Simultaneously accounting for winner's curse and sample structure in Mendelian randomization: bivariate rerandomized inverse variance weighted estimator

TL;DR

The bivariate RIVW (BRIVW) estimator is proposed, which extends the RIVW framework by modeling the joint distribution of SNP-exposure and SNP-outcome associations and provides more accurate causal effect estimates than existing methods.

Abstract

The recently developed rerandomized inverse variance weighted (RIVW) estimator provides a simple and efficient framework to break the winner's curse in two-sample Mendelian randomization (MR). However, this method has ignored the possible presence of sample structure (e.g., residual population stratification and sample overlap), a common confounding factor in MR studies. Sample structure can not only distort SNP-exposure and SNP-outcome association estimates but also induce correlation between them, leading exposure-side instrument selection to propagate bias to the outcome side. To address this challenge, we propose the bivariate RIVW (BRIVW) estimator that can simultaneously account for the winner's curse and sample structure. The BRIVW estimator extends the RIVW framework by modeling the joint distribution of SNP-exposure and SNP-outcome associations, first adjusting their covariance matrix via linkage disequilibrium score regression to account for sample structure, and then applying randomized instrument selection and Rao-Blackwellization to obtain unbiased post-selection association estimates as well as their covariance matrix. Under appropriate conditions, we show that the BRIVW estimator is consistent and asymptotically normal. Extensive simulations and real data analyses demonstrate that the BRIVW estimator provides more accurate causal effect estimates than existing methods.

Figures (5)

• Figure 1: Comparison between $\hat{\Gamma}_{j,\mathrm{RB}}$ and $\hat{\Gamma}_j$ after IV selection under different levels of sample structure ($\rho$) and IV strength ($\gamma_j/\sigma_{\hat{\gamma}_j}$). Weak, moderate, and strong instruments correspond to $\gamma_j/\sigma_{\hat{\gamma}_j}\in\{0.1\lambda,\,0.5\lambda,\,2\lambda\}$, where $\lambda=\Phi^{-1}(1-5\times10^{-5}/2)$. The red dotted lines indicate the true value of $\Gamma_j$.
• Figure 2: Bias proportion of IVW, dIVW, RIVW, and BRIVW estimators under different levels of sample structure ($\rho$) and IV proportion around the selection cutoff value. The IV proportion (x-axis) is calculated as the number of IVs with p-values lying between $5 \times 10^{-8}$ and $5 \times 10^{-10}$ divided by the number of selected IVs with p-value $< 5 \times 10^{-8}$.
• Figure 3: Quantile-quantile plots of $-log_{10}(p)$ from different MR methods across 265 exposure–outcome pairs in the negative control outcome analysis. The red dots represent all 265 pairs, while the blue triangles represent the 93 pairs with insignificant $c_{12}$ values. The black diagonal line indicates the expected distribution under the null hypothesis.
• Figure 4: Same trait results for different methods under the revised sigma-based pruning procedure
• Figure 5: Results of causal relationships among complex traits. (A) Number of significant causal effects detected by different methods under Bonferroni correction ($<0.05/152 \approx 0.0003$). (B) Significant causal effects identified by BRIVW, MR-APSS, Weighted-mode, and Egger under Bonferroni correction.

Theorems & Definitions (5)

• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 1
• Theorem 2