Bias can arise in case-control studies of genotype effects if the underlying population is structured (genetically stratified or admixed). Nuclear-family-based studies enjoy robustness against such bias, provided that inference conditions properly on the parents. Investigators have extended family-based methods to study gene-by-environment interactions, regarding such extensions as retaining robustness. We demonstrate via simulations that, if population structure involves the exposure, nuclear-family-based analyses of gene-by-exposure interaction remain vulnerable to inflated Type I error rates through subtle dependencies that investigators have failed to appreciate. Motivated by the Two Sister Study, an ongoing study of families affected by young-onset breast cancer, we consider a design that supplements the case-parents design with a sibling who is not genotyped but provides exposure data. If, in the population at large, inheritance is Mendelian and genotypes do not influence propensity for exposure, then this 4-person (or tetrad) structure permits the study of genetic effects, exposure effects, and genotype-by-exposure interactions. We show for a dichotomous exposure that, when exposure of an unaffected sibling is available, a modification to the analysis of case-sib or tetrad data re-establishes robustness for tests of multiplicative gene-by-environment interaction. We also use simulations to assess the power for detecting interaction across a range of scenarios, designs, and analytic methods.