A data set on pregnancy rates among girls under 18 years of age in 13 north central Florida counties has information on a 3-year total for each county i on n_i = number of births and y_i = number of those for which mother had age under 18.

a. A beta-binomial model states that given {π_i}, {Y_i} are independent {bin(n_i­, π_i)} variates, and (π_i) are independent from a beta(α, β) distribution. The ML estimated parameters are α̂ = 9.9 and β̂ = 240.8. Use the mean and variance to describe the estimated beta distribution and the estimated marginal distribution of Y_i (as a function of n_i).

b. Quasi-likelihood using variance function (13.10) for the model logit(µ_i) = α has α̂ = –3.18 and ρ̂ = 0.005. Describe the estimated mean and variance of Y_i.

c. Quasi-likelihood using variance (13.11) for the model logit(µ_i) = α has α̂ = –3.35 and ϕ̂ = 8.3. Describe the estimated mean and variance of Y_i.

d. The logistic-normal GLMM, logit(π_i) = α + u_i, yields α̂ = – 3.24 and σ̂ = 0.33. Describe the estimated mean of Y_i [Recall (12.8)].