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Statistics in MedicineGeneticsPeer-reviewed journal article

Testing and Correcting for Weak and Pleiotropic Instruments in Two-Sample Multivariable Mendelian Randomisation

Eleanor C. M. SandersoniD, Wes SpilleriD, Jack BowdeniD

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In-depth summary

Claude Opus 5.5 · max effort · 8 Nov 2026 · from the full text of version 1 (3 Apr 2020) · CC BY 4.0

Summary-data conditional F-statistics detect conditionally weak instruments in multivariable Mendelian randomisation, and Q-statistic minimisation gives weak-instrument and balanced-pleiotropy robust estimates plus an exact pleiotropy test.

Multivariable Mendelian randomisation (MVMR) needs variants predicting each exposure given the others; individually strong instruments can be conditionally weak, biasing inverse-variance-weighted (IVW) estimates and making pleiotropy tests over-reject. The Sanderson-Windmeijer conditional F-test needed individual-level data and pleiotropy-robust MVMR methods were scarce; this work extends testing and robust estimation to two-sample summary data with any number of exposures.

Key results

  1. ShownDividing a summary-data Q-statistic by the instrument count (adjusted for the number of exposures) gives a two-sample conditional F-statistic, FTS, equivalent to the individual-level Sanderson-Windmeijer FSW, so Stock-Yogo critical values or the F > 10 rule of thumb can be applied.

    Equivalence proven in Supplementary S.1 assuming uncorrelated instruments; simulated FSW and FTS distributions for 25 and 100 SNPs (two exposures, large individual F, FTS of 10) support it; Stock-Yogo values stop at 30 instruments, so Table 1 extends critical values for 5%, 10% and 20% relative bias, and F > 10 sits slightly under the 10%-bias thresholds. · 2 A Test for Weak Instruments; Critical values

  2. SuggestedRe-purposing the QA heterogeneity statistic as an estimating equation, analogous to limited-information maximum likelihood (LIML), gives weak-instrument-robust effects (βQ); adding a pleiotropy variance τ^2 (βQ,het) additionally accommodates balanced pleiotropy.

    Two exposures, 200 SNPs, balanced zero-mean pleiotropy, shared confounder: βQ,het tracked true effects where βQ did not, at the cost of larger standard errors, and showed no standard-error penalty versus βQ without pleiotropy; three exposures, 200 SNPs: MVMR-IVW biased by conditional weakness, QA minimisation unbiased (Tables 2 and 3). · Estimation in the presence of weak instruments; Estimation in the presence pleiotropy and weak…; MVMR model with two exposures; Three exposure model

  3. ShownThe minimised QA statistic, referred to a chi-squared distribution with L minus K degrees of freedom (SNPs minus exposures), tests for pleiotropic heterogeneity at nominal size despite weak instruments, unlike IVW-based Q-statistics that over-detect pleiotropy.

    Table 4 (two-exposure, conditionally weak design used for the FTS density simulations): with no heterogeneity QA rejected roughly 5% of the time, whereas QIVW and QIVW,up (IVW estimates with uncertainty-adjusted weights) both over-rejected, QIVW,up less severely than QIVW. · Testing for pleiotropy in the presence of weak…; Heterogeneity Testing; 8 Discussion

  4. SuggestedApplied to blood lipid fractions and age-related macular degeneration (AMD), FTS exposed conditional weakness hidden by strong individual F-statistics, QA flagged potential pleiotropy, and robust re-estimation pointed to no causal effect of the examined fractions on AMD.

    Kettunen GWAS (118 metabolites, 150 SNPs) reduced to 14 metabolites and 78 SNPs with F > 5: most had Fall (mean F over all SNPs) > 10 but very small FTS; within-group joint prediction failed except for small VLDL and IDL.PL; a one-per-group model still had FTS 4.2 to 8.3, and QA = 118 exceeded the 84.7 critical value (chi-squared, 64 df, 5% level). · Application to two-sample summary data MR; 6 Application

  5. SuggestedSetting to zero the covariance between a SNP's estimated associations with different exposures (seldom reported by GWAS) makes FTS unreliable and biases estimates when SNP effects correlate across exposures; approximating it from phenotypic correlation restores agreement with FSW.

    Two-exposure, 200-SNP simulation (Table 2), conditional weakness induced by correlated SNP effects: with covariances fixed at zero, FTS,0 misjudged instrument strength and the heterogeneity-robust estimator was biased, while phenotypic-correlation covariances gave FTS matching FSW; zero covariances are valid only under the global null (β = 0). · 4 Estimation of σij; Estimate the phenotypic correlation between the…; MVMR model with two exposures

Methods, data and reproducibility

Builds a per-exposure conditional F-statistic from a summary-data Q-statistic, with SNP-exposure covariances approximated from phenotypic correlation, and estimates direct effects by minimising the QA heterogeneity statistic (optionally with pleiotropy variance τ^2) using jackknife standard errors; evaluated by simulation and a lipid-AMD application.

