A Constrained Maximum Likelihood Method for Pleiotropy-Robust Mendelian Randomization Using Family-Based GWAS Summary Statistics

Mendelian randomization (MR) using genome-wide association study (GWAS) summary statistics is widely used for causal inference. However, conventional MR methods relying on population-based GWAS are vulnerable to biases arising from family-level confounders. While family-based GWAS (FGWAS) designs offer a solution to address these effects, existing MR methodologies are not optimized for the unique structure of FGWAS data, often resulting in reduced statistical power. Therefore, in this paper, we propose the MRcML-Family (MR based on constrained maximum likelihood for families), a novel method specifically tailored for FGWAS summary statistics. The core innovation of MRcML-Family lies in its unified framework which simultaneously integrates genetic information from offspring (direct effects) and parents (indirect effects) as instrumental variables. By applying different constraints, the method effectively identifies and corrects for horizontal pleiotropy. Comprehensive simulation studies demonstrate that MRcML-Family yields nearly unbiased causal estimates and maintains robust Type I error rate control whether or not the instrument strength independent of direct effect assumption holds, even in challenging scenarios with a limited number of SNPs and small FGWAS sample sizes. Furthermore, real-world data analyses using UK Biobank data show that MRcML-Family provides causal estimates highly consistent with existing literature and recovers the established "gold standard" relationship between height and body mass index. Additionally, it showed higher statistical power than the compared methods in the evaluated scenarios, mitigating the efficiency loss typically associated with family-based designs.