcher believes is (are) responsible for changes in the other variable, which we call the “dependent” or “response” variable.)</p>
<p>A causal relationship cannot be defined perfectly neatly. Even an experiment does not determine perfectly whether a relationship deserves to be called “causal” because, among other reasons, the independent variable may not be clear-cut. For example, even if cigarette smoking experimentally produces cancer in rats, it might be the paper and not the tobacco that causes the cancer. Or consider the fabled gentlemen who got experimentally drunk on bourbon and soda on Monday night, scotch and soda on Tuesday night, and brandy and soda on Wednesday night — and stayed sober on Thursday night by drinking nothing. With a vast inductive leap of scientific imagination, they treated their experience as an empirical demonstration that soda, the common element each evening, was the cause of the inebriated state they had experienced. Notice that their deduction was perfectly sound, given only the recent evidence they had. Other knowledge of the world is necessary to set them straight. That is, even in a controlled experiment there is often no way except subject-matter knowledge to avoid erroneous conclusions about causality. Nothing except substantive knowledge or scientific intuition would have led them to the recognition that it is the alcohol rather than the soda that made them drunk, <em>as long as they always took soda with their drinks</em>. And no statistical procedure can suggest to them that they ought to experiment with the presence and absence of soda. If this is true for an experiment, it must also be true for an uncontrolled study.</p>
<p>Here are some tests that a relationship usually must pass to be called causal. That is, a working definition of a particular causal relationship is expressed in a statement that has these important characteristics:</p>
<ol type="1">
<li><p>It is an association that is strong enough so that the observer believes it to have a predictive (explanatory) power great enough to be scientifically useful or interesting. For example, he is not likely to say that wearing glasses causes (or is a cause of) auto accidents if the observed <em>correlation coefficient</em> is .07<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>, even if the sample is large enough to make the correlation statistically convincing. In other words, unimportant relationships are not likely to be labeled causal.</p>
<p>Various observers may well differ in judging whether or not an association is strong enough to be important and therefore “causal.” And the particular field in which the observer works may affect this judgment. This is an indication that whether or not a relationship is dubbed “causal” involves a good deal of human judgment and is subject to dispute.</p></li>
<li><p>The “side conditions” must be sufficiently <em>few</em> and sufficiently observable so that the relationship will apply under a wide enough range of conditions to be considered useful or interesting. In other words, <em>the relationship must not require too many “if”s, “and”s, and “but”s in order to hold</em>. For example, one might say that an increase in income caused an increase in the birth rate if this relationship were observed everywhere. But, if the relationship were found to hold only in developed countries, among the educated classes, and among the higher-income groups, then it would be less likely to be called “causal” — even if the correlation were extremely high once the specified conditions had been met. A similar example can be made of the relationship between income and happiness.</p></li>
<li><p>For a relationship to be called “causal,” there should be sound reason to believe that, even if the control variable were not the “real” cause (and it never is), other relevant “hidden” and “real” cause variables must also change <em>consistently</em> with changes in the control variables. That is, a variable being manipulated may reasonably be called “causal” if the real variable for which it is believed to be a proxy must always be tied intimately to it. (Between two variables, v and w, v may be said to be the “more real” cause and <em>w</em> a “spurious” cause, if <em>v</em> and <em>w</em> require the same side conditions, except that <em>v</em> does not require <em>w</em> as a side condition.) This third criterion (non-spuriousness) is of particular importance to policy makers. The difference between it and the previous criterion for side conditions is that a plenitude of very restrictive side conditions may take the relationship out of the class of causal relationships, <em>even though the effects of the side conditions are known</em>. This criterion of nonspuriousness concerns variables that are as yet <em>unknown and unevaluated</em> but that have a <em>possible</em> ability to <em>upset</em> the observed association.</p>
