The GENMOD Procedure

Example 44.1 Logistic Regression

In an experiment comparing the effects of five different drugs, each drug is tested on a number of different subjects. The outcome of each experiment is the presence or absence of a positive response in a subject. The following artificial data represent the number of responses r in the n subjects for the five different drugs, labeled A through E. The response is measured for different levels of a continuous covariate x for each drug. The drug type and the continuous covariate x are explanatory variables in this experiment. The number of responses r is modeled as a binomial random variable for each combination of the explanatory variable values, with the binomial number of trials parameter equal to the number of subjects n and the binomial probability equal to the probability of a response.

The following DATA step creates the data set:

data drug;
   input drug$ x r n @@;
A  .1   1  10   A  .23  2  12   A  .67  1   9
B  .2   3  13   B  .3   4  15   B  .45  5  16   B  .78  5  13
C  .04  0  10   C  .15  0  11   C  .56  1  12   C  .7   2  12
D  .34  5  10   D  .6   5   9   D  .7   8  10
E  .2  12  20   E  .34 15  20   E  .56 13  15   E  .8  17  20

A logistic regression for these data is a generalized linear model with response equal to the binomial proportion r/n. The probability distribution is binomial, and the link function is logit. For these data, drug and x are explanatory variables. The probit and the complementary log-log link functions are also appropriate for binomial data.

PROC GENMOD performs a logistic regression on the data in the following SAS statements:

proc genmod data=drug;
   class drug;
   model r/n = x drug / dist = bin
                        link = logit

Since these data are binomial, you use the events/trials syntax to specify the response in the MODEL statement. Profile likelihood confidence intervals for the regression parameters are computed using the LRCI option. General model and data information is produced in Output 44.1.1.

Output 44.1.1: Model Information

The GENMOD Procedure

Model Information
Distribution Binomial
Link Function Logit
Response Variable (Events) r
Response Variable (Trials) n

The five levels of the CLASS variable DRUG are displayed in Output 44.1.2.

Output 44.1.2: CLASS Variable Levels

Class Level Information
Class Levels Values
drug 5 A B C D E

In the "Criteria For Assessing Goodness Of Fit" table displayed in Output 44.1.3, the value of the deviance divided by its degrees of freedom is less than 1. A p-value is not computed for the deviance; however, a deviance that is approximately equal to its degrees of freedom is a possible indication of a good model fit. Asymptotic distribution theory applies to binomial data as the number of binomial trials parameter n becomes large for each combination of explanatory variables. McCullagh and Nelder (1989) caution against the use of the deviance alone to assess model fit. The model fit for each observation should be assessed by examination of residuals. The OBSTATS option in the MODEL statement produces a table of residuals and other useful statistics for each observation.

Output 44.1.3: Goodness-of-Fit Criteria

Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 12 5.2751 0.4396
Scaled Deviance 12 5.2751 0.4396
Pearson Chi-Square 12 4.5133 0.3761
Scaled Pearson X2 12 4.5133 0.3761
Log Likelihood   -114.7732  
Full Log Likelihood   -23.7343  
AIC (smaller is better)   59.4686  
AICC (smaller is better)   67.1050  
BIC (smaller is better)   64.8109  

In the "Analysis Of Parameter Estimates" table displayed in Output 44.1.4, chi-square values for the explanatory variables indicate that the parameter values other than the intercept term are all significant. The scale parameter is set to 1 for the binomial distribution. When you perform an overdispersion analysis, the value of the overdispersion parameter is indicated here. See the section Overdispersion for a discussion of overdispersion.

Output 44.1.4: Parameter Estimates

Analysis Of Maximum Likelihood Parameter Estimates
Parameter   DF Estimate Standard
Likelihood Ratio 95%
Confidence Limits
Wald Chi-Square Pr > ChiSq
Intercept   1 0.2792 0.4196 -0.5336 1.1190 0.44 0.5057
x   1 1.9794 0.7660 0.5038 3.5206 6.68 0.0098
drug A 1 -2.8955 0.6092 -4.2280 -1.7909 22.59 <.0001
drug B 1 -2.0162 0.4052 -2.8375 -1.2435 24.76 <.0001
drug C 1 -3.7952 0.6655 -5.3111 -2.6261 32.53 <.0001
drug D 1 -0.8548 0.4838 -1.8072 0.1028 3.12 0.0773
drug E 0 0.0000 0.0000 0.0000 0.0000 . .
Scale   0 1.0000 0.0000 1.0000 1.0000    

Note: The scale parameter was held fixed.

The preceding table contains the profile likelihood confidence intervals for the explanatory variable parameters requested with the LRCI option. Wald confidence intervals are displayed by default. Profile likelihood confidence intervals are considered to be more accurate than Wald intervals (see Aitkin et al. (1989)), especially with small sample sizes. You can specify the confidence coefficient with the ALPHA= option in the MODEL statement. The default value of 0.05, corresponding to 95% confidence limits, is used here. See the section Confidence Intervals for Parameters for a discussion of profile likelihood confidence intervals.