Simple logistic regression generalizes to multiple logistic regression in the same way that simple linear regression generalizes to multiple linear regression. We regress a dichotomous response variable, such as survival, against several covariates. This allows us to either adjust for confounding variables or account for covariates that have a synergistic effect on the response variable. We can add interaction terms to our model in exactly the same way as in linear regression.

Before discussing multiple logistic regression we will first describe a traditional method for adjusting an odds ratio estimate for a confounding variable.

Mantel–Haenszel estimate of an age-adjusted odds ratio

In Section 4.19.1 we introduced the Ille-et-Vilaine study of esophageal cancer and alcohol (Breslow and Day, 1980). Table 5.1 shows these data stratified by ten-year age groups. It is clear from this table that the incidence of esophageal cancer increases dramatically with age. There is also some evidence that the prevalence of heavy drinking also increases with age; the prevalence of heavy drinking among controls increases from 7.8% for men aged 25–30 to 17.3% for men aged 45–54. Thus, age may confound the alcohol–cancer relationship, and it makes sense to calculate an age-adjusted odds ratio for the effect of heavy drinking on esophageal cancer. Mantel and Haenszel (1959) proposed the following method for adjusting an odds ratio in the presence of a confounding variable.

Suppose that study subjects are subdivided into a number of strata by a confounding variable.