# General linear model for counts which are "correlated"

The typical general linear model (GLM) for count data uses the Poisson link function. The counts there are assumed to be "independent". Now suppose the counts are not "independent" in a sense illustrated by the following toy example.

There are data points on 2100 students who each take 7 courses. The response is the number of "A" grades they earn. The histogram below illustrates the observed "A" count. I'm interested in a GLM for this kind of response distribution with some predictors (for example, # hours spent studying + household income).

From a modeling perspective, it is reasonable to believe that students who get an "A" in one course are likely to get an "A" in other courses (and vice versa). So I am unsure as to what an appropriate link function would be for a GLM. It is clear that the responses don't follow a Poisson distribution in this example. But would a logarithmic link function (i.e. Poisson regression) still be valid in this scenario? Any thoughts would be much appreciated.

• I have a question about getting significance of a predictor in ZIP models. Suppose $Y_i$ is a response. I want to model a single predictor $X_i$ in both the "binomial" and "Poisson" parts of the model. That is, there are two parameters $\beta_{Bin}$ and $\beta_{Pois}$ corresponding to $X_i$ in this ZIP model (Q1.) Is this valid to use the same predictor in both parts of the model? Commented Dec 15, 2015 at 15:38
• (Q2) Fitting the ZIP model here gives an estimate and standard error for $\beta_{Bin}$ and $\beta_{Pois}$. Is there way to combine them to get a total $p$-value corresponding to the significance of $X_i$? Is there an analytical way? For example, $F$-tests or linear restrictions of some sort. Commented Dec 15, 2015 at 15:39
• For the first question, you should be able to use the same x-variable in both parts, as they would be estimating the effect of that predictor on two different processes. I rarely use p-values for statistical inference (I tend to prefer bayesian estimation), so I am not able to directly answer your second question. However, I'm not sure why it wouldn't be preferable to look at the significance of the two $\beta$ coefficients separately. Commented Dec 15, 2015 at 19:39