Missing data arise in genetic association studies when genotypes are unknown or when haplotypes are of direct interest. We provide a general likelihood-based framework for making inference on genetic effects and gene–environment interactions with such missing data. We allow genetic and environmental variables to be correlated while leaving the distribution of environmental variables completely unspecified. We consider 3 major study designs—cross-sectional, case–control, and cohort designs—and construct appropriate likelihood functions for all common phenotypes (e.g. case–control status, quantitative traits, and potentially censored ages at onset of disease). The likelihood functions involve both finite- and infinite-dimensional parameters. The maximum likelihood estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Expectation–Maximization (EM) algorithms are developed to implement the corresponding inference procedures. Extensive simulation studies demonstrate that the proposed inferential and numerical methods perform well in practical settings. Illustration with a genome-wide association study of lung cancer is provided.
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