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<!DOCTYPE html>
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<head>
<title>Historical side note, Regression to Mediocrity</title>
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<h1>Historical side note, Regression to Mediocrity</h1>
<<<<<<< HEAD
<h2>Regression</h2>
<p>Brian Caffo, Jeff Leek and Roger Peng<br/>Johns Hopkins Bloomberg School of Public Health</p>
=======
<h2>Regression to the mean</h2>
<p>Brian Caffo, Jeff Leek, Roger Peng PhD<br/>Johns Hopkins Bloomberg School of Public Health</p>
>>>>>>> 3e5b14bbb8f101fc2a8573beb037d5f1b6f6fe47
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<h2>A historically famous idea, Regression to the Mean</h2>
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<article data-timings="">
<ul>
<li>Why is it that the children of tall parents tend to be tall, but not as tall as their parents? </li>
<li>Why do children of short parents tend to be short, but not as short as their parents?</li>
<li>Why do parents of very short children, tend to be short, but not a short as their child? And the same with parents of very tall children?</li>
<li>Why do the best performing athletes this year tend to do a little worse the following?</li>
</ul>
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<slide class="" id="slide-2" style="background:;">
<hgroup>
<h2>Regression to the mean</h2>
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<article data-timings="">
<ul>
<li>These phenomena are all examples of so-called regression to the mean</li>
<li>Invented by Francis Galton in the paper "Regression towards mediocrity in hereditary stature" The Journal of the Anthropological Institute of Great Britain and Ireland , Vol. 15, (1886).</li>
<li>Think of it this way, imagine if you simulated pairs of random normals
<ul>
<li>The largest first ones would be the largest by chance, and the probability that there are smaller for the second simulation is high.</li>
<li>In other words \(P(Y < x | X = x)\) gets bigger as \(x\) heads into the very large values.</li>
<li>Similarly \(P(Y > x | X = x)\) gets bigger as \(x\) heads to very small values.</li>
</ul></li>
<li>Think of the regression line as the intrisic part.
<ul>
<li>Unless \(Cor(Y, X) = 1\) the intrinsic part isn't perfect</li>
</ul></li>
</ul>
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<!-- Presenter Notes -->
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<slide class="" id="slide-3" style="background:;">
<hgroup>
<h2>Regression to the mean</h2>
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<article data-timings="">
<ul>
<li>Suppose that we normalize \(X\) (child's height) and \(Y\) (parent's height) so that they both have mean 0 and variance 1. </li>
<li>Then, recall, our regression line passes through \((0, 0)\) (the mean of the X and Y).</li>
<li>If the slope of the regression line is \(Cor(Y,X)\), regardless of which variable is the outcome (recall, both standard deviations are 1).</li>
<li>Notice if \(X\) is the outcome and you create a plot where \(X\) is the horizontal axis, the slope of the least squares line that you plot is \(1/Cor(Y, X)\). </li>
</ul>
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<hgroup>
<h2>Plot of the results</h2>
</hgroup>
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<article data-timings="">
<div class="rimage center"><img src="fig/unnamed-chunk-1.png" title="plot of chunk unnamed-chunk-1" alt="plot of chunk unnamed-chunk-1" class="plot" /></div>
</article>
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<slide class="" id="slide-5" style="background:;">
<hgroup>
<h2>Discussion</h2>
</hgroup>
<article data-timings="">
<ul>
<li>If you had to predict a son's normalized height, it would be
\(Cor(Y, X) * X_i\) </li>
<li>If you had to predict a father's normalized height, it would be
\(Cor(Y, X) * Y_i\)</li>
<li>Multiplication by this correlation shrinks toward 0 (regression toward the mean)</li>
<li>If the correlation is 1 there is no regression to the mean (if father's height perfectly determine's child's height and vice versa)</li>
<li>Note, regression to the mean has been thought about quite a bit and generalized </li>
</ul>
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