I have searched a lot, and I can only find tables that show critical values up to n=30. Can someone provide, or point me to, a simple method of estimating this value for different $\alpha$?
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asked Feb 19, 2011 at 18:41
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2 Answers
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For values over thirty the approximation (for a two-tailed test) is $$\frac{\Phi^{-1}\left(1-\tfrac{\alpha}{2}\right)}{\sqrt{n-1}}$$ so for example with $\alpha = 0.05$ and $n=100$ the numerator is about 1.96 and the denominator about 9.95, giving a critical value of about 0.197.
This comes from $\rho$ having approximately a normal distribution for large $n$, with mean $0$ and variance $1/(n − 1)$, assuming independence of the observations.
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$\begingroup$ Thanks. Can you provide a quotable source for this? $\endgroup$ Feb 19, 2011 at 19:10
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$\begingroup$ It seems to be standard in education, e.g. books.google.co.uk/books?id=3LhPwUhrVIcC&pg=PA397 or umich.edu/~exphysio/MVS250/SpearmanRankCorr.doc $\endgroup$– HenryFeb 19, 2011 at 21:54
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See Wikipedia: Spearman's rank correlation coefficient#Determining significance:
"One can test for significance using $$t = r \sqrt{\frac{n-2}{1-r^2}},$$ which is distributed approximately as Student's $t$ distribution with $n − 2$ degrees of freedom under the null hypothesis."
Here $r$ is the sample estimate of Spearman's rank correlation coefficient. The reason critical values often aren't tabulated for $n > 30$ is that this approximation gets better as $n$ gets larger, and is very good for $n > 30$. The Stata statistical software package uses this formula to calculate $p$-values for all values of $n$.
