Unbiased estimation and asymptotically valid inference in multivariable Mendelian randomization with many weak instrumental variables

TL;DR

This paper established the statistical theories of multivariable IVW and MRBEE with many weak IVs and investigated the asymptotic properties of multi-modal MRs, showing that MRBee outperforms multivariables IVW regarding unbiasedness of causal effect estimation and asymPTotic validity of causal inference.

Abstract

Mendelian randomization (MR) is an instrumental variable (IV) approach to infer causal relationships between exposures and outcomes with genome-wide association studies (GWAS) summary data. However, the multivariable inverse-variance weighting (IVW) approach, which serves as the foundation for most MR approaches, cannot yield unbiased causal effect estimates in the presence of many weak IVs. To address this problem, we proposed the MR using Bias-corrected Estimating Equation (MRBEE) that can infer unbiased causal relationships with many weak IVs and account for horizontal pleiotropy simultaneously. While the practical significance of MRBEE was demonstrated in our parallel work (Lorincz-Comi (2023)), this paper established the statistical theories of multivariable IVW and MRBEE with many weak IVs. First, we showed that the bias of the multivariable IVW estimate is caused by the error-in-variable bias, whose scale and direction are inflated and influenced by weak instrument bias and sample overlaps of exposures and outcome GWAS cohorts, respectively. Second, we investigated the asymptotic properties of multivariable IVW and MRBEE, showing that MRBEE outperforms multivariable IVW regarding unbiasedness of causal effect estimation and asymptotic validity of causal inference. Finally, we applied MRBEE to examine myopia and revealed that education and outdoor activity are causal to myopia whereas indoor activity is not.

Figures (6)

• Figure 1: DAG of MR and multivariable MR. Panel (a): causal path digram with valid genetic IVs. Panel (b): causal path digram with UHP and CHP. Panel (c): causal path digram for multivariable MR methods. $G$: genetic IVs; $X$: exposure; $Y$: outcome; $C$: confounders; $\beta$: association between $G$ and $X$; $\theta$: causal effect of $X$ on $Y$; $\gamma_c$: direct correlation between $G$ and $C$; $\gamma_u$: direct correlation between $G$ and $Y$.
• Figure 2: Investigation of MR methods for univarate MR with sample sizes $n_0=n_1=20000$, in terms of overlapping fraction and number of instrumental variants.
• Figure 3: Investigations of MRBEE and IVW in terms of asymptotic bias and covariance matrix.
• Figure 4: Investigation of MR methods for multivariable MR with sample sizes $n_0=\cdots=n_6=20000$ and overlap-sample sizes $n_{01}=\cdots=n_{65}=20000$, in terms of number of instrumental variants.
• Figure 5: Investigation of MR methods for multivariable MR with sample sizes $n_0=\cdots=n_6=20000$ and overlap-sample sizes $n_{01}=\cdots=n_{65}=20000$, in terms of number of specified exposures.

Theorems & Definitions (29)

• Definition 1: Sub-Gaussian variable
• Definition 2: Well-conditioned covariance matrix
• Definition 3: Strongly asymptotically unbiased estimate
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma A.1: Equivalent characterizations of sub-Guassian variables