> >Inference for Pearson's moment correlation relies on normality of the
> >data. Spearman rank correlation is free of any assumptions, but there
> >is no population characteristic that it estimates, which makes
> >interpretation and asymptotic inference somewhat weird. If one is
> >significant and the other is not, you are making either type I or type
> >II error somewhere.
> In the angels on the head of a pin vein:
> Of possible interest in this regard is that the Spearman coefficient is the
> same as the Pearson calculated on the ranked values of the variables (ties
> getting the average rank). I would agree that this is not a terribly
> interesting population parameter, but isn't this nevertheless an
> estimable/testable population characteristic?
If you have a finite population, then of course you will have Spearman
correlation for it. Although if you want to set up any asymptotic
framework, you will be trying to hit a moving target. I don't think
there is a meaningful definition of Spearman correlation for infinite
populations/continuous variables, although I might be mistaken. On the
other hand, Kendall's tau, as Nick Cox quoted from Roger Newson, has
explicit population analogues in probabilities of concordant and
discordant pairs of observations.
The question is: if the correlation estimate is 0.5, what does it say?
For Pearson moment correlation, it means that the proportion of
explained variance in a bivariate regression is 0.25. For Kendall's
tau, it means that for every discordant pair of observations, there
are three concordant pairs (i.e., Prob[ concordant ] = 3 Prob[
discordant ] = 3/4 ). For Spearman rank correlation, you can only say
that the variables are positively associated, but not much more.
--
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: I use this email account for mailing lists only.
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