A study has n_i independent binary observations {y_i1,...,y_in_i} when X = x_i, i = 1,..., N, with n = ∑_i n_i. Consider the model logit(π_i) = α + βx_i, where π_i = P(Y_ij = 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 n_i = 1. Show that the deviance depends on but not y_i. Hence, it is not useful for checking model fit.