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I have a situation where my current model looks like MODEL 1: C ~ A * B + (1|random effect, that is C is predicted by A and B and their interaction and all 3 variables are continuous variables.

I am primarily analyzing it as a multilevel mixed model. However, I noticed that A & B when aggregated over all levels of the random effect turn out to be highly negatively correlated (about -.90), raising concerns of multicollinearity, however, the predicted correlation in the multilevel model ranges from -.60 to -.75 and when I compute a vif measure of MODEL 1, all vif indicators are bellow 2.

The estimates of model 1 show an effect of A (estimate 0.364, p<.001) but not B (0.0331, p=.4) and a significant A*B interaction: estimate 0.132, p<.001)

However, when I test them individually e.g. (MODEL 2: C ~ A + (1|random effect or MODEL 3: C ~ B + (1|random effect both A and B appear significant and with opposite directions as expected. standardized estimate for A is 0.33571 and for B is -0.15296

This is primarily a correlational analysis, as there is no real causal relation or order between A, B and C in my case and I am interested in all 3 variables (theoretically i. Given that the correlation of A or B with C is negligible <.2, I am wondering if it would be sound to reverse the predictor of interest (A) into a DV position, so running a model like the one below

MODEL 4: A ~ C*B + (1|random effect)

Would this be a sound way to deal with collinearity in this case, and would a result from this model allow me to more confidently accept or interpret the results from model 1, mainly the interaction and the relation between A & C without concerns for the collinear predictors in model 1?

Thanks!

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1 Answer 1

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A couple of points:

  1. The fact that most of your diagnostics come back without issue is exactly why multicollinearity is so dangerous-- it can make your model do strange things that really can't be "caught" by standard diagnostics.

  2. Multiple Regression of any kind is by definition directional. E.g (a ~ b * c) is fundamentally different than (b~a * c). The just aren't the same thing and have totally different interpretations. If you care about A~ C * B, then sure, run the model. But it isn't the same as your initial analysis.

If really all you care about is the pairwise relationships, then it probably makes more sense to just do pairwise t-tests with a p-value adjustment, and maybe visualize the correlation. Without some sort of hypothesized directional relationship, linear models aren't really your friend here in my opinion.

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  • $\begingroup$ Thank you for your answer. Just to add (if it is useful to clarify) that in this case, A & C theoretically represent the same underlying construct just measured in different ways, that's why I wondered whether an interaction between AB or CB could be taken as "equivalent". For example, A and C are supposed to represent the intensity of a subjective experience, but A is measured subjectively (via self-report), whereas C is measured using bodily signals and sensors. $\endgroup$
    – Myriad
    Commented Sep 7, 2021 at 13:35
  • $\begingroup$ In this case, you may consider an additive dependent variable. In other worse, model (A+C) ~ B + Random Effect $\endgroup$ Commented Sep 7, 2021 at 14:08

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