Many environmental factors are multi-dimensional; air pollution, for example, is a complex mixture of gases and particles with differing biological effects. Most environmental agents have degrees of exposure intensity, usually varying over time. Even if an exposure is not time-dependent, the resulting disease risk is likely to be modified by temporal factors like age at or duration of exposure12. Seldom are accurate measurements of exposure over a lifetime available on all participants in a large epidemiologic study, but more detailed information may be obtainable on a stratified subsample to allow correction for measurement error13. Exposures may not even be measured on individuals, but assigned on the basis of ecologic-level exposures or a prediction model. Two-phase case-control designs that leverage readily available exposure surrogates to select individuals for more in-depth exposure assessment and/or genotyping might be used. Uncertainties in exposure assignments can be large and lead to unpredictable biases, particularly if differential with respect to disease, and can induce spurious interactions9. Although methods of correction for exposure or genotype measurement errors are well established for main effects, they have seldom been
