Apparently Pearson's correlation coefficient is parametric and Spearman's rho is non-parametric.
I'm having trouble understanding this. As I understand it Pearson is computed as
$$
r_{xy} = \frac{cov(X,Y)}{\sigma_x\sigma_y}
$$
and Spearman is computed in the same way, except we substitute all values with their ranks.
Wikipedia says
The difference between parametric model and non-parametric model is that the former has a fixed number of parameters, while the latter grows the number of parameters with the amount of training data.
But I do not see any parameters except for the samples themselves.
Some say that parametric tests assume normal distributions and go on to say that Pearson does assume normal distributed data, but I fail to see why Pearson would require that.
So my question is what do parametric and non-parametric mean in the context of statistics? And how do Pearson and Spearman fit in there?
Best Answer
The problem is that "nonparametric" really has two distinct meanings these days. The definition in Wikipedia applies to things like nonparametric curve fitting, eg via splines or local regression. The other meaning, which is older, is more along the lines of "distribution-free" -- that is, techniques that can be applied regardless of the assumed distribution of the data. The latter is the one that applies to Spearman's rho, since the rank-transformation implies it will give the same result no matter what your original distribution was.