Chao Zhang writes:

When I want to know the contribution of a predictor in a multilevel model, I often calculate how much of the total variance is reduced in the random effects by the added predictor. For example, the between-group variance is 0.7 and residual variance is 0.9 in the null model, and by adding the predictor the residual variance is reduced to 0.7, then VPC = (0.7 + 0.9 – 0.7 – 0.7) / (0.7 + 0.9) = 0.125. Then I assume that the new predictor explained 12.5% more of the total variance than the null model. I guess this is sometimes done by some researchers when they need a measure of sort of an effect size.

However, now I have a case in which adding a new predictor (X) greatly increased the between-group variance. After some inspection, I realized that this was because although X correlate with Y positively overall, it correlate with Y negatively within each group, and X and Y vary in the same direction regarding the grouping variable. Under this situation, the VPC as computed above becomes negative! I am puzzled by this because how could the total variance increase? And this seems to invalidate the above method, at least in some situations.

My reply: this phenomenon is discussed in Section 21.7 of my book with Jennifer Hill. The section is entitled, “Adding a predictor can increase the residual variance!”

It’s great when I can answer a question so easily!

This is just a sophisticated example of the ecological fallacy, right?