Example 39.7: Goodness-of-Fit Tests and Subpopulations

The LOGISTIC Procedure

Example 39.7: 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 entered into data set A.

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

A full model is fit to examine the main effects of A and B as well as the interaction effect of A and B.

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

Output 39.7.1: Full Model Fit

The LOGISTIC Procedure

Model Information
Data Set	WORK.A
Response Variable	Y
Number of Response Levels	2
Number of Observations	8
Frequency Variable	F
Sum of Frequencies	528
Link Function	Logit
Optimization Technique	Fisher's scoring

Response Profile
Ordered Value	Y	Total Frequency
1	1	191
2	2	337

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

Model Fit Statistics
Criterion	Intercept Only	Intercept and Covariates
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 Error	Chi-Square	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
*AB**	1	-0.1883	0.3933	0.2293	0.6321

Association of Predicted Probabilities and Observed Responses
Percent Concordant	42.2	Somers' D	0.118
Percent Discordant	30.4	Gamma	0.162
Percent Tied	27.3	Tau-a	0.054
Pairs	64367	c	0.559

Pearson and Deviance goodness-of-fit tests cannot be obtained for this model since 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.

Results of the model fit are shown in Output 39.7.1. Notice that neither the A*B interaction nor the B main effect is significant. If a reduced model containing only the A effect is fit, 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=a;
      freq F;
      model Y=A / scale=none aggregate=(A B);
   run;

Output 39.7.2: Reduced Model Fit

The LOGISTIC Procedure

Model Information
Data Set	WORK.A
Response Variable	Y
Number of Response Levels	2
Number of Observations	8
Frequency Variable	F
Sum of Frequencies	528
Link Function	Logit
Optimization Technique	Fisher's scoring

Response Profile
Ordered Value	Y	Total Frequency
1	1	191
2	2	337

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

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

Number of unique profiles: 4

Model Fit Statistics
Criterion	Intercept Only	Intercept and Covariates
AIC	693.061	688.268
SC	697.330	696.806
-2 Log L	691.061	684.268

Testing Global Null Hypothesis: BETA=0
Test	Chi-Square	DF	Pr > ChiSq
Likelihood Ratio	6.7937	1	0.0091
Score	6.6779	1	0.0098
Wald	6.6210	1	0.0101

Analysis of Maximum Likelihood Estimates
Parameter	DF	Estimate	Standard Error	Chi-Square	Pr > ChiSq
Intercept	1	-0.9013	0.1614	31.2001	<.0001
A	1	0.5032	0.1955	6.6210	0.0101

Association of Predicted Probabilities and Observed Responses
Percent Concordant	28.3	Somers' D	0.112
Percent Discordant	17.1	Gamma	0.246
Percent Tied	54.6	Tau-a	0.052
Pairs	64367	c	0.556

The goodness-of-fit tests (Output 39.7.2) 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.

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