Example 28.6: Computing Cochran-Mantel-Haenszel Statistics for a Stratified Table

The FREQ Procedure

Example 28.6: Computing Cochran-Mantel-Haenszel Statistics for a Stratified Table

The data set Migraine contains hypothetical data for a clinical trial of migraine treatment. Subjects of both genders receive either a new drug therapy or a placebo. Their response to treatment is coded as 'Better' or 'Same'. The data are recorded as cell counts, and the number of subjects for each treatment and response combination is recorded in the variable Count.

   data Migraine;
      input Gender $ Treatment $ Response $ Count @@;
      datalines;
   female Active  Better 16   female Active  Same 11
   female Placebo Better  5   female Placebo Same 20
   male   Active  Better 12   male   Active  Same 16
   male   Placebo Better  7   male   Placebo Same 19
   ;

The following statements create a three-way table stratified by Gender, where Treatment forms the rows and Response forms the columns. The CMH option produces the Cochran-Mantel-Haenszel statistics. For this stratified 2×2 table, estimates of the common relative risk and the Breslow-Day test for homogeneity of the odds ratios are also displayed. The NOPRINT option suppresses the display of the contingency tables. These statements produce Output 28.6.1 through Output 28.6.3.

   proc freq data=Migraine;
      weight Count;
      tables Gender*Treatment*Response / cmh noprint;
      title1 'Clinical Trial for Treatment of Migraine Headaches';
   run;

Output 28.6.1: Cochran-Mantel-Haenszel Statistics

Clinical Trial for Treatment of Migraine Headaches

The FREQ Procedure

Summary Statistics for Treatment by Response
Controlling for Gender

Cochran-Mantel-Haenszel Statistics (Based on Table Scores)
Statistic	Alternative Hypothesis	DF	Value	Prob
1	Nonzero Correlation	1	8.3052	0.0040
2	Row Mean Scores Differ	1	8.3052	0.0040
3	General Association	1	8.3052	0.0040

Total Sample Size = 106

For a stratified 2×2 table, the three CMH statistics displayed in Output 28.6.1 test the same hypothesis. The significant p-value (0.004) indicates that the association between treatment and response remains strong after adjusting for gender.

Output 28.6.2: CMH Option: Relative Risks

Clinical Trial for Treatment of Migraine Headaches

The FREQ Procedure

Summary Statistics for Treatment by Response
Controlling for Gender

Estimates of the Common Relative Risk (Row1/Row2)
Type of Study	Method	Value	95% Confidence Limits
Case-Control	Mantel-Haenszel	3.3132	1.4456	7.5934
(Odds Ratio)	Logit	3.2941	1.4182	7.6515
Cohort	Mantel-Haenszel	2.1636	1.2336	3.7948
(Col1 Risk)	Logit	2.1059	1.1951	3.7108
Cohort	Mantel-Haenszel	0.6420	0.4705	0.8761
(Col2 Risk)	Logit	0.6613	0.4852	0.9013

Total Sample Size = 106

The CMH option also produces a table of relative risks, as shown in Output 28.6.2. Because this is a prospective study, the relative risk estimate assesses the effectiveness of the new drug; the "Cohort (Col1 Risk)" values are the appropriate estimates for the first column, or the risk of improvement. The probability of migraine improvement with the new drug is just over two times the probability of improvement with the placebo.

Output 28.6.3: CMH Option: Breslow-Day Test

Clinical Trial for Treatment of Migraine Headaches

The FREQ Procedure

Summary Statistics for Treatment by Response
Controlling for Gender

Breslow-Day Test for Homogeneity of the Odds Ratios
Chi-Square	1.4929
DF	1
Pr > ChiSq	0.2218

Total Sample Size = 106

The large p-value for the Breslow-Day test (0.2218) in Output 28.6.3 indicates no significant gender difference in the odds ratios.

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