A post-hoc test for linear regressions with interaction?

Question

I've been looking for an answer for a question (which I thought was rather trivial) for a while now, but can't seem to get the correct solution for it.

I'm working on data of blood parameters (e.g. hematocrit, HCT; a continuous variable) from fish reared under different temperature treatments (treatment; a factor). These measurements were made at several time points throughout the trial, however are independent from each other, as we did not sample the same fish twice. During this time the fish put on body mass, which we measured too (weight, a continuous variable). I'd like to know whether there is a significant effect of treatment on HCT after controlling for weight. I'm not really interested in the effect weight has on HCT and treatment, I just want to adjust for it. The problem I have is that treatment and weight are not independent from each other, as fish grew differently within each treatment, so I can't go down the ANCOVA route. From my understanding I can simply fit a linear regression in form HCT~treatment*weight, but this would allow me to only compare each treatment to the treatment set as baseline. My question is: Is there a more elegant way to statistically compare the intercepts of all treatment groups with each other including adjustments for multiple comparisons (essentially a post-hoc test for lm's with interaction), instead of running the same model with different base line treatments?

Thanks,

Hanna

Answer 1

I don't think you need to include an interaction in your model. In essence you have a mediational situation: your treatment affects both weight and HCT, and weight affects HCT as well. If the initial weights (pre-treatment) were the same, i.e., the fish were randomized into the treatment conditions, then there are two different effects of treatment on HCT that could be estimated: the total effect or the direct effect. You need to decide which you are interested in.

Controlling for weight allows you to estimate the direct effect. You don't need an interaction for this, just include weight as a covariate in the model. Because weight is correlated with treatment, you will have lower power, but there isn't too much you can do about that.

To estimate the total effect, you can ignore weight (i.e., don't include it in the model). This will give you the direct effect plus the indirect effect (treatment -> weight -> HCT).

I don't think you need a post-hoc test, either. 'Post-hoc' means you looked at your data and decided what you wanted to test based on what you saw. I'm guessing when you designed the study, you wanted to compare all the treatment groups. If so, Tukey contrasts would be appropriate. If not, whichever multiple comparisons strategy matches your situation would be fine here.

If the fish are measured after different amounts of time / exposure to the treatments, you'll probably want to control for that, although if the times are uncorrelated with the treatments, you wouldn't have to.