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arxiv: 2605.28471 · v1 · pith:QQUXW26Jnew · submitted 2026-05-27 · 📊 stat.ME

The Modified Egger Intercept Tests for Detecting Horizontal Pleiotropy in Two-Sample Summary-Data Mendelian Randomization

Yilei Ma , Youpeng Su , Xin Liu , Xuanye Cui , Ping Yin , Peng Wang This is my paper

Pith reviewed 2026-06-29 10:36 UTC · model grok-4.3

classification 📊 stat.ME
keywords Mendelian randomizationhorizontal pleiotropyEgger interceptbias correctiontype I errorstatistical powerinstrumental variablessummary data
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The pith

A bias-corrected Egger intercept test, combined across allele codings, controls type I error and gains power for detecting horizontal pleiotropy.

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

The paper develops a modified Egger intercept test that corrects for bias in the standard version caused by measurement error and winner's curse. It uses the rerandomized inverse-variance weighted estimator to produce an unbiased intercept under the null of no directional or correlated pleiotropy, then proves the test's asymptotic validity. Because power still varies with how SNPs are oriented, the authors combine the modified test statistics from two allele coding schemes. Simulations and real data applications show the combined test maintains nominal error rates more accurately and detects pleiotropy more often than the original Egger intercept test.

Core claim

The modified Egger intercept (MEI) test employs a bias-corrected estimator of the intercept obtained from the rerandomized IVW estimator under the null of no directional or correlated pleiotropy. The paper establishes the asymptotic properties of this test and shows that its power is sensitive to SNP orientation. Combining the MEI statistics computed under two specific allele coding schemes produces a single test that exhibits improved type I error control and higher power relative to the conventional EI test, as confirmed by both simulation studies and real data examples.

What carries the argument

The bias-corrected Egger intercept estimator constructed from the rerandomized IVW estimator, which removes the effects of measurement error and winner's curse under the null.

If this is right

  • The MEI test supplies a more trustworthy check on whether the IVW causal estimator is biased by pleiotropy.
  • Combining results across the two allele coding schemes stabilizes detection power regardless of arbitrary SNP orientation.
  • The test can replace the standard EI procedure in routine two-sample summary-data MR analyses.
  • Under realistic conditions without pleiotropy the combined test preserves correct type I error rates.

Where Pith is reading between the lines

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

  • Previous MR findings that relied on the uncorrected EI test may warrant re-examination with the modified version.
  • The same bias-correction idea could be applied to other intercept-based or regression-based sensitivity analyses in summary-data MR.
  • Routine reporting of both allele coding schemes might become standard practice to avoid orientation-dependent power loss.

Load-bearing premise

The rerandomized IVW estimator supplies an accurate bias correction for the Egger intercept when no directional or correlated pleiotropy is present.

What would settle it

A simulation under the null hypothesis of no pleiotropy in which the combined MEI test rejects at a rate materially above the nominal significance level after the rerandomized IVW correction is applied.

Figures

Figures reproduced from arXiv: 2605.28471 by Peng Wang, Ping Yin, Xin Liu, Xuanye Cui, Yilei Ma, Youpeng Su.

Figure 1
Figure 1. Figure 1: Causal model with SNP Gj , exposure X, outcome Y , and unmeasured confounder U. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Estimated power of various methods to detect directional pleiotropy (r = 1) based on 10, 000 replicates. There are no balanced pleiotropic IVs (q = 0). 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Estimated power of various methods to detect correlated pleiotropy (r = 0) based on 10, 000 replicates. There are no balanced pleiotropic IVs (q = 0). 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Estimated power of various methods in the presence of directional and correlated pleiotropy based on 10, 000 replicates. The total proportion of directional and correlated pleiotropic IVs is set at π3 = 1.5%, and there are no balanced pleiotropic IVs (q = 0). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original abstract

The Egger intercept (EI) test is a widely used tool to detect horizontal pleiotropy in two-sample summary-data Mendelian randomization. A significant EI test suggests that either the average pleiotropic effect differs from zero (i.e., directional pleiotropy) or the InSIDE (Instrument Strength Independent of Direct Effect) assumption is violated (i.e., correlated pleiotropy) or both. As such, the EI test provides an assessment of the validity of the instrumental variable assumptions, with a non-zero EI indicating that the commonly used inverse-variance weighted (IVW) estimator will be biased. However, the EI test may exhibit inaccurate type one error rates due to biased estimation in Egger regression caused by the measurement error and winner's curse. In this article, we propose a modified EI (MEI) test based on a bias-corrected EI estimator under the null hypothesis of no directional or correlated pleiotropy, leveraging the recently developed rerandomized IVW estimator. We then prove the asymptotic properties of the MEI test under realistic conditions. Like the EI test, we find that the power of the MEI test is also affected by the orientation of SNPs. To enhance the robustness of power, we further combine the MEI test statistics obtained under two specific allele coding schemes. Both simulation and real data studies show that the combined test outperforms the EI test in terms of type one error control and power.

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.

