How can I measure linear correlation of non-normally distributed variables?
Pearson coefficient is not valid for non-normally distributed data, and Spearman's rho does not capture linear correlation.
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Solved – Measuring linear correlation of non-normally distributed variables
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The Pearson correlation coefficient measures linear association. Being based on empirical second central moments, it is influenced by extreme values. Therefore:
Evidence of nonlinearity in a scatterplot of actual-vs-predicted values would suggest using an alternative such as the rank correlation (Spearman) coefficient;
If the relationship looks monotonic on average (as in the upper row of the illustration), a rank correlation coefficient will be effective;
Otherwise, the relationship is curvilinear (as in some examples from the lower row of the illustration, such as the leftmost or the middle u-shaped one) and likely any measure of correlation will be an inadequate description; using a rank correlation coefficient won't fix this.
The presence of outlying data in the scatterplot indicates the Pearson correlation coefficient may be overstating the strength of the linear relationship. It might or might not be correct; use it with due caution. The rank correlation coefficient might or might not be better, depending on how trustworthy the outlying values are.
(Image copied from the Wikipedia article on Pearson product-moment correlation coefficient.)
Pearson's r and Spearman's rho are both already effect size measures. Spearman's rho, for example, represents the degree of correlation of the data after data has been converted to ranks. Thus, it already captures the strength of relationship.
People often square a correlation coefficient because it has a nice verbal interpretation as the proportion of shared variance. That said, there's nothing stopping you from interpreting the size of relationship in the metric of a straight correlation.
It does not seem to be customary to square Spearman's rho. That said, you could square it if you wanted to. It would then represent the proportion of shared variance in the two ranked variables.
I wouldn't worry so much about normality and absolute precision on p-values. Think about whether Pearson or Spearman better captures the association of interest. As you already mentioned, see the discussion here on the implication of non-normality for the choice between Pearson's r and Spearman's rho.
Best Answer
Why do you require normality for computing a correlation? How about a simple scatterplot? As long as the data are continuous, ordinary (Pearson) correlation should be fine. All that it is measuring is the strength of the linear relationship between two variables (if indeed there is such a relationship).