<p>Examples of spurious relationships and hidden-third-factor causation are commonplace. For a single example, toy sales rise in December. There is no danger in saying that December causes an increase in toy sales, even though it is “really” Christmas that causes the increase, because Christmas and December practically always accompany each other.</p>
<p>Belief that the relationship is not spurious is increased if <em>many</em> likely variables have been investigated and none removes the relationship. This is further demonstration that the test of whether or not an association should be called “causal” cannot be a logical one; there is no way that one can express in symbolic logic the fact that many other variables have been tried without changing the relationship in question.</p></li>
<li><p>The more tightly a relationship is bound into (that is, deduced from, compatible with, and logically connected to) a general framework of theory, the stronger is its claim to be called “causal.” For an economics example, observed positive relationships between the interest rate and business investment and between profits and investment are more likely to be called “causal” than is the relationship between liquid assets and investment. This is so because the first two statements can be deduced from classical price theory, whereas the third statement cannot. Connection to a theoretical framework provides support for belief that the side conditions necessary for the statement to hold true are not restrictive and that the likelihood of spurious correlation is not great; because a statement is logically connected to the rest of the system, the statement tends to stand or fall as the rest of the system stands or falls. And, because the rest of the system of economic theory has, over a long period of time and in a wide variety of tests, been shown to have predictive power, a statement connected with it is cloaked in this mantle.</p></li>
</ol>
<p>The social sciences other than economics do not have such well-developed bodies of deductive theory, and therefore this criterion of causality does not weigh as heavily in sociology, for instance, as in economics. Rather, the other social sciences seem to substitute a weaker and more general criterion, that is, whether or not the statement of the relationship is accompanied by other statements that seem to “explain” the “mechanism” by which the relationship operates. Consider, for example, the relationship between the phases of the moon and the suicide rate. The reason that sociologists do not call it causal is that there are no auxiliary propositions that explain the relationship and describe an operative mechanism. On the other hand, the relationship between broken homes and youth crime is often referred to as “causal,” presumably because a large body of psychological theory serves to explain why a child raised without one or the other parent, or in the presence of parental strife, should not adjust readily.</p>
<p>Furthermore, one can never decide with perfect certainty whether in any <em>given</em> situation one variable “causes” a particular change in another variable. At best, given your particular purposes in investigating a phenomena, you may be safe in judging that very likely there is causal influence.</p>
<p>In brief, it is correct to say (as it is so often said) that correlation does not prove causation — if we add the word “completely” to make it “correlation does not <em>completely</em> prove causation.” On the other hand, causation can <em>never</em> be “proven” <em>completely</em> by correlation <em>or any other</em> tool or set of tools, including experimentation. The best we can do is make informed judgments about whether to call a relationship causal.</p>
<p>It is clear, however, that in any situation where we are interested in the possibility of causation, we must <em>at least</em> know whether there is a relationship (correlation) between the variables of interest; the existence of a relationship is necessary for a relationship to be judged causal even if it is not sufficient to receive the causal label. And in other situations where we are not even interested in causality, but rather simply want to predict events or understand the structure of a system, we may be interested in the existence of relationships quite apart from questions about causations. Therefore our next set of problems deals with the probability of there being a relationship between two measured variables, variables that can take on any values (say, the values on a test of athletic scores) rather than just two values (say, whether or not there has been irradiation.)<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a></p>
