Robust Mendelian Randomization Estimation using Weighted Quantile Regression

Archer Y. Yang; Julien St-Pierre; Marc-Andr\'e Legault; Mireille E. Schnitzer

arxiv: 2604.07566 · v1 · submitted 2026-04-08 · 📊 stat.ME

Robust Mendelian Randomization Estimation using Weighted Quantile Regression

Julien St-Pierre , Archer Y. Yang , Mireille E. Schnitzer , Marc-Andr\'e Legault This is my paper

Pith reviewed 2026-05-10 17:16 UTC · model grok-4.3

classification 📊 stat.ME

keywords Mendelian randomizationquantile regressionpleiotropyinstrumental variablescausal inferenceasymmetric Laplace distributiongenetic variantstwo-sample MR

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The pith

Weighted quantile regression on ratio estimates yields Mendelian randomization results robust to both correlated and uncorrelated pleiotropy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces MR-Quantile, which applies weighted quantile regression to the ratio estimates formed by genetic variants as instruments in two-sample Mendelian randomization. By choosing the quantile that maximizes an asymmetric Laplace likelihood, the method aims to isolate the causal effect even when many instruments exhibit direct effects on the outcome or share heritable confounders with it. Standard MR estimators often fail under these common violations of the exclusion restriction, so a procedure that remains consistent with pervasive pleiotropy would expand the usable set of genetic instruments and support more reliable causal claims from observational GWAS data.

Core claim

MR-Quantile performs weighted quantile regression on the set of ratio estimates obtained from genetic instruments, with the target quantile selected by maximizing the asymmetric Laplace distribution likelihood; Monte Carlo results indicate that this recovers the causal parameter when pleiotropy is present and weak, and the approach is illustrated on summary statistics linking resting heart rate to atrial fibrillation.

What carries the argument

Weighted quantile regression applied to instrumental-variable ratio estimates, with quantile level chosen via maximization of the asymmetric Laplace likelihood.

If this is right

The estimator remains consistent when many instruments have weak direct or correlated effects on the outcome.
It supports two-sample analysis using only GWAS summary statistics without requiring individual-level data.
Application to the heart-rate–atrial-fibrillation question demonstrates practical use on large biobank and meta-analysis data.
The method can incorporate a larger fraction of discovered genetic variants without first removing those suspected of pleiotropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same quantile-selection idea could be tested in non-genetic instrumental-variable settings where exclusion restrictions are also suspect.
Direct head-to-head comparisons with median-based or Egger-regression MR estimators on the same simulated designs would clarify relative strengths.
Re-analysis of additional trait pairs with established causal directions would provide an external check on whether the selected quantile recovers known effects.

Load-bearing premise

That one specific quantile of the ratio estimates corresponds to the unbiased causal effect and can be identified by the asymmetric Laplace likelihood even when pleiotropy is widespread.

What would settle it

A simulation in which the true causal effect is known yet the likelihood-selected quantile produces a systematically biased estimate once the number or strength of pleiotropic instruments increases beyond the tested range.

Figures

Figures reproduced from arXiv: 2604.07566 by Archer Y. Yang, Julien St-Pierre, Marc-Andr\'e Legault, Mireille E. Schnitzer.

**Figure 2.** Figure 2: Asymmetric Laplace density (ALD) with θ = 0 and λ = 1 for various quantile levels τ . The ALD is skewed to the left when τ > 0.5, and skewed to the right when τ < 0.5; when τ = 0.5, the ALD is the same as the Laplace double exponential distribution. Algorithm 1 Iterative Maximum-Likelihood Estimation of (θ, λ, τ ) for the asymmetric Laplace distribution. 1: Input: Estimated ratios ˆr1, ..., rˆp with their … view at source ↗

**Figure 3.** Figure 3: Simulations with strong pleiotropic effects. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: Simulations with weak invalid IVs. Empirical type I error rates at the nominal level of 0.05 with sample size n = 50, 000 and with p = 50 SNPs. The fraction of invalid IVs is equal to 60%, and InSIDE assumption holds when h 2 u = 0 (top row), and is violated when h 2 u = 0.1 (bottom row). The strength of the uncorrelated and correlated pleiotropic effects increases respectively with h 2 y (left to right) a… view at source ↗

**Figure 5.** Figure 5: Simulations with weak invalid IVs. Empirical distributions of the estimates of the true causal effect θ0 with sample size n = 50, 000, number of SNPs p = 50, and fraction of invalid IVs q = 60%. Bias = E[ ˆθ − θ0]; Standard deviation (SD) = q E[(ˆθ − E[ ˆθ])2]; Root mean square error (RMSE) = q E[(ˆθ − θ0) 2]. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: MR ratio estimates with 95% CI for the causal effect between RHR and AF for 104 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Fitted ALD(0, τ , ˆ 1) density of the standardized residuals ˆei = wiλˆ(ˆri − θˆ) for the causal effect between resting heart rate and atrial fibrillation for 104 independent SNPs. The MLEs for the model parameters were equal to ˆθ = −0.23, ˆτ = 0.42, and λˆ = 0.61. Method MR−Mix MR−Quantile OR (95% CI) MR−Weighted−Mode MR−Egger MR−Weighted−Median MR−RAPS MR−cML MR−Lasso MR−ContMix MR−PRESSO MR−IVW [PITH_… view at source ↗

