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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.

Example of Count distribution

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If you treat "student" as a random effect (e.g. by including a random "intercept" to represent the student effect), that's one way of modelling this sort of dependence.

Given a GLM to begin with, this would result in a generalized linear mixed model (GLMM) on which topic you can find quite a few posts here.

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Your histogram looks a lot like a zero-inflated Poisson (ZIP) distribution. In short, the ZIP is a mixture of a Bernoulli (which in this case would model something like probability a student has the potential to get an A) and a Poisson (which would model the number of A's a student receives given they have the potential to get one). If 7 is the maximum number of classes, however, a zero-inflated Binomial model would be more appropriate.

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  • $\begingroup$ 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? $\endgroup$
    – user29620
    Commented Dec 15, 2015 at 15:38
  • $\begingroup$ (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. $\endgroup$
    – user29620
    Commented Dec 15, 2015 at 15:39
  • $\begingroup$ 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. $\endgroup$
    – C.R. Peterson
    Commented Dec 15, 2015 at 19:39

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