[Paper Review] Weak-Instrument Robust Tests in Two-Sample Summary-Data Mendelian Randomization

Sheng Wang, Hyunseung Kang|arXiv (Cornell University)|Sep 16, 2019

Advanced Causal Inference Techniques12 references4 citations

TL;DR

This paper proposes weak-instrument robust tests—mrAR, mrK, and mrCLR—for two-sample summary-data Mendelian randomization by extending econometric methods (Anderson-Rubin, Kleibergen, and conditional likelihood ratio tests) to summary statistics. The proposed tests maintain correct size and coverage under weak instruments, with mrCLR uniquely satisfying Dufour's (1997) necessary condition for valid confidence intervals by producing infinite intervals when needed, outperforming existing methods in simulation and empirical studies.

ABSTRACT

Mendelian randomization (MR) has been a popular method in genetic epidemiology to estimate the effect of an exposure on an outcome using genetic variants as instrumental variables (IV), with two-sample summary-data MR being the most popular. Unfortunately, instruments in MR studies are often weakly associated with the exposure, which can bias effect estimates and inflate Type I errors. In this work, we propose test statistics that are robust under weak instrument asymptotics by extending the Anderson-Rubin, Kleibergen, and the conditional likelihood ratio test in econometrics to two-sample summary-data MR. We also use the proposed Anderson-Rubin test to develop a point estimator and to detect invalid instruments. We conclude with a simulation and an empirical study and show that the proposed tests control size and have better power than existing methods with weak instruments.

Motivation & Objective

• To address the problem of weak instruments in two-sample summary-data Mendelian randomization, which can bias effect estimates and inflate Type I error rates.
• To extend weak-instrument robust econometric tests—Anderson-Rubin, Kleibergen, and conditional likelihood ratio—to the two-sample summary-data setting where individual-level data are unavailable.
• To ensure valid inference under weak instruments by maintaining correct size and coverage, even when instruments are weakly associated with the exposure.
• To develop a point estimator and a test for invalid instruments using the mrAR statistic, enhancing robustness to pleiotropy and confounding.
• To provide practical recommendations for using the proposed tests in real-world MR analyses, especially when instrument strength is uncertain.

Proposed method

• Extend the Anderson-Rubin (AR), Kleibergen (K), and conditional likelihood ratio (CLR) tests from econometrics to two-sample summary-data MR by leveraging recent work on two-sample summary statistics.
• Derive test statistics—mrAR, mrK, and mrCLR—that are asymptotically valid under weak instrument asymptotics, ensuring correct Type I error control.
• Use the mrAR test statistic to construct a point estimator and a test for invalid instruments, analogous to the Q statistic in MR-Egger, enabling detection of pleiotropy.
• Ensure that confidence intervals from mrCLR satisfy Dufour’s (1997) necessary condition for valid inference by allowing infinite intervals when instrument strength is weak.
• Apply the tests to simulated data and real-world MR datasets, including replication of Zhao et al. (2020), to evaluate performance under varying instrument strength and invalidity.
• Assess robustness to correlated instruments by simulating scenarios with unknown pairwise correlations and recommending conservative analysis under correlation.

Experimental results

Research questions

• RQ1 Can weak-instrument robust tests from econometrics be successfully adapted to two-sample summary-data Mendelian randomization where individual-level data are unavailable?
• RQ2 Do the proposed mrAR, mrK, and mrCLR tests maintain correct size and coverage under weak instrument asymptotics?
• RQ3 Can the mrAR test detect invalid instruments (e.g., pleiotropy) in a manner similar to existing methods like the Q statistic in MR-Egger?
• RQ4 Does mrCLR satisfy Dufour’s (1997) necessary condition for valid confidence intervals by producing infinite intervals when needed?
• RQ5 How do the proposed tests compare in power and coverage to existing methods like IVW, MR-Egger, and weighted median estimators under weak instruments?

Key findings

• The proposed mrAR, mrK, and mrCLR tests maintain correct size and 95% coverage across all scenarios, including under weak instruments, while existing methods like IVW and MR-Egger suffer from coverage inflation.
• mrCLR uniquely satisfies Dufour’s (1997) necessary condition for valid confidence intervals by producing infinite intervals approximately 95% of the time when instruments are weak, ensuring robust coverage.
• In simulations, mrCLR outperformed other weak-instrument-robust methods in terms of power and coverage, especially when there was no evidence of pleiotropy.
• The mrAR test successfully detected invalid instruments in both simulations and empirical data, with performance comparable to the Q statistic in MR-Egger, enabling pre-screening for instrument validity.
• When instruments were invalid or had strong direct effects, mrK and mrCLR showed increasing bias as the number and magnitude of invalid instruments increased, though they remained more robust than standard methods.
• Empirical replication of Zhao et al. (2020) showed that confidence intervals from mrCLR and mrK were similar to those from MR-RAPS with robust loss functions, suggesting consistency and practical utility.