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.

引言：病例母亲-对照母亲设计（case-mother control-mother design）可用于探究胎儿与母亲的遗传因素，以及环境暴露对早期生命结局的影响。孟德尔约束以及子代基因型与环境因素间的条件独立性，使得半参数似然方法能够构建比标准逻辑回归效率更高的逻辑斯蒂模型。由于子代基因型的采集存在实际困难，亟需可处理缺失子代基因型的分析方法。
方法：本文综述了一种分层回顾似然方法，以及两种半参数似然方法：一种为前瞻性似然方法，另一种为改良回顾似然方法；后者可将母亲基因型建模为协变量的函数，或未指定其联合分布（稳健版本）。本文同时综述了实现上述建模方案的软件，通过模拟研究对比了各方法的统计学性能，并以基因-环境交互作用及部分缺失子代基因型为研究重点，展示了这些方法的应用实例。
结果：稳健回顾似然方法通常能提供无偏估计，其标准误仅略高于以暴露因素建模母亲基因型的情况。前瞻性似然方法则存在极大化求解问题。在针对小于胎龄儿与细胞色素P450 2E1（CYP2E1）及饮用水消毒副产物的关联分析实例中，回顾似然方法可纳入全协变量集，而前瞻性似然方法仅能纳入少量协变量。
结论：我们推荐使用改良回顾似然方法的稳健版本。