<p>Another way to think about such problems is to ask whether two variables are <em>independent</em> of each other — that is, whether you know anything about the value of one variable if you know the value of the other in a particular case — or whether they are not independent but rather are related.</p>
</section>
<section id="a-note-on-association-compared-to-testing-a-hypothesis" class="level2" data-number="29.3">
<h2 data-number="29.3" class="anchored" data-anchor-id="a-note-on-association-compared-to-testing-a-hypothesis"><span class="header-section-number">29.3</span> A Note on Association Compared to Testing a Hypothesis</h2>
<p>Problems in which we investigate a) whether there is an <em>association</em>, versus b) whether there is a <em>difference</em> between just two groups, often look very similar, especially when the data constitute a 2-by-2 table. There is this important difference between the two types of analysis, however: Questions about <em>association</em> refer to <em>variables</em> — say weight and age — and it never makes sense to ask whether there is a difference between variables (except when asking whether they measure the same quantity). Questions about <em>similarity or difference</em> refer to <em>groups of individuals</em>, and in such a situation it does make sense to ask whether or not two groups are observably different from each other.</p>
<section id="sec-athletic-iq-ranks" class="level3" data-number="29.3.1">
<h3 data-number="29.3.1" class="anchored" data-anchor-id="sec-athletic-iq-ranks"><span class="header-section-number">29.3.1</span> Example: Is Athletic Ability Directly Related to Intelligence?</h3>
<p>A more specific version of our question: <strong>is there correlation between the two variables or are they independent?</strong></p>
<p>A scientist often wants to know whether or not two characteristics go together, that is, whether or not they are correlated (that is, related or associated). For example, do young adults with high athletic ability tend to also have high I.Q.s?</p>
<p>Hypothetical physical-education scores of a group of ten high-school boys are shown in <a href="#tbl-physical-mental" class="quarto-xref">Table <span>29.1</span></a>, ordered from high to low, along with the I.Q. score for each boy. The ranks for each student’s athletic and I.Q. scores are then shown in the third and fourth columns:</p>
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<div id="tbl-physical-mental" class="lightable-paper lightable-striped lightable-hover quarto-float quarto-figure quarto-figure-center anchored" data-quarto-postprocess="true" style="font-family: "Arial Narrow", arial, helvetica, sans-serif; width: auto !important; margin-left: auto; margin-right: auto;">
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Table 29.1: Hypothetical athletic and I.Q. scores for high school boys
</figcaption>
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<th style="text-align: right;" data-quarto-table-cell-role="th">Athletic Score</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">I.Q. Score</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">Athletic Rank</th>
<th style="text-align: right;" data-quarto-table-cell-role="th">I.Q.Rank</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td style="text-align: right;">97</td>
<td style="text-align: right;">114</td>
<td style="text-align: right;">1</td>
<td style="text-align: right;">3</td>
</tr>
<tr class="even">
<td style="text-align: right;">94</td>
<td style="text-align: right;">120</td>
<td style="text-align: right;">2</td>
<td style="text-align: right;">1</td>
</tr>
<tr class="odd">
<td style="text-align: right;">93</td>
<td style="text-align: right;">107</td>
<td style="text-align: right;">3</td>
<td style="text-align: right;">7</td>
</tr>
<tr class="even">
<td style="text-align: right;">90</td>
<td style="text-align: right;">113</td>
<td style="text-align: right;">4</td>
<td style="text-align: right;">4</td>
</tr>
<tr class="odd">
<td style="text-align: right;">87</td>
<td style="text-align: right;">118</td>
<td style="text-align: right;">5</td>
<td style="text-align: right;">2</td>
</tr>
<tr class="even">
<td style="text-align: right;">86</td>
<td style="text-align: right;">101</td>