Referee Report

2 major / 2 minor

Summary. The paper proposes a modified Egger intercept (MEI) test for detecting horizontal pleiotropy (directional or correlated) in two-sample summary-data Mendelian randomization. It constructs a bias-corrected EI estimator using the rerandomized IVW estimator to mitigate measurement-error and winner's-curse bias under the null of no directional or correlated pleiotropy, proves asymptotic properties of the resulting test, and combines MEI statistics across two allele-coding orientations to improve power robustness. Simulations and real-data applications are reported to show superior type-I error control and power relative to the standard EI test.

Significance. If the asymptotic derivations are correct and the finite-sample bias correction is reliable under realistic instrument selection, the MEI and combined tests would offer a more trustworthy diagnostic for IV validity than the existing EI test, with direct implications for downstream causal-effect estimation in MR. The explicit asymptotic proofs and the orientation-robust combination are concrete strengths that distinguish the contribution.

major comments (2)
  1. [Abstract; §3 (asymptotic properties)] The central claim that the MEI (and combined) test controls type I error at nominal levels rests on the rerandomized IVW supplying an unbiased correction for the EI estimator's finite-sample bias. The manuscript states that asymptotics hold under realistic conditions, but does not demonstrate that the rerandomization procedure replicates the exact winner's-curse mechanism induced by selecting the strongest SNPs from a finite discovery GWAS; residual bias would invalidate the type-I guarantee even when InSIDE holds.
  2. [Simulation section (referenced in abstract)] Simulations are invoked to support superior type-I control and power, yet the design details (instrument selection threshold, discovery-sample size, correlation structure between pleiotropy and strength) are not shown to match the data-generating process that produces the winner's-curse bias the correction is intended to remove. Without this match, the reported performance gains cannot be taken as evidence that the correction works under the conditions where the EI test is known to fail.
minor comments (2)
  1. The precise definition of the two allele-coding schemes used for the combined test should be stated explicitly (e.g., reference allele choice relative to the exposure or outcome GWAS) so that readers can reproduce the orientation-robust procedure.
  2. Table or figure captions for the real-data applications should list the exact number of instruments retained after clumping and the discovery-sample sizes, to allow assessment of how close the settings are to the finite-sample regime where bias correction is most needed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the conditions under which the proposed MEI test provides reliable type I error control. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract; §3 (asymptotic properties)] The central claim that the MEI (and combined) test controls type I error at nominal levels rests on the rerandomized IVW supplying an unbiased correction for the EI estimator's finite-sample bias. The manuscript states that asymptotics hold under realistic conditions, but does not demonstrate that the rerandomization procedure replicates the exact winner's-curse mechanism induced by selecting the strongest SNPs from a finite discovery GWAS; residual bias would invalidate the type-I guarantee even when InSIDE holds.

    Authors: The rerandomized IVW estimator is constructed precisely to replicate the finite-sample selection bias arising from choosing the strongest instruments in a discovery GWAS, by resampling the summary statistics conditional on the observed selection event. Section 3 derives the asymptotic normality of the bias-corrected MEI estimator under the null (InSIDE holding with no directional or correlated pleiotropy) and shows that the correction term converges to the exact bias induced by this selection mechanism. We will add an expanded paragraph in §2.2 and §3 explicitly linking the rerandomization steps to the winner's-curse distribution and stating the regularity conditions under which the replication holds. revision: yes

  2. Referee: [Simulation section (referenced in abstract)] Simulations are invoked to support superior type-I control and power, yet the design details (instrument selection threshold, discovery-sample size, correlation structure between pleiotropy and strength) are not shown to match the data-generating process that produces the winner's-curse bias the correction is intended to remove. Without this match, the reported performance gains cannot be taken as evidence that the correction works under the conditions where the EI test is known to fail.

    Authors: The simulation design in §4 uses selection thresholds (p < 5×10^{-8}) and discovery-sample sizes (N_disc = 10^5–5×10^5) that induce the same order of winner's-curse bias as typical two-sample MR applications, with the correlation between instrument strength and pleiotropy set to zero under the null. To make the match explicit, we will insert a new subsection 4.1 that tabulates the simulation parameters against the theoretical DGP, reports the realized bias of the uncorrected EI estimator, and adds sensitivity analyses varying the selection threshold and discovery-sample size to confirm that type I error remains controlled only when the rerandomized correction is applied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external bias correction and independent evaluation

full rationale

The paper proposes the MEI test by applying a bias correction drawn from the rerandomized IVW estimator (described as recently developed and external to this work), states and proves asymptotic properties under the null, and evaluates type I error and power via separate simulation studies and real-data applications. No equation reduces a claimed prediction or result to a quantity fitted from the same data used to define the test statistic, and no load-bearing premise rests solely on a self-citation chain. The central performance claims are therefore not equivalent to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method depends on the validity of the rerandomized IVW estimator for bias correction and on standard Mendelian randomization assumptions such as the InSIDE condition for interpreting pleiotropy tests.

axioms (2)
  • domain assumption The rerandomized IVW estimator provides an appropriate bias correction for the Egger intercept under the null of no directional or correlated pleiotropy
    This is the basis for the modified EI estimator as described in the abstract.
  • domain assumption Asymptotic properties of the MEI test hold under realistic conditions
    Invoked to support the theoretical justification of the test.

pith-pipeline@v0.9.1-grok · 5806 in / 1401 out tokens · 36044 ms · 2026-06-29T10:36:20.564523+00:00 · methodology

discussion (0)

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Reference graph

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