In short, my question is as follows:
- Why is it common to assume normally distributed random effects (especially in generalized linear mixed models)?
A longer version:
Under some circumstances, an approximately normally distributed random effect makes sense. For example, say we measure individuals' weight ($y$) depending on the type of diet ($x$) they were on, once before and once a month after dieting. If individuals ($\upsilon$) are measured twice, then the following LMM:
$$y_{ij} = \beta_0+ \beta_1 x + \upsilon_i + \epsilon_{ij} \\ \upsilon \sim \mathcal{N}(0,\,\sigma_\upsilon^2), \; \epsilon \sim \mathcal{N}(0,\,\sigma_\epsilon^2)$$
basically assumes that individuals ($\upsilon$) come from some larger population, which causes a random, normally distributed offset in their initial weight. One could argue that whatever (unknown) differences are present between individuals (genetic, environmental, lifestyle), might sum up to a normal distribution as many sums of independent random variables would. In fact, we could use almost the same argument for the errors of the outcome variable ($\epsilon$).
However, say we count birds ($y$) in different terrain types ($x$) on different islands ($\upsilon$) and use a Poisson GLMM, why, if at all, is the normality assumption still defensible? Surely the sum of random variables differing between two islands can cause a normally distributed offset for an outcome with normally distributed errors, but how can we justify this for a non-normal error structure?
I understand that a GLMM models the random effect in the linear part, but isn't this linear part still not assumed to have a normal error structure? (Sorry for the double negative.)
Bonus question:
- Are there any simple examples of non-normal random effects (e.g. Poisson distributed)?
lme4
!). I can imagine the computational convenience, but I still don't quite understand why normality on the linear predictor scale is defensible on the same grounds. Perhaps I'm thinking in the wrong direction, but aren't the 'residuals' of a GLMM non-normal like described here: stats.stackexchange.com/a/139624/176202)? Even after applying the link function, I can't see how the normal distribution arises. $\endgroup$ – Frans Rodenburg Sep 13 '18 at 3:13