I am investigating whether the predictive validity of the Big Five personality traits can be improved by controlling for sources of self-report error (e.g. memory, interpretation biases, ect.). I am specifically interested in identifying the confounding variables with the strongest effects.
To ensure that I am testing general differences in predictive validity, I would like to consider three outcome variables simultaneously (a,b,c), known to correlate strongly with two of the Big Five traits (Neuroticism and Extraversion)
Thus, I conducted several hierarchical multivariate regressions in the following form:
BaseModel <- lm(cbind(a, b, c) ~ Neuroticism + Extraversion, data)
Confound.1.Model <- cbind(a, b, c) ~ Neuroticism + Extraversion + Confound1, data)
Confound.2.Model <- cbind(a, b, c) ~ Neuroticism + Extraversion + Confound2, data)
Now, both of the ANOVAs tell me that the confounds explain incremental variance in the set of outcomes, over and above personality. However, I do not know how much more variance is explained -- that is to say, the R-Square change metric is not provided, as it normally would be in a multiple regression with one outcome variable.
Is there some valid way to calculate the R-square change for a multivariate regression? (e.g. averaging the R-square changes for the individual outcomes).
Is there some other statistic that is better suited for my question?
Thank you for your time.