  • Simulated two-sample summary data: 2-exposure models (weak or conditionally weak instruments, balanced pleiotropy, confounder) and a 3-exposure model with correlated SNP effects · n = 200 SNPs per model
  • Simulated two-exposure designs with strong individual but weak joint prediction (FTS of 10), comparing FSW and FTS distributions and Q-statistic rejection rates · n = 25 and 100 SNPs
  • Kettunen et al 2016 metabolite GWAS summary statistics used as exposures, restricted to 14 lipid fractions in IDL, LDL, small VLDL and very small VLDL groups · human (same metabolites measured in the ALSPAC human cohort) · n = 118 metabolites, 150 SNPs; 14 metabolites and 78 SNPs used
  • Genome-wide association study of age-related macular degeneration supplying the SNP-outcome associations · human (AMD described as a leading cause of blindness)
  • Avon Longitudinal Study of Mothers and Children (ALSPAC) metabolite data used to estimate phenotypic correlations among the analysed lipid fractions · human (ALSPAC mother-child cohort)
  • MVMR package tutorial only (not analysed in the paper): lipid fractions and systolic blood pressure using Global Lipids Genetics Consortium and UK Biobank data · human (UK Biobank; Global Lipids Genetics Consortium)
Methods & tools
MVMR R package, MVMR helpers snpcov_mvmr, phenocov_mvmr and format_mvmr (covariance estimation, data formatting), MVMR functions strength_mvmr, pleiotropy_mvmr, ivw_mvmr and qhet_mvmr, Two-sample conditional F-statistic (FTS), Sanderson-Windmeijer conditional F-statistic (FSW), Stock-Yogo weak-instrument critical values, Multivariable inverse-variance weighted estimation (MVMR-IVW), Q-statistic minimisation estimators (βQ, βQ,het), Heterogeneity statistics QA, QIVW and QIVW,up, Jackknife standard errors
Software & environment
R; MVMR package implementing a five-step workflow (covariance estimation, formatting, strength, pleiotropy, estimation); no package, R or dependency versions or containers are stated.
Reproducibility gaps
GWAS rarely report the SNP-exposure covariances the methods need, so external ALSPAC metabolite correlations were substituted; the text gives no download location or access route for ALSPAC, the Kettunen metabolite GWAS or the AMD GWAS.
Limitations (6)
  • Requires the covariance between each SNP's estimated effects on every pair of exposures, which GWAS rarely release; the phenotypic-correlation substitute is only a reasonable approximation when each SNP explains a small share of exposure variance. · stated by the authors
  • Q-minimisation yields no analytic standard errors, so jackknife intervals add computational burden, and point estimates become unreliable with very weak instruments or few SNPs; how weak or how few is acceptable remains unresolved. · stated by the authors
  • βQ,het accommodates only balanced pleiotropy and stays biased under directional or unbalanced pleiotropy; dropping conditionally weak exposures to regain strength can itself induce directional pleiotropy, as when the third simulated exposure was omitted. · stated by the authors
  • FTS gauges bias for the model as a whole using standardised exposures, so the bias of any single exposure's estimate may depart from 10% and may even differ in sign from the model-averaged bias. · stated by the authors
  • FTS-FSW equivalence assumes uncorrelated instruments, and simulations use at most three exposures (25, 100 or 200 SNPs), so simulated performance with linkage-disequilibrium-correlated variants or larger exposure sets is not reported despite claims for arbitrary exposure numbers. · noted in this summary
  • The AMD analysis relies on external ALSPAC correlations in place of GWAS-sample covariances and an illustrative subset of 14 metabolites, and the text describes no sensitivity analysis to either choice. · noted in this summary

Read this if you run multivariable Mendelian randomisation with GWAS summary statistics and need to check conditional instrument strength or obtain estimates robust to weak instruments and balanced pleiotropy.

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Abstract

Multivariable Mendelian Randomisation (MVMR) is a form of instrumental variable analysis which estimates the direct effect of multiple exposures on an outcome using genetic variants as instruments. Mendelian Randomisation and MVMR are frequently conducted using two-sample summary data where the association of the genetic variants with the exposures and outcome are obtained from separate samples. If the genetic variants are only weakly associated with the exposures either individually or conditionally, given the other exposures in the model, then standard inverse variance weighting will yield biased estimates for the effect of each exposure. Here we develop a two-sample conditional F-statistic to test whether the genetic variants strongly predict each exposure conditional on the other exposures included in a MVMR model. We show formally that this test is equivalent to the individual level data conditional F-statistic, indicating that conventional rule-of-thumb critical values of F > 10, can be used to test for weak instruments. We then demonstrate how reliable estimates of the causal effect of each exposure on the outcome can be obtained in the presence of weak instruments and pleiotropy, by re-purpousing a commonly used heterogeneity Q-statistic as an estimating equation. Furthermore, the minimised value of this Q-statistic yields an exact test for heterogeneity due to pleiotropy. We illustrate our methods with an application to estimate the causal effect of blood lipid fractions on age related macular degeneration.

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