Introduction to Categorical Data Analysis

146 BUILDING AND APPLYING LOGISTIC REGRESSION MODELS
freedom, called the residual df for the model, subtract the number of parameters in
the model from the number of parameters in the saturated model. The number of
parameters in the saturated model equals the number of settings of the predictors,
which is the number of binomial observations for the data in the grouped form of the
contingency table. Large X 2 (M) or G 2 (M) values provide evidence of lack of fit.
The P -value is the right-tail probability.
We illustrate by checking the model Section 4.3.2 used for the data on AIDS
symptoms (y = 1, yes), AZT use, and race, shown again in Table 5.4. Let x = 1 for
those who took AZT immediately and x = 0 otherwise, and let z = 1 for whites and
z = 0 for blacks. The ML fit is
logit( ˆπ) =−1.074 − 0.720x + 0.056z
The model assumes homogeneous association (Section 2.7.6), with odds ratio between
each predictor and the response the same at each category of the other variable. Is
this assumption plausible?
Table 5.4. Development of AIDS Symptoms by AZT Use and Race
Symptoms
Race AZT Use Yes No Total
White Yes 14 93 107
No 32 81 113
Black Yes 11 52 63
No 12 43 55
For this model fit, white veterans with immediate AZT use had estimated probability
0.150 of developing AIDS symptoms during the study. Since 107 white veterans
tookAZT, the fitted number developing symptoms is 107(0.150) = 16.0, and the fitted
number not developing symptoms is 107(0.850) = 91.0. Similarly, one can obtain
fitted values for all eight cells in Table 5.4. Substituting these and the cell counts
into the goodness-of-fit statistics, we obtain G 2 (M) = 1.38 and X 2 (M) = 1.39. The
model applies to four binomial observations, one at each of the four combinations of
AZT use and race. The model has three parameters, so the residual df = 4 − 3 = 1.
The small G 2 and X 2 values suggest that the model fits decently (P = 0.24).
5.2.3 Checking Fit: Grouped Data, Ungrouped Data, and
Continuous Predictors
The beginning of Section 4.2 noted that, with categorical predictors, the data file
can have the form of ungrouped data or grouped data. The ungrouped data are the