Flexible maximum likelihood methods for assessing joint effects in case-control studies with complex sampling

Case-control studies can often be made more efficient by using frequency matching, randomized recruitment, stratified sampling, or two-stage sampling. These designs share two common features: (1) some "first-stage" variables are ascertained for all study subjects, while complete variable ascertainment is carried out for only a selected subsample, and (2) the subsampling of subjects for "second-stage" variable ascertainment depends jointly on their disease status and their observed first-stage variables. Because first-stage variables alter the subsampling fractions, standard analyses require a multiplicative specification of any joint effects of a second- and a first-stage variable. We show that by making use of missing data methods, maximum likelihood estimates can be obtained for risk parameters of interest, even those characterizing interactions between first- and second-stage variables. Joint effects can thus be modelled flexibly, with allowance for both additive and multiplicative models. Preliminary data from a case-control study of lung cancer as related to age, sex, and smoking provide an example, leading to the suggestion that the combined effect of age and smoking is multiplicative.