Introduction to Categorical Data Analysis

114 LOGISTIC REGRESSION
4.3.4 The Cochran–Mantel–Haenszel Test for 2 × 2 × K
Contingency Tables ∗
In many examples with two categorical predictors, X identifies two groups to compare
and Z is a control variable. For example, in a clinical trial X might refer to two
treatments and Z might refer to several centers that recruited patients for the study.
Problem 4.20 shows such an example. The data then can be presented in several 2 × 2
tables.
With K categories for Z, model (4.7) refers to a 2 × 2 × K contingency table.
That model can then be expressed as
logit[P(Y = 1)] =α + βx + β Z k
(4.8)
where x is an indicator variable for the two categories of X. Then, exp(β) is the
common XY odds ratio for each of the K partial tables for categories of Z. This
is the homogeneous association structure for multiple 2 × 2 tables, introduced in
Section 2.7.6.
In this model, conditional independence between X and Y , controlling for Z,
corresponds to β = 0. When β = 0, the XY odds ratio equals 1 for each partial table.
Given that model (4.8) holds, one can test conditional independence by the Wald test
or the likelihood-ratio test of H0: β = 0.
The Cochran–Mantel–Haenszel test is an alternative test of XY conditional independence
in 2 × 2 × K contingency tables. This test conditions on the row totals and
the column totals in each partial table. Then, as in Fisher’s exact test, the count in
the first row and first column in a partial table determines all the other counts in that
table. Under the usual sampling schemes (e.g., binomial for each row in each partial
table), the conditioning results in a hypergeometric distribution (Section 2.6.1) for
the count n11k in the cell in row 1 and column 1 of partial table k. The test statistic
utilizes this cell in each partial table.
In partial table k, the row totals are {n1+k,n2+k}, and the column totals are
{n+1k,n+2k}. Given these totals, under H0,
μ11k = E(n11k) = n1+kn+1k/n++k
Var(n11k) = n1+kn2+kn+1kn+2k/n 2 ++k (n++k − 1)
The Cochran–Mantel–Haenszel (CMH) test statistic summarizes the information
from the K partial tables using
CMH =
��
k (n11k − μ11k) � 2
�
k Var(n11k)
(4.9)
This statistic has a large-sample chi-squared null distribution with df = 1. The approximation
improves as the total sample size increases, regardless of whether the number
of strata K is small or large.