Sometimes a safety and efficacy study is stratified according to some factor, such as clinical center, disease severity, gender, etc. In such a situation, it still may be desirable to estimate the odds ratio while accounting for strata effects. The Mantel-Haenszel test for the odds ratio assumes that the odds ratio is equal across all strata, although the rates, \(p_1\) and \(p_2\), may differ across strata. This procedure calculates the odds ratio within each stratum and then combines the strata estimates into one estimate of the common odds ratio. For example,
| Stratum | \(p_1\) | \(p_2\) | \(\theta\) |
| 1 | 0.50 | 0.25 | 3.00 |
| 2 | 0.40 | 0.18 | 3.00 |
| 3 | 0.30 | 0.12 | 3.00 |
| 4 | 0.20 | 0.08 | 3.00 |
Example
Using PROC FREQ for conducting a Mantel-Haenszel test Section
A company performed a multi-center safety and efficacy study at six sites, with a binary outcome (success/failure), for comparing placebo and treatment.
***********************************************************************
* This is a program that illustrates the use of PROC FREQ for *
* conducting a Mantel-Haenszel test within the setting of a multi- *
* center clinical trial. *
***********************************************************************;
proc format;
value centfmt 1='Phoenix'
2='Denver'
3='Miami'
4='New York'
5='Atlanta'
6='Chicago';
value groupfmt 0='Placebo'
1='Treatment';
value respfmt 0='Failure'
1='Success';
run;
data one;
input center group response count;
format center centfmt. group groupfmt. response respfmt.;
cards;
1 0 0 24
1 0 1 8
1 1 0 20
1 1 1 12
2 0 0 44
2 0 1 28
2 1 0 40
2 1 1 32
3 0 0 25
3 0 1 5
3 1 0 20
3 1 1 10
4 0 0 14
4 0 1 16
4 1 0 12
4 1 1 18
5 0 0 32
5 0 1 20
5 1 0 24
5 1 1 28
6 0 0 45
6 0 1 5
6 1 0 32
6 1 1 18
;
run;
proc freq data=one;
tables center*group*response/cmh;
exact comor;
weight count;
title "Example of the Mantel-Haenszel Test Applied to a Multi-Center Trial";
run;
SAS PROC FREQ yields an estimated odds ratio of 1.84 with an approximate 95% confidence interval is (1.28, 2.66).
The exact 95% confidence interval is (1.26, 2.69). The exact and asymptotic confidence intervals are nearly identical due to the large sample size across the six clinical centers.
\(H_0 \colon θ = 1\) is rejected at the 0.05 significance level (\(p = 0.0013\), which is consistent with the 95% confidence interval not containing 1.0. (Later in this chapter we discuss the construction of the Mantel-Haenszel test statistic.)