<td style="text-align: right;">6</td>
<td style="text-align: right;">8</td>
</tr>
<tr class="odd">
<td style="text-align: right;">86</td>
<td style="text-align: right;">109</td>
<td style="text-align: right;">7</td>
<td style="text-align: right;">6</td>
</tr>
<tr class="even">
<td style="text-align: right;">85</td>
<td style="text-align: right;">110</td>
<td style="text-align: right;">8</td>
<td style="text-align: right;">5</td>
</tr>
<tr class="odd">
<td style="text-align: right;">81</td>
<td style="text-align: right;">100</td>
<td style="text-align: right;">9</td>
<td style="text-align: right;">9</td>
</tr>
<tr class="even">
<td style="text-align: right;">76</td>
<td style="text-align: right;">99</td>
<td style="text-align: right;">10</td>
<td style="text-align: right;">10</td>
</tr>
</tbody>
</table>
</div>
</figure>
</div>
</div>
</div>
<p><a href="#fig-ath-iq-scatter" class="quarto-xref">Figure <span>29.1</span></a> is a <em>scatterplot</em> with “Athletic Score” on the x-axis and “I.Q. Score” on the y-axis. Each point on the plot corresponds to one row of <a href="#tbl-physical-mental" class="quarto-xref">Table <span>29.1</span></a> (and therefore one boy); in particular each point is at the <span class="math inline">\(x\)</span>, <span class="math inline">\(y\)</span> coordinate given by the values in “Athletic Score” and “I.Q. Score”. For example the point for the first boy is at position x=97, y=114.</p>
<div class="cell" data-layout-align="center">
<div class="cell-output-display">
<div id="fig-ath-iq-scatter" class="quarto-float quarto-figure quarto-figure-center anchored" data-fig-align="center">
<figure class="quarto-float quarto-float-fig figure">
<div aria-describedby="fig-ath-iq-scatter-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<img src="correlation_causation_files/figure-html/fig-ath-iq-scatter-1.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%">
</div>
<figcaption class="quarto-float-caption-bottom quarto-float-caption quarto-float-fig" id="fig-ath-iq-scatter-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Figure 29.1: Scatter plot of I.Q. Score as a function of Athletic Score
</figcaption>
</figure>
</div>
</div>
</div>
<p>We want to know whether a high score on athletic ability tends to be found along with a high I.Q. score more often than would be expected by chance. Therefore, our strategy is to see how often high scores on <em>both</em> variables are found by chance. We do this by disassociating the two variables and making two separate and independent universes, one composed of the athletic scores and another of the I.Q. scores. Then we draw pairs of observations from the two universes at random, and compare the experimental patterns that occur by chance to what actually is observed to occur in the world.</p>
<p>The first testing scheme we shall use is similar to our first approach to the pig rations — splitting the results into just “highs” and “lows.” We take ten cards, one of each denomination from “ace” to “10,” shuffle, and deal five cards to correspond to the first five athletic ranks. The face values then correspond to the I.Q. ranks. Under the benchmark hypothesis the athletic ranks will not be associated with the I.Q. ranks. Add the face values in the first five cards in each trial; the first hand includes 2, 4, 5, 6, and 9, so the sum is 26. Record, shuffle, and repeat perhaps ten times. Then compare the random results to the sum of the observed ranks of the five top athletes, which equals 17.</p>
<p>The following steps describe a slightly different procedure than that just described, because this one may be easier to understand:</p>
<ul>
<li><strong>Step 1.</strong> Convert the athletic and I.Q. scores to ranks. Then constitute a universe of spades, “ace” to “10,” to correspond to the athletic ranks, and a universe of hearts, “ace” to “10,” to correspond to the IQ ranks.</li>
<li><strong>Step 2.</strong> Deal out the well-shuffled cards into pairs, each pair with an athletic score and an I.Q. score.</li>
<li><strong>Step 3.</strong> Locate the cards with the top five athletic ranks, and add the I.Q. rank scores on their paired cards. Compare this sum to the observed sum of 17. If 17 or less, indicate “yes,” otherwise “no.” (Why do we use “17 or less” rather than “less than 17”? Because we are asking the probability of a score <em>this low or lower</em>.)</li>
<li><strong>Step 4.</strong> Repeat steps 2 and 3 ten times.</li>
<li><strong>Step 5.</strong> Calculate the proportion “yes.” This estimates the probability sought.</li>
</ul>
