Example 39.9: Conditional Logistic Regression for Matched Pairs Data

The LOGISTIC Procedure

Example 39.9: Conditional Logistic Regression for Matched Pairs Data

In matched case-control studies, conditional logistic regression is used to investigate the relationship between an outcome of being a case or a control and a set of prognostic factors. When each matched set consists of a single case and a single control, the conditional likelihood is given by

$\prod_i(1+\exp(-{\beta}'(x_{i1}-x_{i0}))^{-1}$

where x_i1 and x_i0 are vectors representing the prognostic factors for the case and control, respectively, of the ith matched set. This likelihood is identical to the likelihood of fitting a logistic regression model to a set of data with constant response, where the model contains no intercept term and has explanatory variables given by d_i = x_i1 - x_i0 (Breslow 1982).

The data in this example are a subset of the data from the Los Angeles Study of the Endometrial Cancer Data in Breslow and Days (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.

Before PROC LOGISTIC is used for the logistic regression analysis, each matched pair is transformed into a single observation, where the variables Gall and Hyper contain the differences between the corresponding values for the case and the control (case - control). The variable Outcome, which will be used as the response variable in the logistic regression model, is given a constant value of 0 (which is the Outcome value for the control, although any constant, numeric or character, will do).

   data Data1;
      drop id1 gall1 hyper1;
      retain id1 gall1 hyper1 0;
      input ID Outcome Gall Hyper @@ ;
      if (ID = id1) then do;
         Gall=gall1-Gall; Hyper=hyper1-Hyper;
         output;
      end;
      else do;
         id1=ID; gall1=Gall; hyper1=Hyper;
      end;
      datalines;
    1  1 0  0  1  0  0  0   2  1 0  0  2  0  0  0
    3  1 0  1  3  0  0  1   4  1 0  0  4  0  1  0
    5  1 1  0  5  0  0  1   6  1 0  1  6  0  0  0
    7  1 1  0  7  0  0  0   8  1 1  1  8  0  0  1
    9  1 0  0  9  0  0  0  10  1 0  0 10  0  0  0
   11  1 1  0 11  0  0  0  12  1 0  0 12  0  0  1
   13  1 1  0 13  0  0  1  14  1 1  0 14  0  1  0
   15  1 1  0 15  0  0  1  16  1 0  1 16  0  0  0
   17  1 0  0 17  0  1  1  18  1 0  0 18  0  1  1
   19  1 0  0 19  0  0  1  20  1 0  1 20  0  0  0
   21  1 0  0 21  0  1  1  22  1 0  1 22  0  0  1
   23  1 0  1 23  0  0  0  24  1 0  0 24  0  0  0
   25  1 0  0 25  0  0  0  26  1 0  0 26  0  0  1
   27  1 1  0 27  0  0  1  28  1 0  0 28  0  0  1
   29  1 1  0 29  0  0  0  30  1 0  1 30  0  0  0
   31  1 0  1 31  0  0  0  32  1 0  1 32  0  0  0
   33  1 0  1 33  0  0  0  34  1 0  0 34  0  0  0
   35  1 1  1 35  0  1  1  36  1 0  0 36  0  0  1
   37  1 0  1 37  0  0  0  38  1 0  1 38  0  0  1
   39  1 0  1 39  0  0  1  40  1 0  1 40  0  0  0
   41  1 0  0 41  0  0  0  42  1 0  1 42  0  1  0
   43  1 0  0 43  0  0  1  44  1 0  0 44  0  0  0
   45  1 1  0 45  0  0  0  46  1 0  0 46  0  0  0
   47  1 1  1 47  0  0  0  48  1 0  1 48  0  0  0
   49  1 0  0 49  0  0  0  50  1 0  1 50  0  0  1
   51  1 0  0 51  0  0  0  52  1 0  1 52  0  0  1
   53  1 0  1 53  0  0  0  54  1 0  1 54  0  0  0
   55  1 1  0 55  0  0  0  56  1 0  0 56  0  0  0
   57  1 1  1 57  0  1  0  58  1 0  0 58  0  0  0
   59  1 0  0 59  0  0  0  60  1 1  1 60  0  0  0
   61  1 1  0 61  0  1  0  62  1 0  1 62  0  0  0
   63  1 1  0 63  0  0  0
   ;

Note that there are 63 observations in the data set, one for each matched pair. The variable Outcome has a constant value of 0.

