# Reasonableness of assumptions for a generalized linear mixed-effects model

This is an extension of Goodness of fit for a logit-transformed linear random-effects model?. Here is the setting of the problem, inserted below for convenience.

There is an examination that students (indexed by $$i$$) can take once annually in a three-year program. Students are not obligated to take this examination each year, but are required to take it at least once over the three years.

A student could (but is not obligated to) take an exam-prep class of sorts either before or after an exam at time $$t$$ (indicated by $$\alpha_{it}$$) and may be in years 1, 2, or 3 of the program (indicated by $$\beta_{it}$$). For our purposes, if student $$i$$ never takes the prep class, $$\alpha_{it} = 0$$ for all $$t$$. The value $$t$$ indicates the calendar year in which the exam was taken, distinct from the year the student is in the program.

This was the original way I had considered modeling the problem:

Suppose I have a linear mixed-effects model $$\log\left(\dfrac{y_{it}}{1 - y_{it}} \right) = \mu + b_1\alpha_{it} + b_2\beta_{it} + \gamma_i+\epsilon_{it}$$

where $$y_{it} \in (0, 1)$$ is the exam score expressed as a percentage of the $$i$$th student at time $$t$$(truncated from above at 0.999), $$\mu$$ is an intercept, $$\alpha_{it} \in \{0, 1\}$$ (a binary indicator equalling $$1$$ if an exam-prep class was taken before taking the exam at time $$t$$), $$\beta_{it} \in \{1, 2, 3\}$$, $$\gamma_i \sim \mathcal{N}(0, \sigma^2_{\gamma})$$ is a random effect used to incorporate student-to-student variability, and $$\epsilon_{it} \sim \mathcal{N}(0, \sigma^2_{\epsilon})$$ is the usual noise term. $$b_1$$ and $$b_2$$ are your usual coefficients estimated through least squares (taking into account the random effect).

Based on one of the comments I received in the original question, one suggestion was to use a Gamma generalized linear mixed-effects model instead, so something like $$\log(y_{it}) = \mu + b_1\alpha_{it} + b_2\beta_{it} + \gamma_i$$ where we assume $$y_{it}$$ is Gamma distributed. This is great for me, because I can use deviance testing to compare models, but this makes me wonder the following:

• How do I know that a Gamma assumption for $$y_{it}$$ is reasonable, other than that $$y_{it}$$ must be a positive value?
• Why would I choose a log link over a different link?
• How would I know that $$\gamma_i$$ being normally distributed is a reasonable assumption?

Journal articles and textbooks are appreicated.

• Thank you for another informative answer. I've never run a Beta regression myself before: is it standard to use the $\log$ link with a Beta GLM? I'm working on this primarily for a non-statistician audience, so any resources that specifically outline such a procedure would be great. – Clarinetist Jun 19 at 16:01