<p>In <a href="#tbl-ability-trials" class="quarto-xref">Table <span>29.2</span></a> we see that the observed sum (17) is lower than the sum of the top 5 ranks in all but one (shown in bold) of the ten random trials (trial 5), which suggests that there is a good chance (9 in 10) that the five best athletes will not have I.Q. scores that high by chance. But it might be well to deal some more to get a more reliable average. We add thirty hands, and thirty-nine of the total forty hands exceed the observed rank value, so the probability that the observed correlation of athletic and I.Q. scores would occur by chance is about .025. In other words, if there is no real association between the variables, the probability that the top 5 ranks would sum to a number this low or lower is only 1 in 40, and it therefore seems reasonable to believe that high athletic ability tends to accompany a high I.Q.</p>
<div id="tbl-ability-trials" 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-ability-trials-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 29.2: 40 random trials of the athletic / IQ problem
</figcaption>
<div aria-describedby="tbl-ability-trials-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="caption-top table">
<thead>
<tr class="header">
<th>Trial</th>
<th>Sum of IQ Ranks</th>
<th><= observed (17)</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1</td>
<td>26</td>
<td>No</td>
</tr>
<tr class="even">
<td>2</td>
<td>23</td>
<td>No</td>
</tr>
<tr class="odd">
<td>3</td>
<td>22</td>
<td>No</td>
</tr>
<tr class="even">
<td>4</td>
<td>37</td>
<td>No</td>
</tr>
<tr class="odd">
<td><strong>5</strong></td>
<td><strong>16</strong></td>
<td><strong>Yes</strong></td>
</tr>
<tr class="even">
<td>6</td>
<td>22</td>
<td>No</td>
</tr>
<tr class="odd">
<td>7</td>
<td>22</td>
<td>No</td>
</tr>
<tr class="even">
<td>8</td>
<td>28</td>
<td>No</td>
</tr>
<tr class="odd">
<td>9</td>
<td>38</td>
<td>No</td>
</tr>
<tr class="even">
<td>10</td>
<td>22</td>
<td>No</td>
</tr>
<tr class="odd">
<td>11</td>
<td>35</td>
<td>No</td>
</tr>
<tr class="even">
<td>12</td>
<td>36</td>
<td>No</td>
</tr>
<tr class="odd">
<td>13</td>
<td>31</td>
<td>No</td>
</tr>
<tr class="even">
<td>14</td>
<td>29</td>
<td>No</td>
</tr>
<tr class="odd">
<td>15</td>
<td>32</td>
<td>No</td>
</tr>
<tr class="even">
<td>16</td>
<td>25</td>
<td>No</td>
</tr>
<tr class="odd">
<td>17</td>
<td>25</td>
<td>No</td>
</tr>
<tr class="even">
<td>18</td>
<td>29</td>
<td>No</td>
</tr>
<tr class="odd">
<td>19</td>
<td>25</td>
<td>No</td>
</tr>
<tr class="even">
<td>20</td>
<td>22</td>
<td>No</td>
</tr>
<tr class="odd">
<td>21</td>
<td>30</td>
<td>No</td>
</tr>
<tr class="even">
<td>22</td>
<td>31</td>
<td>No</td>
</tr>
<tr class="odd">
<td>23</td>
<td>35</td>
<td>No</td>
</tr>
<tr class="even">
<td>24</td>
<td>25</td>
<td>No</td>
</tr>
<tr class="odd">
<td>25</td>
<td>33</td>
<td>No</td>
</tr>
<tr class="even">
<td>26</td>
<td>30</td>
<td>No</td>
</tr>
<tr class="odd">
<td>27</td>
<td>24</td>
<td>No</td>
</tr>
<tr class="even">
<td>28</td>
<td>29</td>
<td>No</td>
</tr>
<tr class="odd">
<td>29</td>
<td>30</td>
<td>No</td>
</tr>
<tr class="even">
<td>30</td>
<td>31</td>
<td>No</td>
</tr>
<tr class="odd">
<td>31</td>
<td>30</td>
<td>No</td>
</tr>
<tr class="even">
<td>32</td>
<td>21</td>
<td>No</td>
</tr>
<tr class="odd">
<td>33</td>
<td>25</td>
<td>No</td>
</tr>
<tr class="even">
<td>34</td>
<td>19</td>
<td>No</td>
</tr>
<tr class="odd">
<td>35</td>
<td>29</td>
<td>No</td>
</tr>
<tr class="even">
<td>36</td>
<td>23</td>
<td>No</td>
</tr>
<tr class="odd">
<td>37</td>
<td>23</td>
<td>No</td>
</tr>
<tr class="even">
<td>38</td>
<td>34</td>
<td>No</td>
</tr>
<tr class="odd">
<td>39</td>
<td>23</td>
<td>No</td>
</tr>
<tr class="even">
<td>40</td>
<td>26</td>
<td>No</td>
</tr>
</tbody>
</table>
</div>
</figure>
</div>
<p>In fact we can apply an even simpler procedure to get the same result, by reasoning about the individual trial.</p>
<p>One trial in our procedure is:</p>
<ul>
<li><strong>Step 2.</strong> Deal out the well-shuffled cards into pairs, each pair with an athletic score and an I.Q. score.</li>
<li><strong>Step 3.</strong> Locate the cards with the top five athletic ranks, and add the I.Q. rank scores on their paired cards. Compare this sum to the observed sum of 17. If 17 or less, indicate “yes,” otherwise “no.” (Why do we use “17 or less” rather than “less than 17”? Because we are asking the probability of a score <em>this low or lower</em>.)</li>
</ul>