In the following SAS statements, PROC LOGISTIC is invoked with the NOINT option to obtain the conditional logistic model estimates. Two models are fitted. The first model contains Gall as the only predictor variable, and the second model contains both Gall and Hyper as predictor variables. Because the option CLODDS=PL is specified, PROC LOGISTIC computes a 95% profile likelihood confidence interval for the odds ratio for each predictor variable.

   proc logistic data=Data1;
      model outcome=Gall / noint CLODDS=PL;
   run;

   proc logistic data=Data1;
      model outcome=Gall Hyper / noint CLODDS=PL;
   run;

Results from the two conditional logistic analyses are shown in Output 39.9.1 and Output 39.9.2. Note that there is only one response level listed in the "Response Profile" tables and there is no intercept term in the "Analysis of Maximum Likelihood Estimates" tables.

Output 39.9.1: Conditional Logistic Regression (Gall as risk factor)

The LOGISTIC Procedure

Model Information
Data Set	WORK.DATA1
Response Variable	Outcome
Number of Response Levels	1
Number of Observations	63
Link Function	Logit
Optimization Technique	Fisher's scoring

Response Profile
Ordered Value	Outcome	Total Frequency
1	0	63

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

Model Fit Statistics
Criterion	Without Covariates	With Covariates
AIC	87.337	85.654
SC	87.337	87.797
-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 Maximum Likelihood Estimates
Parameter	DF	Estimate	Standard Error	Chi-Square	Pr > ChiSq
Gall	1	0.9555	0.5262	3.2970	0.0694

NOTE:

Since there is only one response level, measures of association between the observed and predicted values were not calculated.

Profile Likelihood Confidence Interval for Adjusted Odds Ratios
Effect	Unit	Estimate	95% Confidence Limits
Gall	1.0000	2.600	0.981	8.103

Output 39.9.2: Conditional Logistic Regression (Gall and Hyper as risk factors)

The LOGISTIC Procedure

Model Information
Data Set	WORK.DATA1
Response Variable	Outcome
Number of Response Levels	1
Number of Observations	63
Link Function	Logit
Optimization Technique	Fisher's scoring

Response Profile
Ordered Value	Outcome	Total Frequency
1	0	63

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

Model Fit Statistics
Criterion	Without Covariates	With Covariates
AIC	87.337	86.788
SC	87.337	91.074
-2 Log L	87.337	82.788

Testing Global Null Hypothesis: BETA=0
Test	Chi-Square	DF	Pr > ChiSq
Likelihood Ratio	4.5487	2	0.1029
Score	4.3620	2	0.1129
Wald	4.0060	2	0.1349

Analysis of Maximum Likelihood Estimates
Parameter	DF	Estimate	Standard Error	Chi-Square	Pr > ChiSq
Gall	1	0.9704	0.5307	3.3432	0.0675
Hyper	1	0.3481	0.3770	0.8526	0.3558

NOTE:

Since there is only one response level, measures of association between the observed and predicted values were not calculated.

Profile Likelihood Confidence Interval for Adjusted Odds Ratios
Effect	Unit	Estimate	95% Confidence Limits
Gall	1.0000	2.639	0.987	8.299
Hyper	1.0000	1.416	0.682	3.039

In the first model, where Gall is the only predictor variable (Output 39.9.1), the odds ratio estimate for Gall is 2.60, which is an estimate of the relative risk for gall bladder disease. A 95% confidence interval for this relative risk is (0.981, 8.103).

In the second model, where both Gall and Hyper are present (Output 39.9.2), the odds ratio estimate for Gall is 2.639, which is an estimate of the relative risk for gall bladder disease adjusted for the effects of hypertension. A 95% confidence interval for this adjusted relative risk is (0.987, 8.299). Note that the adjusted values (accounting for hypertension) for gall bladder disease are not very different from the unadjusted values (ignoring hypertension). This is not surprising since the prognostic factor Hyper is not statistically significant. The 95% profile likelihood confidence interval for the odds ratio for Hyper is (0.682, 3.039), which contains unity.

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