James Waters writes:

I’m a biology graduate student working on the physiology of whole ant colonies and I’ve been struggling with planning out the statistics for an experiment I’ve been working on. The main question I have, is once a linear mixed model has been calculated, how do you test for significant differences among the regression coefficients attributed to the random effects (i.e., subjects in a longitudinal study)? I’ve been using R, SAS, or Statistica to do the model, but I haven’t been able to figure out how to test for differences among random-level coefficients in any of these programs, but that is what the goal of my experiment is, I’m not actually interested in the grand or whole population-level equations.

In the terms of my experiments, I have 20 ant colonies that I keep in the lab and have been measuring the metabolic rates of once a month as they grow in mass. So there are three variables in my experiment: Colony ID, Colony mass, and Colony metabolic rate.

There is usually a good regression of metabolic rate on mass. My question is do the regression coefficients (both slope/intercept) differ between different colonies?

Usually, this would be easy to do with ANCOVA, but since this is a longitudinal study, the independence assumption is violated and so that’s how I came to think about doing this with mixed models instead. Mixed models seem to pretty easily give you the regression coefficients both for the whole population and for the random effects, but it’s completely unclear to me how to test for statistical differences among these coefficients.

My reply: It sounds like you have these models under control. I’d recommend forgetting about the testing. Everything differs, so you know ahead of time that these differences are not zero. Using the multilevel model, you can get inferences (simulations, estimates, uncertainty intervals) for any comparisons you might be interested in.

Waters replied:

You wrote, “….I’d recommend forgetting about the testing. Everything differs, so you know ahead of time that these differences are not zero.”

But the goal is to quantify and compare the differences… to know whether the population of colonies all show more or less the same slopes or different ones. The only way to know this is to estimate the variance or standard error of those slopes… i.e., the random-effects.

My reply: Yes, that’s fine. You can get se’s, no problem. (In R, fit model using lmer(), then se.ranef()). I just wouldn’t frame it as “testing.” I’d also recommend plotting the estimates and se’s vs. relevant colony-level predictors.