This is a general question about using mean centering to handle time-varying predictors in a hierarchical Bayesian model for longitudinal data. This is NOT a question about reparameterizing a model using the ‘centering’ trick to deal with funnels in joint distributions.
To motivate this question, imagine you have a dataset where a group of people rated their sleep quality and recorded the number of alcoholic drinks they had the day before for 30 days. I want to estimate the relationship between alcohol consumption and sleep quality over time. I land on a multilevel model with random intercepts and slopes where time points are at level 1 and people are at level 2.
One recommendation for dealing with a time-varying predictor like alcohol consumption would be to create two versions using mean centering: one version is person-centered, where you subtract a person’s mean alcohol consumption from all their values (Xi - X_bar); the other is grand mean-centered, where you take the person’s mean alcohol consumption and subtract the grand mean of alcohol consumption (X_bar - X_gm). You would then enter the person-centered version as a level 1 predictor and use the grand-mean centered version to explain random intercept and slope variance. Person-centering is designed to isolate the within-person variation and grand mean centering isolates between-person variation. If instead you entered alcohol consumption into the model as a level 1 fixed effect without mean centering, the resulting estimate would capture a mixture of within person and between person variance. This has been referred to as a conflated effect (Preacher, Zhang, & Zyphur, 2011) or smushed effect (Hoffman, 2015).
A lot has been written about mean centering in the MLM literature, and there is a lot of debate and argument about when and how to use mean centering for substantive reasons beyond just making the intercept interpretable. However, I’ve not seen any discussion of this topic in Pymc materials/examples or the larger Bayesian literature on MLMs for longitudinal data. @drbenvincent recently mentioned this topic in his wonderful tutorial on moderation analysis but the central issue there is slightly different.
So, my question is: Why isn’t mean centering discussed in relation to Bayesian MLMs for longitudinal data? Is it because mean centering isn’t really needed in a Bayesian MLM? Or, is it just an artifact of the way people think about and approach MLMs in Bayesian stats?
I would be really interested to hear others perspectives on this topic and why it doesn’t get much attention in the Bayesian world. I’m also interested to hear if, in fact, mean centering isn’t really necessary in a Bayesian model of the sort described above, and why that is.