A study has ni independent binary observations {yi1,…,yini} when X = xi, i = 1,…, N, with n = ∑i ni. Consider the model logit(πi) = α + βxi, where πi = P(Yij = 1).
a. Show that the kernel of the likelihood function is the same treating the data as n Bernoulli observations or N binomial observations.
b. For the saturated model, explain why the likelihood function is different for these two data forms. Hence, the deviance reported by software depends on the form of data entry.
c. Explain why the difference between deviances for two unsaturated models does not depend on the form of data entry.
d. Suppose that each ni = 1. Show that the deviance depends on but not yi. Hence, it is not useful for checking model fit.
