The LOGISTIC Procedure

Example 72.11 Conditional Logistic Regression for Matched Pairs Data

Subsections:

Conditional Analysis Using the STRATA Statement
Exact Analysis Using the STRATA Statement

In matched pairs, or case-control, studies, conditional logistic regression is used to investigate the relationship between an outcome of being an event (case) or a nonevent (control) and a set of prognostic factors.

The following data are a subset of the data from the Los Angeles Study of the Endometrial Cancer Data in Breslow and Day (1980). There are 63 matched pairs, each consisting of a case of endometrial cancer (Outcome=1) and a control (Outcome=0). The case and corresponding control have the same ID. Two prognostic factors are included: Gall (an indicator variable for gall bladder disease) and Hyper (an indicator variable for hypertension). The goal of the case-control analysis is to determine the relative risk for gall bladder disease, controlling for the effect of hypertension.

data Data1;
   do ID=1 to 63;
      do Outcome = 1 to 0 by -1;
         input Gall Hyper @@;
         output;
      end;
   end;
   datalines;
0 0  0 0    0 0  0 0    0 1  0 1    0 0  1 0    1 0  0 1
0 1  0 0    1 0  0 0    1 1  0 1    0 0  0 0    0 0  0 0
1 0  0 0    0 0  0 1    1 0  0 1    1 0  1 0    1 0  0 1
0 1  0 0    0 0  1 1    0 0  1 1    0 0  0 1    0 1  0 0
0 0  1 1    0 1  0 1    0 1  0 0    0 0  0 0    0 0  0 0
0 0  0 1    1 0  0 1    0 0  0 1    1 0  0 0    0 1  0 0
0 1  0 0    0 1  0 0    0 1  0 0    0 0  0 0    1 1  1 1
0 0  0 1    0 1  0 0    0 1  0 1    0 1  0 1    0 1  0 0
0 0  0 0    0 1  1 0    0 0  0 1    0 0  0 0    1 0  0 0
0 0  0 0    1 1  0 0    0 1  0 0    0 0  0 0    0 1  0 1
0 0  0 0    0 1  0 1    0 1  0 0    0 1  0 0    1 0  0 0
0 0  0 0    1 1  1 0    0 0  0 0    0 0  0 0    1 1  0 0
1 0  1 0    0 1  0 0    1 0  0 0
;

There are several ways to approach this problem with PROC LOGISTIC:

Specify the STRATA statement to perform a conditional logistic regression.
Specify EXACT and STRATA statements to perform an exact logistic regression on the original data set, if you believe the data set is too small or too sparse for the usual asymptotics to hold.
Transform each matched pair into a single observation, and then specify a PROC LOGISTIC statement on this transformed data without a STRATA statement; this also performs a conditional logistic regression and produces essentially the same results.
Specify an EXACT statement on the transformed data.

SAS statements and selected results for the first two approaches are given in the remainder of this example.

Conditional Analysis Using the STRATA Statement

In the following statements, PROC LOGISTIC is invoked with the ID variable declared in the STRATA statement to obtain the conditional logistic model estimates for a model containing Gall as the only predictor variable:

proc logistic data=Data1;
   strata ID;
   model outcome(event='1')=Gall;
run;

Results from the conditional logistic analysis are shown in Output 72.11.1. Note that there is no intercept term in the "Analysis of Maximum Likelihood Estimates" tables.

The odds ratio estimate for Gall is 2.60, which is marginally significant (p = 0.0694) and which is an estimate of the relative risk for gall bladder disease. A 95% confidence interval for this relative risk is (0.927, 7.293).

Output 72.11.1: Conditional Logistic Regression (Gall as Risk Factor)

The LOGISTIC Procedure

Conditional Analysis

Model Information
Data Set	WORK.DATA1
Response Variable	Outcome
Number of Response Levels	2
Number of Strata	63
Model	binary logit
Optimization Technique	Newton-Raphson ridge

Number of Observations Read	126
Number of Observations Used	126

Response Profile
Ordered Value	Outcome	Total Frequency
1	0	63
2	1	63

Probability modeled is Outcome=1.

Strata Summary
Response Pattern	Outcome		Number of Strata	Frequency
Response Pattern	0	1	Number of Strata	Frequency
1	1	1	63	126

Newton-Raphson Ridge Optimization

Without Parameter Scaling

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

Model Fit Statistics
Criterion	Without Covariates	With Covariates
AIC	87.337	85.654
SC	87.337	88.490
-2 Log L	87.337	83.654

Testing Global Null Hypothesis: BETA=0
Test	Chi-Square	DF	Pr > ChiSq
Likelihood Ratio	3.6830	1	0.0550
Score	3.5556	1	0.0593
Wald	3.2970	1	0.0694

Analysis of Conditional Maximum Likelihood Estimates
Parameter	DF	Estimate	Standard Error	Wald Chi-Square	Pr > ChiSq
Gall	1	0.9555	0.5262	3.2970	0.0694

Odds Ratio Estimates
Effect	Point Estimate	95% Wald Confidence Limits
Gall	2.600	0.927	7.293

Exact Analysis Using the STRATA Statement

When you believe there are not enough data or that the data are too sparse, you can perform a stratified exact logistic regression. The following statements perform stratified exact logistic regressions on the original data set by specifying both the STRATA and EXACT statements:

proc logistic data=Data1 exactonly;
   strata ID;
   model outcome(event='1')=Gall;
   exact Gall / estimate=both;
run;

Output 72.11.2: Exact Logistic Regression (Gall as Risk Factor)

The LOGISTIC Procedure

Exact Conditional Analysis

Exact Conditional Tests
Effect	Test	Statistic	p-Value
Effect	Test	Statistic	Exact	Mid
Gall	Score	3.5556	0.0963	0.0799
	Probability	0.0327	0.0963	0.0799

Exact Parameter Estimates
Parameter	Estimate	Standard Error	95% Confidence Limits		Two-sided p-Value
Gall	0.9555	0.5262	-0.1394	2.2316	0.0963

Exact Odds Ratios
Parameter	Estimate	95% Confidence Limits		Two-sided p-Value
Gall	2.600	0.870	9.315	0.0963

Note that the score statistic in the "Conditional Exact Tests" table in Output 72.11.2 is identical to the score statistic in Output 72.11.1 from the conditional analysis. The exact odds ratio confidence interval is much wider than its conditional analysis counterpart, but the parameter estimates are similar. The exact analysis confirms the marginal significance of Gall as a predictor variable.