Effect of treatment on correlation between two sets of variables

Question

Imagine I wish to see the effect of some treatment on the correlation between two groups of variables, which I measure with Spearman's rank correlation coefficient.

Treatment A gives a Spearman's rank of 0.4, 0.44, 0.43 for my 3 replicates.

Treatment B gives a Spearman's rank of 0.48, 0.45, 0.46 for my 3 replicates

Treatment C gives a Spearman's rank of 0.47, 0.50, 0.51 for my 3 replicates.

Can I compare the groups using e.g. the Mann-Whitney U test?

N.B. I'm not looking for the correlation between the sets of variables, I'm looking to see whether a specific treatment improves the correlation.

Apologies if this question is similar to this Significance test on the difference of Spearman's correlation coefficient

Answer 1

It is certainly possible to test if two Pearson product-moment correlations are equivalent. The most basic approach is to use a t-test. This assumes that the distribution of the sample statistic is approximately normal under the Fisher transform. Dividing the difference of the two transformed values by the standard error of the difference (which is: $\sqrt{1/(N_1-3)+1/(N_2-3)}$) constitutes a valid t-test used like any other. (Here is a good, basic overview of tests of correlations by one of CV's esteemed contributors.) However, I don't know if the Fisher transform of Spearman's rho is normally distributed. Hence, your suggestion of the Mann-Whitney U test is reasonable. However, since you have more than 2 groups, the Kruskal-Wallis test is more appropriate, as 3 pairwise comparisons leads to a multiplicity problem.