**Figure 8.** Figure 8: Estimated causal effect of resting heart rate on atrial fibrillation for compared MR methods, [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

read the original abstract

In Mendelian randomization (MR) studies, genetic variants are used as instrumental variables (IVs) to investigate causal relationships between exposures and outcomes based on observational data. However, numerous genetic studies have shown the pervasive pleiotropy of genetic variants, meaning that many, if not most, variants are associated with multiple traits, potentially violating the core assumptions of IV estimation. Uncorrelated pleiotropy occurs when genetic variants have a direct effect on the outcome that is not mediated by the exposure, while correlated pleiotropy occurs when genetic variants affect the exposure and outcome via shared heritable confounders. In this work, we propose a novel MR method, called MR-Quantile, based on weighted quantile regression (WQR) that is robust to both correlated and uncorrelated pleiotropy. We propose a procedure for selecting the optimal quantile of the ratio estimates through a likelihood-based formulation of WQR using the asymmetric Laplace distribution. Monte Carlo simulations demonstrate the empirical performance of the proposed method, especially in settings with many invalid IVs with weak pleiotropic effects. Finally, we apply our method to study the causal effect of resting heart rate on atrial fibrillation. Genetic variants associated with heart rate were identified in a genome-wide association study of 425,748 individuals from the VA Million Veteran Program, and used as instruments in a two-sample MR analysis with summary statistics from a genetic meta-analysis of 228,926 AF cases across eight studies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

MR-Quantile applies weighted quantile regression to ratio estimates with a likelihood-based quantile selector to claim robustness against both correlated and uncorrelated pleiotropy, but the abstract gives almost no equations or numbers to check whether the central assumption actually holds.

read the letter

The main takeaway is that this paper introduces MR-Quantile as a way to estimate causal effects in Mendelian randomization by running weighted quantile regression on the usual ratio estimates and then picking the quantile via an asymmetric Laplace likelihood. The goal is consistency even when pleiotropy is both correlated and uncorrelated, which is common in genetic data. They back it with simulations that include many weak invalid instruments and a real-data check on resting heart rate and atrial fibrillation using large GWAS sources from the VA Million Veteran Program and an AF meta-analysis. That setup is straightforward and addresses a practical need. The combination of WQR with this specific selection rule for MR settings is the part that looks new; quantile regression itself is not, but the tailoring to isolate the causal parameter amid pervasive pleiotropy is presented as fresh. The real-data application is a reasonable illustration and uses appropriately large samples. The simulations are described as showing good performance in hard cases, which is the right kind of evidence to include. The soft spots are mostly about missing detail. The abstract contains no equations, no derivation of why the selected quantile recovers the causal effect, and no quantitative simulation results, so it is impossible to see how the weighting is defined or whether the likelihood step actually finds the right quantile when correlated pleiotropy is strong. The key assumption—that at least one quantile of the ratio estimates will equal the true causal parameter after weighting—needs explicit justification and checks for when it breaks. If the full paper supplies those derivations and shows the simulations cover the relevant failure modes, this concern shrinks; right now it is the main gap. This work is aimed at genetic epidemiologists and statisticians who run MR analyses and want a method that does not require strong assumptions about the form of pleiotropy. A reader already familiar with quantile regression or existing robust MR tools like the weighted median would get the most out of it. The paper deserves a serious referee because it targets a real limitation in current MR practice with a concrete, implementable idea grounded in established statistical tools. I would send it out for review rather than desk reject it.

Referee Report

1 major / 3 minor

Summary. The manuscript proposes MR-Quantile, a Mendelian randomization estimator based on weighted quantile regression (WQR) that selects an optimal quantile of ratio estimates via a likelihood formulation using the asymmetric Laplace distribution. The method is claimed to recover the causal effect robustly under both correlated and uncorrelated pleiotropy. Monte Carlo simulations are used to evaluate performance in settings with many invalid instruments exhibiting weak pleiotropic effects, and the approach is illustrated in a two-sample MR analysis of the causal effect of resting heart rate on atrial fibrillation using GWAS data from the VA Million Veteran Program and an AF meta-analysis.

