Supplementary Material for: Methods and software to analyze gene-environment interactions under a case-mother - control-mother design with partially missing child genotype

Introduction: The case-mother - control-mother design allows to study fetal and maternal genetic factors together with environmental exposures on early-life outcomes. Mendelian constraints and conditional independence between child genotype and environmental factors enabled semiparametric likelihood methods to estimate logistic models with greater efficiency than standard logistic regression. Difficulties in child genotype collection require methods handling missing child genotype. Methods: We review a stratified retrospective likelihood and two semiparametric likelihood approaches: a prospective one and a modified retrospective one, the latter either modeling the maternal genotype as a function of covariates or leaving their joint distribution unspecified (robust version). We also review software implementing these modelling alternatives, compare their statistical properties in a simulation study, and illustrate their application, focusing on gene-environment interactions and partially missing child genotype. Results; The robust retrospective likelihood provides generally unbiased estimates, with standard errors only slightly larger than when modelling maternal genotype based on exposure. The prospective likelihood encounters maximization problems. In the application to the association of small-for-gestational-age babies with CYP2E1 and drinking water disinfection by-products, the retrospective likelihood allowed a full array of covariates, while the prospective likelihood was limited to few covariates. Conclusion: We recommend the robust version of the modified retrospective likelihood.