# Testing Spearman's rho between-groups

I am analysing ordinal, non-normally distributed dependent measures using Spearman's rho. Specifically, we have an independent groups design, and I have found a significant relationship between the variables in one group but not the other (running separate Spearman's analyses for each group).

I wish to test the difference in associations between the two groups, but am unable to find information concerning what procedure/test to use. I understand the Fisher r to z transformation may be used in the case of a Pearson's Product Moment Correlation, allowing differences in associations to be tested between-groups - is there a means of running a similar test of between-groups association with Spearman's rho?

Spearman rank correlation is just Pearson correlation applied to ranks, a point often obscured by the emphasis on the simple computational short-cut formula for Spearman that is found in many books. So, I wouldn't rule out Fisher's z procedures for Spearman. There is a caution that the sampling distribution will differ at least a bit with Spearman -- as the sampling distribution of Spearman is irregular in detail -- and indeed that could bite hard with small sample sizes. But most things are problematic with small sample sizes. The caution that everything hinges on the data being treated as ranks is already the caution that applies to Spearman correlation.

I've got to suggest, however, that this line of enquiry may not prove very fruitful for you.

For a start, it is usually better to bring two groups into the same model if you can. Also, one rank correlation being stronger than another leaves the more important question of what the relationships and differences are, quantitatively, a bit in the background.

The data have to be really irregular for there to be no transformation or link function (logarithm? square root? reciprocal?) that won't bring them into reasonable shape for some kind of ANOVA or more general(ized) linear model. As it seems you have some kind of experiment, that would probably mesh better with your scientific objectives too.

(LATER) You did say "ordinal" and that is important. Much depends on what that means precisely. If ordinal means a five-point scale, something like an ordered logit or probit model may be appropriate. If ordinal means judgment-based scores of some kind, much depends on how they behave.

The relevant wikipedia page suggests that one approach is to apply the Fisher transformation to approximate normality. The difference in z's for the two groups would under the null be approximately normal with mean $0$ and variance $(n_1 + n_2 - 6 / 1.06)$

Personally, I'd probably just turn it into a permutation test and (depending on sample size) either enumerate the permutation distribution or sample from it (a randomization test).