The LOGISTIC Procedure

Example 72.9 Goodness-of-Fit Tests and Subpopulations

A study is done to investigate the effects of two binary factors, A and B, on a binary response, Y. Subjects are randomly selected from subpopulations defined by the four possible combinations of levels of A and B. The number of subjects responding with each level of Y is recorded, and the following DATA step creates the data set One:

data One;
   do A=0,1;
      do B=0,1;
         do Y=1,2;
            input F @@;
23 63 31 70 67 100 70 104

The following statements fit a full model to examine the main effects of A and B as well as the interaction effect of A and B:

proc logistic data=One;
   freq F;
   model Y=A B A*B;

Results of the model fit are shown in Output 72.9.1. Notice that neither the A*B interaction nor the B main effect is significant.

Output 72.9.1: Full Model Fit

The LOGISTIC Procedure

Model Information
Response Variable Y
Number of Response Levels 2
Frequency Variable F
Model binary logit
Optimization Technique Fisher's scoring

Number of Observations Read 8
Number of Observations Used 8
Sum of Frequencies Read 528
Sum of Frequencies Used 528

Response Profile
Y Total
1 1 191
2 2 337

Probability modeled is Y=1.

Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics
Criterion Intercept Only Intercept and
AIC 693.061 691.914
SC 697.330 708.990
-2 Log L 691.061 683.914

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 7.1478 3 0.0673
Score 6.9921 3 0.0721
Wald 6.9118 3 0.0748

Analysis of Maximum Likelihood Estimates
Parameter DF Estimate Standard
Pr > ChiSq
Intercept 1 -1.0074 0.2436 17.1015 <.0001
A 1 0.6069 0.2903 4.3714 0.0365
B 1 0.1929 0.3254 0.3515 0.5533
A*B 1 -0.1883 0.3933 0.2293 0.6321

Pearson and deviance goodness-of-fit tests cannot be obtained for this model because a full model containing four parameters is fit, leaving no residual degrees of freedom. For a binary response model, the goodness-of-fit tests have $m-q$ degrees of freedom, where m is the number of subpopulations and q is the number of model parameters. In the preceding model, $m=q=4$, resulting in zero degrees of freedom for the tests.

The following statements fit a reduced model containing only the A effect, so two degrees of freedom become available for testing goodness of fit. Specifying the SCALE=NONE option requests the Pearson and deviance statistics. With single-trial syntax, the AGGREGATE= option is needed to define the subpopulations in the study. Specifying AGGREGATE=(A B) creates subpopulations of the four combinations of levels of A and B. Although the B effect is being dropped from the model, it is still needed to define the original subpopulations in the study. If AGGREGATE=(A) were specified, only two subpopulations would be created from the levels of A, resulting in $m=q=2$ and zero degrees of freedom for the tests.

proc logistic data=One;
   freq F;
   model Y=A / scale=none aggregate=(A B);

The goodness-of-fit tests in Output 72.9.2 show that dropping the B main effect and the A*B interaction simultaneously does not result in significant lack of fit of the model. The tests’ large p-values indicate insufficient evidence for rejecting the null hypothesis that the model fits.

Output 72.9.2: Reduced Model Fit

The LOGISTIC Procedure

Deviance and Pearson Goodness-of-Fit Statistics
Criterion Value DF Value/DF Pr > ChiSq
Deviance 0.3541 2 0.1770 0.8377
Pearson 0.3531 2 0.1765 0.8382

Number of unique profiles: 4