Performing Exact Logistic Regression with the SAS System — Revised 2009

Exact logistic regression has become an important analytical technique, especially in the pharmaceutical industry, since the usual asymptotic methods for analyzing small, skewed, or sparse data sets are unreliable. Inference based on enumerating the exact distributions of sufficient statistics for parameters of interest in a logistic regression model, conditional on the remaining parameters, is computationally infeasible for many problems. Efficient algorithms for generating the required conditional distributions were introduced in Hirji, Mehta, and Patel (1987) and Mehta, Patel, and Senchaudhuri (1992, 2000), thus making these methods computationally available. This paper discusses the theory and methods for exact logistic regression and illustrates their application with the LOGISTIC procedure in SAS/STAT® 9.2 software. INTRODUCTION Many clinical trials deal with the comparison of populations of subjects with categorical responses. Historically, statistical inference for such studies involved large-sample approximations, and fitting logistic regression models to such data was performed through the unconditional likelihood function. However, asymptotic methods might be inadequate when sample sizes are small or the data are sparse, skewed, or heavily tied. Exact conditional inference remains valid in such situations. The LOGISTIC, GENMOD, GLIMMIX, PROBIT, and CATMOD procedures perform unconditional likelihood inference for logit models, and the LOGISTIC and PHREG procedures can perform asymptotic conditional likelihood inference for logit models. SAS users have requested the ability to perform exact tests for logistic regression modeling. Many exact statistical tests have already been added to the FREQ and NPAR1WAY procedures, and as of SAS 8.1, SAS/STAT software includes exact logistic regression for binary (dichotomous) response variables in the LOGISTIC procedure. Exact methods for generalized logit (GLOGIT) models have been available in the LOGISTIC procedure since SAS 9. The “METHODOLOGY” section in this paper presents the logistic regression model and the different likelihoods, and then explains how the exact analysis algorithm implemented in PROC LOGISTIC works. Details about the reported statistics are available in the appendix. The “SYNTAX” section describes the statements and options in the LOGISTIC procedure for the exact methods. The “EXAMPLES” section provides several examples to illustrate the syntax and the usefulness of the method. Dose-Response Study To demonstrate the usefulness of exact logistic regression, consider a small dose-response study. Researchers are interested in analyzing how mortality rates change with respect to dosage of a drug. The dose data set contains life and death outcomes for six levels of drug dosage (0 to 5). Three subjects are given each specific dose of the drug, and the number of deaths are recorded. 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;