In Section 13.5.1 and Problem 13.42 we saw that for Poisson GLMMs, the marginal effects are the same as the cluster-specific effects. This does not imply that ML estimates of effects are the same for a Poisson GLMM and a Poisson GLM. Explain why. (For the GLMM, is the marginal distribution Poisson?)

Data from Problem 13.42:

Consider the loglinear random effects model

log[E(Y_it | u_i)] = x’_it β + z’_it u_i,

where {µ_i} are independent N(0, ∑). Show that this implies the marginal loglinear model

with the same fixed effects but with offset term. For the random-intercept case, indicate the role of σ on the size of the offset. Explain what happens when σ = 0.