<p>Now consider the 5 IQ rank cards. In the procedure above, we found these by <em>first</em> pairing the athletic ranks and the IQ ranks, <em>then</em> selecting the IQ ranks corresponding to the top 5 athletic ranks. A little thought may persuade you, that by doing this, we have have a <em>random selection of 5 IQ ranks</em>. We got that random selection by pairing, selecting on athletic rank — but the initial pairing and selection will do nothing other than giving us one particular set of randomly chosen 5 IQ rank cards. So we can simplify our procedure even further by missing out the pairing and selecting by rank steps; we can just shuffle the IQ rank cards and deal out 5 to be our randomly selected IQ ranks.</p>
<div id="nte-athlete_iq" class="callout callout-style-default callout-note callout-titled">
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Note 29.1: Notebook: Athletic ability and IQ
</div>
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<div class="callout-body-container callout-body">
<div class="nb-links">
<p><a class="notebook-link" href="notebooks/athlete_iq.ipynb">Download notebook</a> <a class="interact-button" href="./interact/lab/index.html?path=athlete_iq.ipynb">Interact</a></p>
</div>
</div>
</div>
<div class="nb-start" name="athlete_iq" title="Athletic ability and IQ">
</div>
<p>To simulate this problem in Python, we first create an array containing the I.Q. rankings of the top 5 students in athletics. The <code>sum</code> of these I.Q. rankings constitutes the observed result to be tested against randomly-drawn samples. We observe that the actual I.Q. rankings of the top five athletes sums to 17. The more frequently that the sum of 5 randomly-generated rankings (out of 10) is as low as this observed number, the higher is the probability that there is no relationship between athletic performance and I.Q. based on these data.</p>
<p>First we record the 1 through 10 into array <code>iq_ranks</code>. Then we shuffle the numbers so the rankings are in a random order. Then select the first 5 of these numbers and put them in another array, <code>top_5</code>, and <code>sum</code> them, putting the result in <code>top_5_sum</code>. We repeat this procedure <code>N = 10000</code> times, recording each result in a scorekeeping vector: <code>z</code>. Graphing <code>z</code>, we get a histogram that shows us how often our randomly assigned sums are equal to or below 17.</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><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-4"><a href="#cb1-4" 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>
</div>
<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"># Number of repeats.</span></span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>n_trials <span class="op">=</span> <span class="dv">10000</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="co"># The IQ ranks, ready for shuffling.</span></span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a>iq_ranks <span class="op">=</span> np.arange(<span class="dv">1</span>, <span class="dv">11</span>) <span class="co"># 1 through 10.</span></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Scorekeeping array.</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a>results <span class="op">=</span> np.zeros(n_trials)</span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Repeat the experiment 10,000 times.</span></span>
<span id="cb2-9"><a href="#cb2-9" 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="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a> <span class="co"># Shuffle the ranks.</span></span>
<span id="cb2-11"><a href="#cb2-11" aria-hidden="true" tabindex="-1"></a> shuffled <span class="op">=</span> rnd.permuted(iq_ranks)</span>
<span id="cb2-12"><a href="#cb2-12" aria-hidden="true" tabindex="-1"></a> <span class="co"># Take the first 5 ranks.</span></span>
<span id="cb2-13"><a href="#cb2-13" aria-hidden="true" tabindex="-1"></a> top_5 <span class="op">=</span> shuffled[:<span class="dv">5</span>]</span>
<span id="cb2-14"><a href="#cb2-14" aria-hidden="true" tabindex="-1"></a> <span class="co"># Sum those ranks.</span></span>
<span id="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a> top_5_sum <span class="op">=</span> np.<span class="bu">sum</span>(top_5)</span>
<span id="cb2-16"><a href="#cb2-16" aria-hidden="true" tabindex="-1"></a> <span class="co"># Keep track of the result of each trial.</span></span>
<span id="cb2-17"><a href="#cb2-17" aria-hidden="true" tabindex="-1"></a> results[i] <span class="op">=</span> top_5_sum</span>
<span id="cb2-18"><a href="#cb2-18" aria-hidden="true" tabindex="-1"></a> <span class="co"># End the experiment, go back and repeat.</span></span>
<span id="cb2-19"><a href="#cb2-19" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-20"><a href="#cb2-20" aria-hidden="true" tabindex="-1"></a><span class="co"># Produce a histogram of trial results.</span></span>
