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      <text>43.2.09 Significance of Spearman's R</text>
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      <text><![CDATA[<h3>The significance of Spearman’s \(R\)</h3>

Like Pearson's \(r\), the value of \(R\) may be shown to lie in the range \(-1 \leq R \leq 1\) and in order to test the significance of a calculated value of \(R\) we set up the hypotheses
\[ H_0: \rho = 0 \hspace{2cm} H_1: \rho \neq 0 \]

<p>A generalisation of the third law of indices states:</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<p><b>Significance of Spearman's \(R\)</b></p>

<p></p> We wish to know whether our correlation coefficient is significantly different to zero. It can be shown that for large samples, the test statistic
\[ R_{test} = \dfrac{\lvert R \rvert \sqrt{n - 2}}{\sqrt{1-R^2}} \]
calculated from a sample of \(n\) pairs of values, follows a \(t\)-distribution with \(n-2\) degrees of freedom.
</div><p><br></p>


Critical values depending on the level of significance required are read from \(t\)-tables. When dealing with Spearman's coefficient of correlation, the size of the sample is important. Different authors recommend different <i>minimum</i> sample sizes, a common recommendation being a minimum of \(n = 10\). Even though they are not used here, you should note that tables are available which allow us to read critical values corresponding to small sample sizes.

<hr>
<h4 class="HELM_example">Example</h4>
<p>A production engineer is asked to grade, on the basis of \(12\) criteria (say) \(A\) to \(L\)
a junior colleague who has applied for promotion. He repeats his gradings after a
few days. The results (calculated in the example above) gave a value of \(R = 0.87\). Test
at the \(5\%\) level to determine whether the results are consistent.</p>
<h4 class="HELM_solution">Solution</h4>
<p>The calculation is
\[ R_{test} = \dfrac{R \sqrt{n - 2}}{\sqrt{1 - R^2}}  = \dfrac{0.87 \times \sqrt{12 - 2}}{\sqrt{1 - 0.87^2}} = 5.580. \]</p>
The \(5\%\) critical value for a two sided test read from tables is \(2.228\) and since \(5.580 &gt; 2.228\) we conclude that we must reject the null hypothesis that the correlation coefficient is zero.]]></text>
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