I have been reading a lot about Pearson correlation and its alternative Spearman's rank correlation. I realized that in Pearson we are finding the correlation on our observed data. However, Spearman's rank correlation tend to find the correlation on the ranked data (So I am not actually working on my real data anymore).
Why should the data be normally distributed and why Pearson don't work with discrete data? Is this due to the tests used to find the significance level for Pearson correlation and Spearman's rank correlation?
The relevant assumption here is that the two variables are bivariate normal (not just the marginal distribution of each individual variable is normal).
According to Rob Hyndman (see linked stackexchange discussion), Pearsons correlation remains a consistent estimator of the population correlation even when bivariate normality is not present.
However, when the variables are not bivariate normal, the sampling distribution of the coefficient may not be normal. This means that inferential tests that assumes a normal sampling distribution (e.g., via a Fisher transformation, or a t-distribution) may not be trustworthy. One of these methods is usually used to determine statistical significance. So as you seem to have picked up, it's the significance test or confidence interval that may be negatively affected, rather than the point estimate of correlation itself. (Sidenote: Confidence intervals are much more informative than significance tests!)
That said, the sampling distribution of the Pearson correlation will converge toward a normal distribution as the sample gets larger, almost regardless of the distribution of the variables. Furthermore, you could replace inferential statistics based on the Fisher transformation or t-distribution with confidence intervals obtained by bootstrapping, without having to fall back on Spearmans rho. So the normality issue isn't too hard to deal with.
The more important issue is whether the relationship between the variables is actually linear. Pearsons correlation only describes the linear part of the relationship, whereas Spearman's rho is an estimate of any monotonic relationship form.
http://stats.stackexchange.com/questions/3730/pearsons-or-spearmans-correlation-with-non-normal-data
Thank you for your answer. Is this right ?
Pearson use inferential tests which requires normally distributed data, while Spearmans use some kind of transformation to normalize ranked data and then applies inferential tests because it also report P-value and confidence interval
No worries Mona. That isn't completely right - Spearmans rho does use a transformation in the sense that it converts the observations to ranks, but it isn't a normalising transformation. You can see how inferential tests for Spearmans rho can be conducted on the linked Wikipedia page. Hope that helps!
http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient
After Matt Williams's response, there is nothing left to be said :-)
Because the Pearson's correlation is a measure of the linear relationship between two continuous random variables. In addition of the distribution of the test statistic to test the significance for Pearson correlation .Thus, data should be normally distributed and continuous in order to apply Pearson correlation, But when the sample size large enough you can use the Pearson's correlation due to the central limit theory.
I have a similar problem. From the previou answers I could not lunderstand why we cannot apply the pearson correlation to discrete data.
I have a problem. How Can I calcutate correlation coefficient for two contiuous random variables when I have only mean values, standard deviations and probability density functions?
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