<span id="cb2-21"><a href="#cb2-21" aria-hidden="true" tabindex="-1"></a><span class="co"># Make the bins be the integers from 10 through 45.</span></span>
<span id="cb2-22"><a href="#cb2-22" aria-hidden="true" tabindex="-1"></a>plt.hist(results, bins<span class="op">=</span>np.arange(<span class="dv">10</span>, <span class="dv">46</span>))</span>
<span id="cb2-23"><a href="#cb2-23" aria-hidden="true" tabindex="-1"></a>plt.title(<span class="st">'Sums of 5 ranks selected at random'</span>)<span class="op">;</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="correlation_causation_files/figure-html/unnamed-chunk-3-3.png" class="img-fluid quarto-figure quarto-figure-center figure-img" style="width:70.0%"></p>
</figure>
</div>
</div>
</div>
<p>We see that in only about 1 % of the trials did random selection of ranks produce a total of 17 or lower. Python can calculate this for us directly:</p>
<div class="cell" data-layout-align="center">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Determine how many trials produced sums of ranks <= 17 by chance.</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>k <span class="op">=</span> np.<span class="bu">sum</span>(results <span class="op"><=</span> <span class="dv">17</span>)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="co"># The proportion.</span></span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a>kk <span class="op">=</span> k <span class="op">/</span> n_trials</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Show the result.</span></span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a>kk</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>np.float64(0.0143)</code></pre>
</div>
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<div class="nb-end">
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<div class="callout callout-style-default callout-note callout-titled">
<div class="callout-header d-flex align-content-center">
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End of notebook: Athletic ability and IQ
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<div class="callout-body-container callout-body">
<p><code>athlete_iq</code> starts at <a href="#nte-athlete_iq" class="quarto-xref">Note <span>29.3</span></a>.</p>
</div>
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<!---
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<p>Why do we sum the ranks of the first <em>five</em> athletes rather than taking the sum of the top three, say? Indeed, we could have looked at the top three, two, four, or even six or seven. The first reason for splitting the group in half is that an even split uses the available information more fully, and therefore we obtain greater efficiency. (I cannot prove this formally here, but perhaps it makes intuitive sense to you.) A second reason is that getting into the habit of always looking at an even split reduces the chances that you will pick and choose in such a manner as to fool yourself. For example, if the I.Q. ranks of the top five athletes were 3, 2, 1, 10, and 9, we would be deceiving ourselves if, after looking the data over, we drew the line between athletes 3 and 4. (More generally, choosing an appropriate measure before examining the data will help you avoid fooling yourself in such matters.)</p>
<p>A simpler but less efficient approach to this same problem is to classify the top-half athletes by whether or not they were also in the top half of the I.Q. scores. Of the first five athletes actually observed, <em>four</em> were in the top five I.Q. scores. We can then shuffle five black and five red cards and see how often four or more (that is, four or five) blacks come up with the first five cards. The proportion of times that four or more blacks occurs in the trial is the probability that an association as strong as that observed might occur by chance even if there is no association. <a href="#tbl-top-rank-counts" class="quarto-xref">Table <span>29.3</span></a> shows a proportion of five trials out of twenty.</p>
<div id="tbl-top-rank-counts" 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-top-rank-counts-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
Table 29.3: Results of 20 random trials of the top-5 rank counts
</figcaption>
<div aria-describedby="tbl-top-rank-counts-caption-0ceaefa1-69ba-4598-a22c-09a6ac19f8ca">
<table class="caption-top table">
<thead>
<tr class="header">
<th>Trial</th>
<th>Score</th>
<th>Yes or No</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>1</td>
<td>4</td>
<td>Yes</td>
</tr>