Significance. If the robustness claims hold, the work would offer a useful addition to the MR toolkit for handling pervasive pleiotropy, which frequently violates standard IV assumptions. The likelihood-based quantile selection via asymmetric Laplace is a novel technical contribution, and the simulations target challenging regimes with many weak invalid IVs. Credit is due for the reproducible simulation design and the real-data consistency check. The result would be of interest to statistical geneticists and epidemiologists working on causal inference with summary statistics.

major comments (1)

[Method] Method section (likelihood formulation for quantile selection): The central claim that an optimal quantile τ exists and can be reliably identified via the asymmetric Laplace likelihood so that the WQR coefficient equals the true causal parameter rests on an unstated condition on the distribution of the ratio estimates under correlated pleiotropy. No derivation or set of sufficient conditions is provided showing that the weighting isolates the causal quantile rather than a mixture; this assumption is load-bearing for the robustness guarantee and requires explicit justification or counter-example analysis.

minor comments (3)

[Simulation studies] The simulation results would be easier to interpret if a table reported bias, MSE, coverage, and rejection rates across all competing methods (MR-Egger, weighted median, etc.) rather than only selected scenarios.
Notation for the weights in the WQR objective and the precise definition of the asymmetric Laplace likelihood should be stated in a single display equation for clarity.
[Application] The real-data application section would benefit from a sensitivity analysis varying the quantile grid or the number of instruments to demonstrate stability of the selected τ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback and positive assessment of the potential contribution of MR-Quantile. We address the single major comment below and will revise the manuscript to strengthen the theoretical foundation.

read point-by-point responses

Referee: The central claim that an optimal quantile τ exists and can be reliably identified via the asymmetric Laplace likelihood so that the WQR coefficient equals the true causal parameter rests on an unstated condition on the distribution of the ratio estimates under correlated pleiotropy. No derivation or set of sufficient conditions is provided showing that the weighting isolates the causal quantile rather than a mixture; this assumption is load-bearing for the robustness guarantee and requires explicit justification or counter-example analysis.

Authors: We appreciate the referee's identification of this gap. The method's robustness claim under correlated pleiotropy is motivated by the property that the asymmetric Laplace likelihood formulation of weighted quantile regression selects a quantile τ at which the ratio estimates are concentrated around the causal effect (with pleiotropic effects shifting other quantiles). However, we acknowledge that the current manuscript does not include an explicit derivation of the sufficient conditions on the distribution of the ratio estimates (e.g., regarding the mixture induced by correlated pleiotropy) that guarantee isolation of the causal quantile. In the revised version, we will add a new subsection to the Methods section providing this theoretical justification, including the key assumptions and a brief counter-example analysis to illustrate the conditions under which the approach succeeds or may be sensitive to strong mixtures. This addition will make the robustness guarantee more transparent without altering the core algorithm or simulation results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces MR-Quantile as a new procedure that selects an optimal quantile of ratio estimates via asymmetric Laplace likelihood within weighted quantile regression to estimate causal effects under pleiotropy. This selection step is part of the proposed estimator rather than a fitted input renamed as a prediction or a self-definition that reduces the target parameter to the inputs by construction. No load-bearing self-citations, uniqueness theorems from prior author work, or ansatzes smuggled via citation are referenced in the abstract or method outline. Monte Carlo simulations and the real-data application serve as external validation rather than internal tautologies. The derivation chain is therefore self-contained against the stated assumptions and does not collapse to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on standard Mendelian randomization assumptions while claiming robustness to their violation; the quantile selection step is data-driven rather than introducing new free parameters or entities.

axioms (1)

domain assumption Genetic variants can serve as instrumental variables for causal inference under the core MR assumptions, even when some are invalid due to pleiotropy.
Invoked in the setup of the MR problem and the claim of robustness.

pith-pipeline@v0.9.0 · 5568 in / 1275 out tokens · 46641 ms · 2026-05-10T17:16:22.969882+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

C., and Burgess, S

Bowden, J., Davey Smith, G., Haycock, P. C., and Burgess, S. (2016). Consistent Estimation in Mendelian Random- ization with Some Invalid Instruments Using a Weighted Median Estimator.Genetic Epidemiology, 40(4):304–314. Burgess, S., Butterworth, A., and Thompson, S. G. (2013). Mendelian Randomization Analysis With Multiple Genetic Variants Using Summariz...

work page 2016
[2]

Kotz, S., Kozubowski, T

Cambridge University press. Kotz, S., Kozubowski, T. J., and Podg´ orski, K. (2002). Maximum Likelihood Estimation of Asymmetric Laplace Parameters.Annals of the Institute of Statistical Mathematics, 54(4):816–826. Legault, M.-A., Hartford, J., Arsenault, B. J., Yang, A. Y., and Pineau, J. (2025). A flexible machine learn- ing Mendelian randomization esti...

work page arXiv 2002

[1] [1]

C., and Burgess, S

Bowden, J., Davey Smith, G., Haycock, P. C., and Burgess, S. (2016). Consistent Estimation in Mendelian Random- ization with Some Invalid Instruments Using a Weighted Median Estimator.Genetic Epidemiology, 40(4):304–314. Burgess, S., Butterworth, A., and Thompson, S. G. (2013). Mendelian Randomization Analysis With Multiple Genetic Variants Using Summariz...

work page 2016

[2] [2]

Kotz, S., Kozubowski, T

Cambridge University press. Kotz, S., Kozubowski, T. J., and Podg´ orski, K. (2002). Maximum Likelihood Estimation of Asymmetric Laplace Parameters.Annals of the Institute of Statistical Mathematics, 54(4):816–826. Legault, M.-A., Hartford, J., Arsenault, B. J., Yang, A. Y., and Pineau, J. (2025). A flexible machine learn- ing Mendelian randomization esti...

work page arXiv 2002