# R-Square Change in a Multivariate Regression

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)

anova(BaseModel,Confound.1.Model)

Confound.2.Model <- cbind(a, b, c) ~ Neuroticism + Extraversion + Confound2, data)

anova(BaseModel,Confound.1.Model)

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?

Second, if there is a substantive reason to examine the correlated errors across the set of dependent variables (e.g., different temporal measurements of the same construct), then I would recommend using structural equation modeling in which the dependent variables are predicted by a latent variable, and the independent variables serve as predictors for the latent variable. (Note, this almost is a MIMIC model, but in truth, it actually is structurally equivalent to a CFA...happy to speak more to that in comments if relevant.) In this case, you can obtain an $R^2$ for the latent variable, and you can do an F-test with the two different models (2 iv's vs. 3 iv's).
• If they are not theoretically comparable constructs, then this is reasonable. If you want to be judicious in evaluating whether or not the observed confounding is truly a consequence of the confounding variables or some attribute of shared variance across the errors of the dependent variables, (1) you could run separate univariate analyses, (2) examine the $∆ R^2$ for each, (3) run a PCA with the dep. var.s, (4) run the analysis with this as the dep. var., (5) calculate this $∆ R^2$. This is not a rigorous assessment, but it may suggest whether a difference is worth exploring. Mar 18 '18 at 19:35
What I normally see in papers using hierarchical multiple regression is simply the arithmetic difference between the $R^2$s between the models of interest.