I have data from a research that was conducted by me, the data refer to four conditions made of two levels of time (pretest and post-test) and two levels of group (control and intervention). Shapiro-Wilk test have shown that the data for two of three dependent variables (hereafter A,B and C) is non normally distributed in all four conditions and for third variable it's near normal distribution in one of two of the conditions and not normal in the two others. All third dependent variables are of tests with continuous scores (scale order).
I conducted repeated measures ANOVA and main effects were significant, confirming the study first hypothesis that post-tests scores would be higher on all three tests in the intervention group and no change would be found for the control group.
The study second hypothesis is that the same neural mechanism is mediating the scores on all three tests and therefore even if the scores are altered between these 4 different conditions, the size of the correlations between scores (A and B, A and C, B and C) would remain unchanged between the different conditions. To test this hypothesis I used simple Pearson correlations (4 conditions X 3 correlations in each of them), correlations were all medium to large (~0.2-0.8) and significant, Then, I used Steiger's Z tests for dependent samples (for comparing the correlations in the same group pretest and post-test) and for independent samples (for comparing the correlations in the same time between groups). All Zs were insignificant (p>0.05) and smaller than 2. Confirming my second hypothesis. But I been told that this method I used is improper for two reasons. First, my data is not normally distributed. Second, this is not how I should compare between size of correlations. So, the question is what test should I take and how should I conduct and analyse it? I would appreciate any help on that, I should put it on scientific paper and currently it's clear that my data analysis is a pitfall.