# Reporting coefficient of determination using Spearman's rho

I have two non-normally distributed variables (positively-skewed, exhibiting ceiling effects). I would like to calculate the correlation coefficient between these two variables. Due to the non-normal distribution, I used Spearman's rank-order correlation, which returns a correlation coefficient and a significance (p) value. My results (n=400) show a significant ($p = 8 \times 10^{-5}$) but weak correlation (Spearman's $\rho$ = .20). If one uses Pearson's, one could describe the strength of the correlation in terms of shared variance (coefficient of determination, $R^2$ – in my case $R^2$ = .04, ie. 4%). Clearly, for Spearman's, it doesn't seem meaningful to square the $\rho$ value as Spearman ranks data. What is the best way to talk about the effect size using Spearman's $\rho$?

Alternatively, following a discussion here (Pearson's or Spearman's correlation with non-normal data), I interpret the discussions as meaning that Pearson's correlation does not assume normality, but calculating p-values from the correlation coefficients does. Thus, I was wondering whether one could use Spearman to calculate the p-value of the correlation, and Pearson's to calculate the effect size, and thus continue to speak of shared variance between the two variables.

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could you specify the nature of data(continuous or grouped data or count data on each of variable. The two measures of correlation are quite different in terms of purpose. – subhash c. davar Dec 29 '13 at 15:13

The original scale for $\rho$ is the best, and is directly related to a "majority concordance of triplets of observations" index. But I also don't have much of a problem communicating $\rho^2$. A (much) different way to view this is the $\rho$ is almost a special case of a regression test arising from the proportional odds ordinal logistic model, and that model has a generalized $R^2$ measure based on log likelihood explained, that is akin to variance explained. – Frank Harrell Jul 14 '13 at 11:30
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