What generalized linear mixed model do you use for 1) binary 2) count data?

• A. Probit and Negative Binomial with random effects
• B. Logistic regression for binary data only with fixed effects
• C. Logistic and Poisson both with random effects
• D. Poisson and Negative Binomial with fixed effects

Answer: (C)

Generalized linear mixed models handle different outcome types by pairing a link function with a distribution for the data and adding random effects to account for correlation within clusters or repeated measures. For a binary outcome, the natural approach is logistic regression with a logit link, and when data are clustered, you include a random effect to form a logistic mixed model. For count data, the natural approach is Poisson regression with a log link, and similarly you add a random effect to create a Poisson mixed model. Putting those together, the standard choice is a logistic GLMM for binary data and a Poisson GLMM for counts, both incorporating random effects to account for non-independence. (Note: if counts show overdispersion, a negative binomial GLMM with random effects can also be appropriate, but Poisson with random effects is the typical default.)