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I want to investigate the correlation between X and Y, both which have 560 observations. I applied Shapiro-Wilks Normality Test to X and Y and neither of the variables are normally distributed. I calculated Spearman,Kendall and Pearson correlations and here are the results.

Spearman: : 0.697
Kendall : 0.535
Pearson : 0.492

Which one should I rely on?

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In well-behaved datasets those three are usually in agreement but there are some differences in your case, which I believe are due to the presence of outliers.

In fact, I would like to emphasize the lack of robustness of the Pearson product-moment correlation coefficient. Since this is an estimate based on sample means and variances, which are highly non-robust, even one outlier is enough to distort the estimate, or as robust statisticians say the breakdown value of the estimator is zero.

This not the case for Spearman's and Kendall's correlation coefficients. Replacing the observations by their ranks or looking at the product signed differences serve to dampen the effect of outliers so that the estimates are more reliable. Of course, these coefficients are general measures of monotonicity and are not only restricted to linear relationships. This is not a problem for most people using them but it is worth keeping it in mind.

Between these two, Kendall's correlation coefficient seems to be the most efficient one. At the normal distribution its variance is comparable to Pearson's measure and in situations of contamination, i.e. outliers, it is much less variable. So this is what I would recommend for your case.

Here are two nice papers in case you want to look some more into the properties of the correlation coefficients.

Devlin, Susan J., Ramanathan Gnanadesikan, and Jon R. Kettenring. "Robust estimation and outlier detection with correlation coefficients." Biometrika 62.3 (1975): 531-545.

Croux, Christophe, and Catherine Dehon. "Influence functions of the Spearman and Kendall correlation measures." Statistical methods & applications 19.4 (2010): 497-515.

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