I'd like to discuss the centering of continuous predictor variables in multiple linear regression with an interaction term for the sake of "relieving" multicollinearity.
I've read about multicollinearity in many sources, but for the points I bring up here, this paper by Afshartous and Preston (2011) should offer sufficient background/support: http://www.amstat.org/publications/jse/v19n3/afshartous.pdf
For simplicity, I'm considering a multiple regression model with one binary predictor, one continuous and the interaction between the two. This implies I'm not considering a polynomial model where the interaction term is a given predictor multiplied by itself.
Note: In this thread, I'm not interested in the improved interpretability of a model with centered predictors (although I am aware of that benefit).
- In order to avoid model misspecification, we should assess whether or not an interaction term is appropriate for a given set of data (look at graphs and assess with the p-value from the relevant regression model)
- However, inherent in an interaction term is often notable multicollinearity with the lower-order terms from which it was constructed
- Because of this, it's often suggested we center the continuous predictor before fitting a model with an interaction model
- In Afshartous and Preston (2011), they illustrate how the SE and thus test statistic (t) and thus p-value for the interaction term are not affected by centering (or lack thereof)
Now, consider the general approach to fitting a regression model of the sort described above:
- Fit the "full" model: Include both main effects and the interaction term. If the interaction is significant, this is the final model. Additionally, the individual "main effects" are not meaningful on their own (neither their p-values nor parameter estimates), as both variables are interpreted with respect to their involvement in the interaction.
- If, after fitting the "full" model the interaction is not significant, remove it and assess the "main effects" model.
And herein lies the point I'd like to discuss:
Since inference for the interaction term is not affected by multicollinearity, as described above, we may trust its p-value even when the continuous predictor has not been centered
As such, if the interaction is significant, we trust this is a "real" result and may go on to interpret the model as described in point (1) above, since we aren't concerned about the individual p-values from the main effects at this point anyway (each predictor is important because it's involved in an interaction and thus will remain in the model).
Similarly, if the p-value for the interaction term is not significant, we also trust this as a valid result and know that we should remove this term. Upon doing so, we have alleviated any issue with collinearity since the interaction term is no longer in the model, and thus we may trust the inference (p-values) for the main effects in the main effects-only model.
In summary, I feel the two points above indicate there is no motivation from an inferential standpoint to center a predictor in the described type of model, because, no matter what, it will always be correctly specified since the interaction term is not affected by collinearity. What I'm looking for is confirmation that my logic is correct.
Lastly, my other question is this: I understand that, inherently, the parameter estimates for the main effects will be different between a centered and uncentered model solely because you're estimating two different parameters (stated many times in this paper and makes sense intuitively). However, what I'm still uncertain about is: Relatively speaking, are the parameter estimates for the main effects "incorrect" when you don't center the continuous predictor and there's collinearity between the main effects and interaction?
In other words/Why this matters: Say I want to predict the value of the outcome for given values of the predictors (so I literally write out the equation and plug-in specified values of each X). Here, the parameter estimates for the (uncentered) predictors come into play because they're being used in the calculation, but are they "correct" on the uncentered scale? Or have they been affected by multicollinearity and thus are "off" by some amount?
Final summary: I think inference and model specification are not problematic when the predictor (as described here) has not been centered, but from a predictive standpoint, may we trust the main effects parameter estimates from an uncentered model where the main effects are presumably collinear with the interaction term?