Error in specification of GLM in GEE blog post

[staceyhancock]

I came across your blog post on GEEs: https://rlbarter.github.io/Practical-Statistics/2017/05/10/generalized-estimating-equations-gee/.

When defining a GLM, the only situation in which you can write the response variable as a mean plus error is if the response is normally distributed. In the case of other distributions (e.g., binomial, Poisson), the representation $Y_i = \mu_i + \epsilon_i$ is incorrect. For example, if $Y_i$ is binary, then there is no named probability distribution you could put on the error term in order for $Y_i$ to only take on values 0 and 1.

Instead the correct specification of a GLM involves specifying the probability distribution for $Y_i$, e.g., Binomial, Poisson, etc. Then specifying the model equation involving the mean response $\mu_i = E(Y_i)$, as you did in your post: $g(\mu_i) = X_i' \beta$.