A repeated-measures LMM... Am I understanding this correctly?

Question

So I've been reading West et al's book on Linear Mixed Models as I'm trying to figure out how I should analyze my data, and I was wondering if I've correctly understood what I've ready so far. So, I have repeated measures on a response variable: for each subject, a measure under the control condition and under a "treatment." The subjects also differ in sex, which we predict to be an important factor in the response to treatment. Importantly, some subjects (not all) are measured multiple times, because they went through "treatment" multiple times. So, in those cases, I have a new control measure and treatment measure for each individual.

What I think I is a LMM with 3 fixed effects: my response variable, sex, and treatment condition. In addition, I think I will need 1 random effect: subject identification (to correct for the non-independence of some of my data points because some subjects were included twice).

First of all, am I missing anything? The book lost me a bit when it started including all of these interactions in their models... I'm also working with very small sample sizes (because I work with endangered and long-lived animals), so I worry about trying to cram too many variables into the model. Second, do my data need to follow any particular distribution? Or do the residuals need to follow some particular distribution?

Thanks for taking the time to read...

Answer 1

First of all given you are using a standard LMM you definitely look to have normally distributed residuals. ie. $\epsilon \sim N(0, \sigma^2 I)$.

When you are using a LMM you are practically saying that you data have a distribution as : $y \sim N( X\beta, \sigma^2 I + ZDZ^T)$. So in relation with what we wrote above : $y|u \sim N( X\beta + Zu, \sigma^2 I)$;

I believe that your response variable is not a fixed effect. It is actually what you try to predict so it is not part of the $X$ model matrix. I would think that a simple interaction between treatment and sex (something like Treatment*Sex will be adequate for first try coupled with a "intercept-only" random effects structure (eg. (1|Subject)). As you importantly identify yourself, if your total number of samples isn't that great, just feeding in a big number of covariates is a bit "insecure". I would also contemplate not using an intercept when starting mostly because that "should" be taken care of by your interaction term but that's a bit of a personal taste thing. :) Good luck.