Example 36.7 Cochran-Mantel-Haenszel Statistics

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 @@;
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 PROC FREQ statements create a multiway table stratified by Gender, where Treatment forms the rows and Response forms the columns. The RELRISK option in the TABLES statement requests the odds ratio and relative risks for the two-way tables of Treatment by Response. The PLOTS= option requests a relative risk plot, which shows the relative risk and its confidence limits for each level of Gender. The CMH option requests Cochran-Mantel-Haenszel statistics for the multiway table. For this stratified table, the CMH option also produces estimates of the common relative risk and the Breslow-Day test for homogeneity of the odds ratios. The NOPRINT option suppresses the display of the crosstabulation tables.

ods graphics on;
proc freq data=Migraine;
   tables Gender*Treatment*Response / 
          relrisk plots(only)=relriskplot(stats) cmh noprint; 
   weight Count;
   title 'Clinical Trial for Treatment of Migraine Headaches';
ods graphics off;

Output 36.7.1 through Output 36.7.4 show the results of the analysis. The relative risk plot (Output 36.7.1) displays the relative risks and confidence limits for the two levels of Gender. Output 36.7.2 displays the CMH statistics. For a stratified table, the three CMH statistics test the same hypothesis. The significant -value (0.004) indicates that the association between treatment and response remains strong after adjusting for gender.

The CMH option also produces a table of overall relative risks, as shown in Output 36.7.3. 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 (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.

The large -value for the Breslow-Day test (0.2218) in Output 36.7.4 indicates no significant gender difference in the odds ratios.

Output 36.7.1 Relative Risk Plot
 Relative Risk Plot

Output 36.7.2 Cochran-Mantel-Haenszel Statistics
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

Output 36.7.3 CMH Option: Common Relative Risks
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

Output 36.7.4 CMH Option: Breslow-Day Test
Breslow-Day Test for
Homogeneity of the Odds Ratios
Chi-Square 1.4929
DF 1
Pr > ChiSq 0.2218