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I want to perform significance test for correlation coefficient. However my data is not normally distributed. In my books I have found tests only for the normally distributed data. Is there such test I could apply to data having any distribution?

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    $\begingroup$ This is often fudged: people report P-value but flag that strictly it applies only to normally distributed data. In practice, a better procedure would be to bootstrap and get a confidence interval. In most research, correlations measure or indicate strength of relationships. If there is doubt about correlations being not zero, then perhaps the sample size is too small. Note that other assumptions, especially independence of observations, are often dubious but more rarely discussed. Testing the corresponding regression coefficient downplays this problem and is often closer to the science. $\endgroup$ – Nick Cox Sep 9 '14 at 12:17
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I am assuming that you are interested strictly in finding a p-value to test $H_0: \rho=0$ and yet want to control Type I error rates. Then you may want to check out this article that discusses 12 tests of correlation, including familiar Pearson and Spearman as well as less known tests based on transformation and resampling approaches. The authors examine which tests are robust to various non-normal distributions across a range of sample sizes. Judging from the shape of your data's distribution and sample size, you can pick the best test.

Reference

Anthony J. Bishara, James B. Hittner, Testing the significance of a correlation with nonnormal data: comparison of Pearson, Spearman, transformation, and resampling approaches Psychol Methods, 17(3), 399-417 (2012)

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    $\begingroup$ Thanks for pointing out our work. For those who don't have access to the journal, a free copy of Bishara & Hittner (2012) can be found here $\endgroup$ – Anthony Oct 20 '14 at 18:58
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If you're testing that the correlation differs from zero you could do a permutation test (or a randomization test). This is probably the simplest to do.

Alternatively if sample sizes are not small, you could form a bootstrap interval for the correlation and see if it includes whatever value you want to test for a difference from.

If you are prepared to assume a distribution you may be able to derive a test or perhaps simulate the distribution of the correlation under independence.

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