10.22 Problem 3.15 analyzed data from a General Social Survey on responses of 1308 subjects to the question, “Within the past 12 months, how many people have you known personally that were victims of homicide?” It used Poisson and negative binomial GLMs for count data. Here is a possible GLMM: For response yi for subject i of race xi (1 = black, 0 = white),

where conditional on ui , yi has a Poisson distribution, and where {ui } are independent N(0, σ). Like the negative binomial GLM, unconditionally (when σ > 0) this model can allow more variability than the Poisson GLM.

a. The Poisson GLMM has ˆα = −3.69 and ˆ β = 1.90, with ˆσ = 1.6. Show that, for subjects at the mean of the random effects distribution, the estimated expected responses are 0.167 for blacks and 0.025 for whites.

b. Interpret ˆ β.

log[E(Y)]=ui + + xi