Exact Logistic Regression and Exact Poisson Regression

The exact conditional logistic and Poisson regression facilities in the LOGISTIC and GENMOD procedures provide numerous features, including:

data dose;
  input Dose Deaths Total @@;
  datalines;
0 0 3  1 0 3  2 0 3  3 0 3
4 1 3  5 2 3
;
run;

proc logistic data=dose descending;
  model Deaths/Total = Dose;
  exact Dose / estimate = both;
run;

The following SAS statements replicate the previous analysis by using PROC GENMOD:

proc genmod data=dose descending;
  model Deaths/Total = Dose;
  exact Dose / estimate = both;
run;

The output generated by the EXACT statement in GENMOD is identical to that from LOGISTIC. For more information about PROC GENMOD refer to the chapter The GENMOD Procedure in SAS/STAT User's Guide.

For more information about performing exact logistic regression using the LOGISTIC procedure, refer to the paper "Performing Exact Logistic Regression with the SAS System-Revised 2009" by Robert Derr.

The following data, taken from Cox and Snell (1989, pp. 9–10) , consists of the number, Notready, of ingots that are not ready for rolling, out of Total tested, for several combinations of heating time and soaking time:

data ingots;
   input Heat Soak NotReady Total @@;
   lnTotal= log(Total);
   datalines;
7 1.0 0 10  14 1.0 0 31  27 1.0 1 56  51 1.0 3 13
7 1.7 0 17  14 1.7 0 43  27 1.7 4 44  51 1.7 0  1
7 2.2 0  7  14 2.2 2 33  27 2.2 0 21  51 2.2 0  1
7 2.8 0 12  14 2.8 0 31  27 2.8 1 22  51 4.0 0  1
7 4.0 0  9  14 4.0 0 19  27 4.0 1 16
;
run;

The following invocation of PROC GENMOD fits an asymptotic (unconditional) Poisson regression model to the data. The variable NotReady is specified as the response variable, and the continuous predictors Heat and Soak are defined in the CLASS statement as categorical predictors that use reference coding. Specifying the offset variable as lnTotal enables you to model the ratio NotReady/Total.

proc genmod data=ingots;
   class Heat Soak / param=ref;
   model NotReady=Heat Soak / offset=lnTotal dist=poisson link=log;
   exact Heat Soak / joint estimate;
run;

The EXACT statement is specified to additionally fit an exact conditional Poisson regression model. Specifying the LnTotal offset variable models the ratio NotReady/Total; in this case, the Total variable contains the largest possible response value for each observation. The JOINT option produces a joint test for the significance of the covariates, along with the usual marginal tests. The ESTIMATE option produces exact parameter estimates for the covariates.

From the "Analysis of Maximum Likelihood Parameter Estimates" table in Output 3 you can see that only two of the Heat parameters are significantly different than the reference level. Looking at the standard errors you will notice that the unconditional analysis had convergence difficulties with the Heat=7 parameter (Standard Error=264324.6), which implies that the regression parameter for Heat=7 is not estimable for this model thereby precluding inference.

The "Exact Parameter Estimates" table in Output 4 displays parameter estimates and tests of significance for the levels of the CLASS variables versus the reference level. Again, the Heat=7 parameter has some difficulties; however, in the exact analysis, a median unbiased estimate is computed for the parameter instead of a maximum likelihood estimate. The confidence limits show that the Heat variable contains some explanatory power, while the categorical Soak variable is not significant due to wide confidence intervals and p-values equal to 1.000.

The Joint test shown in the "Exact Conditional Analysis" table in Output 5 is produced by specifying the JOINT option in the EXACT statement. The p-values for this test indicate that the parameters for Heat and Soak are jointly significant as explanatory effects in the model.

The rows of this table labeled as “Heat” show the joint significance of all the Heat effect parameters conditional on Soak being in the model. In this case, the Heat parameters explain a significant amount of the variability; however, you can see that the Soak parameters, condional on Heat being in the model, are not significant. A reasonable next step in this modeling process would be to remove the Soak variable from the model as it does not appear to be a significant predictor of the outcome variable NotReady.

Note:If you want to make predictions from the exact results, you can obtain an estimate for the intercept parameter by specifying the INTERCEPT keyword in the EXACT statement. You can also remove the JOINT option to reduce the amount of time and memory consumed.