I have a dataset of ~400 subjects. For each subject, I have about 18 variables (V1...V18), all of those continuous and measured at the same timepoint.
A paper I'm working on would benefit from showing that one of those variables (say V1) has a significant correlation with one of three other variables (say V2, V3, or V4) – all of these are measuring different aspects of the same illness. And I have found some significant correlations between the pairs (V1–V2, V1–V3, V1–V4). However, I have another variable, say V5, that is more strongly correlated to all of them (V1–V4). When I correct (linearly) for V5, the correlations between V1–(V2/V3/V4) is no longer significant/there.
So at this point, any critic could say that the V1 is correlated to V5, V5 is correlated to V2/3/4, and that I can't make the point I want to make. However, I have reason to believe that subgroups of the subjects (more susceptible to the illness), will show strong correlation between V1 and V2–4 that is independent of V5. Is there any way to find these subgroups?
I would like to find some partition(s), A < Vx < B, in which there is a significant correlation between V1 and V2/V3/V4, that I can show is independent of V5. Alternatively, if there's a way to quantify how much of the correlation between V1 and V2–4 is due to correlation between V1–V5 and how much is independent, that would be a good solution as well.
I have been reading a lot about two-way ANOVA, thinking about doing it between V1 as rows, V5 as columns, and one of V2–4 as outcome. But even if I do this (which would take significant effort, since I couldn't find a simple Python implementation), I'm not sure if it would advance me towards my goals. Any help is appreciated.