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<h1>
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Spearman's Rank Correlation Coefficient
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<time class="published" datetime="2018-11-14T22:09:00+01:00">
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14 novembre 2018
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</li>
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<li>Mathematics</li>
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<li>Statistics</li>
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<p>A rank correlation is any of several statistics that measure an ordinal association—the relationship between rankings of different ordinal variables or different rankings of the same variable, where a "ranking" is the assignment of the ordering labels "first", "second", "third", etc. to different observations of a particular variable. A rank correlation coefficient measures the degree of similarity between two rankings, and can be used to assess the significance of the relation between them. </p>
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<p>We have two random variables <span class="math">\(X\)</span> and <span class="math">\(Y\)</span>:
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* <span class="math">\(X=\{x_i, x_2, x_3, ..., x_n\}\)</span>
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* <span class="math">\(Y=\{y_i, y_2, y_3, ..., y_n\}\)</span></p>
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<p>if <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span> denote the respective ranks of each data point, then the Spearman's rank correlation coefficient, <span class="math">\(r_s\)</span>, is the Pearson correlation coefficient of <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span>.</p>
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<h2>What does it means?</h2>
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<p>The Spearman's rank correlation coefficientis is a nonparametric measure of rank correlation (statistical dependence between the rankings of two variables). It assesses how well the relationship between two variables can be described using a monotonic function.
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The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson's correlation assesses linear relationships, Spearman's correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other. </p>
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<p><img alt="Spearman" src="../images/spearman/Spearman_fig1.svg"></p>
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<p>A Spearman correlation of 1 results when the two variables being compared are monotonically related, even if their relationship is not linear. This means that all data-points with greater x-values than that of a given data-point will have greater y-values as well. In contrast, this does not give a perfect Pearson correlation.</p>
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<h2>Example</h2>
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<ul>
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<li><span class="math">\(X=\{0.2, 1.3, 0.2, 1.1, 1.4, 1.5\}\)</span></li>
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<li><span class="math">\(Y=\{1.9, 2.2, 3.1, 1.2, 2.2, 2.2\}\)</span></li>
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</ul>
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<div class="math">$$
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Rank_X
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\quad
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\begin{bmatrix}
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X: & 0.2 & 1.3 & 0.2 & 1.1 & 1.4 & 1.5 \\
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Rank: & 1 & 3 & 1 & 2 & 4 & 5
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\end{bmatrix}
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\quad
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$$</div>
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<p>so, <span class="math">\(Rank_X = \{1, 3, 1, 2, 4, 5\}\)</span></p>
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<p>similarly, <span class="math">\(Rank_Y=\{2,3,4,1,3,3\}\)</span></p>
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<p><span class="math">\(r_s\)</span> equals the Pearson correlation coefficient of <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span>, meaning that <span class="math">\(r=0.158114\)</span></p>
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<h2>Special case : <span class="math">\(X\)</span> and <span class="math">\(Y\)</span> don't contain duplicates</h2>
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<div class="math">$$r_s=1-\frac{6\sum d_i^2}{n(n^2-1)}$$</div>
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<p>Where, <span class="math">\(d_i\)</span> is the difference between the respective values of <span class="math">\(Rank_X\)</span> and <span class="math">\(Rank_Y\)</span>.</p>
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