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-对照母亲研究设计（case-mother - control-mother design）可用于探究胎儿与母亲的遗传因素、环境暴露对早期生命结局的影响。借助孟德尔约束（Mendelian constraints）与子代基因型和环境因素间的条件独立性，半参数似然方法（semiparametric likelihood methods）能够构建比标准逻辑回归（standard logistic regression）效率更高的逻辑回归模型。由于子代基因型样本采集存在实际困难，亟需可处理子代基因型缺失（missing child genotype）的分析方法。
方法：本文综述了分层回顾性似然法（stratified retrospective likelihood）以及两种半参数似然方法：前瞻性似然法（prospective likelihood）与改良回顾性似然法（modified retrospective likelihood）。后者可分为两种实现形式：一种将母亲基因型建模为协变量（covariates）的函数，另一种则不对其联合分布（joint distribution）进行预设（即稳健版本（robust version））。此外，本文还对实现上述建模方案的软件进行了综述，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（gene-environment interactions）及部分子代基因型缺失场景为例，演示了方法的实际应用。
结果：稳健回顾性似然法可获得总体无偏的估计结果，其标准误（standard errors）仅略高于基于暴露因素构建母亲基因型模型的情况；而前瞻性似然法则会遇到极大值求解问题（maximization problems）。在针对小于胎龄儿（small-for-gestational-age babies）与CYP2E1及饮用水消毒副产物（drinking water disinfection by-products）关联的实际应用中，回顾性似然法可纳入全维度协变量（full array of covariates），而前瞻性似然法仅能纳入少量协变量。
结论：综上，我们推荐采用改良回顾性似然法的稳健版本。