Refer to the following artificial data:
Denote by M0 the logistic model with only an intercept term and by M1 the model that also has x as a linear predictor. Denote the maximized log likelihood values by L0 for M0, L1 for M1, and Ls for the saturated model. Recall that G2(Mi) = -2(Li - Ls), i = 0, 1. Create a data file in two ways, entering the data as (i) ungrouped data: 12 individual binary observations, (ii) grouped data: three summary binomial observations each with sample size = 4. The saturated model has 12 parameters for data file (i) but three parameters for data file (ii).
a. Fit M0 and M1 for each data file. Report L0 and L1 (or -2L0 and -2L1) in each case. Note that they do not depend on the form of data entry.
b. Show that the deviances G2(M0) and G2(M1) depend on the form of data entry. Why is this? (Hint: They depend on Ls. Would Ls depend on the form of data entry? Why? Thus, you should group the data to use the deviance to check the fit of a model.)
c. Show that the difference between the deviances, G2(M0 | M1), does not depend on the form of data entry (because Ls cancels in the difference). Thus, for testing the effect of a predictor, it does not matter how you enter the data.