Valid Inference for Two Sample Summary Statistics Mendelian Randomization

Mendelian randomization (MR) studies increasingly rely on summary statistics from genome-wide association studies (GWASs). However, current summary-data methods treat marginal GWAS effect estimates as independent across instrumental single-nucleotide polymorphisms (SNPs), which holds only when the SNPs have no effect. Without imposing this independence assumption, we derive expressions for the two-stage least squares (TSLS), two-sample TSLS (TSTSLS), and generalized method-of-moments estimators together with their variance estimators using only GWAS summary statistics, assuming the instrumental SNPs are in linkage equilibrium. We show that the widely used inverse-variance weighted (IVW) estimator is generally not equivalent to either TSLS or TSTSLS, contrary to a common claim in the MR literature. We further demonstrate that the conventional IVW standard error is generally inconsistent and provide a consistent alternative. Extensive simulations confirm the theoretical results and reveal that Wald confidence intervals based on summary statistics can be anti-conservative. An application to publicly available GWAS summary statistics on body mass index and systolic blood pressure illustrates the practical implications of the proposed methodology.