Introduction to Categorical Data Analysis

4.4 MULTIPLE LOGISTIC REGRESSION 115
When the true odds ratio exceeds 1.0 in partial table k, we expect
(n11k − μ11k) >0. The test statistic combines these differences across all K tables,
and we then expect the sum of such differences to be a relatively large positive number.
When the odds ratio is less than 1.0 in each table, the sum of such differences
tends to be a relatively large negative number. The CMH statistic takes larger values
when (n11k − μ11k) is consistently positive or consistently negative for all tables,
rather than positive for some and negative for others. The test works best when the
XY association is similar in each partial table.
This test was proposed in 1959, well before logistic regression was popular. The
formula for the CMH test statistic seems to have nothing to do with modeling. In fact,
though, the CMH test is the score test (Section 1.4.1) of XY conditional independence
for model (4.8). Recall that model assumes a common odds ratio for the partial tables
(i.e., homogeneous association). Similarity of results for the likelihood-ratio, Wald,
and CMH (score) tests usually happens when the sample size is large.
For Table 4.4 from the AZT and AIDS study, consider H0: conditional independence
between immediate AZT use and AIDS symptom development. Section 4.3.2
noted that the likelihood-ratio test statistic is −2(L0 − L1) = 6.9 and the Wald test
statistic is ( ˆβ1/SE) 2 = 6.6, each with df = 1. The CMH statistic (4.9) equals 6.8,
also with df = 1, giving similar results (P = 0.01).
4.3.5 Testing the Homogeneity of Odds Ratios ∗
Model (4.8) and its special case (4.5) when Z is also binary have the homogeneous
association property of a common XY odds ratio at each level of Z. Sometimes it
is of interest to test the hypothesis of homogeneous association (although it is not
necessary to do so to justify using the CMH test). A test of homogeneity of the odds
ratios is, equivalently, a test of the goodness of fit of model (4.8). Section 5.2.2 will
show how to do this.
Some software reports a test, called the Breslow–Day test, that is a chi-squared
test specifically designed to test homogeneity of odds ratios. It has the form of
a Pearson chi-squared statistic, comparing the observed cell counts to estimated
expected frequencies that have a common odds ratio. This test is an alternative to
the goodness-of-fit tests of Section 5.2.2.
4.4 MULTIPLE LOGISTIC REGRESSION
Next we will consider the general logistic regression model with multiple explanatory
variables. Denote the k predictors for a binary response Y by x1,x2,...,xk. The
model for the log odds is
logit[P(Y = 1)] =α + β1x1 + β2x2 +···+βkxk
(4.10)
The parameter βi refers to the effect of xi on the log odds that Y = 1, controlling
the other xs. For example, exp(βi) is the multiplicative effect on the odds of a 1-unit
increase in xi, at fixed levels of the other xs.