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（case-mother-control-mother）研究设计可用于同时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暴露之间的条件独立性，使得半参数似然法能够构建比标准逻辑回归更高效的逻辑回归模型估计。由于子代基因型采集存在困难，亟需可处理子代基因型缺失的分析方法。
方法：本研究综述了一种分层回顾似然法，以及两种半参数似然分析方法：一种为前瞻性似然法，另一种为改良回顾式似然法；后者可将母亲基因型建模为协变量的函数，或不对其联合分布进行设定（即稳健版本）。此外，本研究还综述了实现上述建模方案的软件工具，通过模拟研究对比了各方法的统计学特性，并以基因-环境交互作用及子代基因型部分缺失为应用场景，展示了各方法的实际应用效果。
结果：改良回顾式似然法的稳健版本通常可得到无偏估计，其标准误仅略高于以暴露因素为基础构建母亲基因型模型时的结果；而前瞻性似然法则会遇到最大化求解问题。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之间关联的应用实例中，回顾似然法可纳入全部协变量，而前瞻性似然法则仅能纳入少量协变量。
结论：综上，本研究推荐使用改良回顾式似然法的稳健版本。