Refer to the following artificial data:

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x number of trials number of successes 04 1 14 2 24 4

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Denote by M0 the logistic null model and by M1 the model that also has x as a predictor. Denote the maximized log-likelihood values by L0 for M0, L1 for M1, and Ls for the saturated model. Create a data file in two ways, entering the data as

(i) ungrouped data: 12 individual binary observations, (ii) grouped data: 3 summary binomial observations each with sample size = 4.

a. Fit M0 and M1 for each data file. Report L0 and L1 (or −2L0 and −2L1) in each case. Do they depend on the form of data entry?

b. Show that the deviances for M0 and M1 depend on the form of data entry. Why is this? (Hint: The saturated model has 12 parameters for data file (i) but 3 parameters for data file (ii).)

c. Show that the difference between the deviances does not depend on the form of data file. Thus, for testing the effect of x, it does not matter how you enter the data.