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GENIUS-MAWII: For Robust Mendelian Randomization with Many Weak Invalid Instruments

Ting Ye, Zhonghua Liu, Baoluo Sun, Eric Tchetgen Tchetgen

TL;DR

A new MR method, G-Estimation under No Interaction with Unmeasured Selection (GENIUS)-MAny Weak Invalid IV, is proposed, which simultaneously addresses the 2 salient challenges in MR: many weak instruments and widespread horizontal pleiotropy.

Abstract

Mendelian randomization (MR) has become a popular approach to study causal effects by using genetic variants as instrumental variables. We propose a new MR method, GENIUS-MAWII, which simultaneously addresses the two salient phenomena that adversely affect MR analyses: many weak instruments and widespread horizontal pleiotropy. Similar to MR GENIUS (Tchetgen Tchetgen et al., 2021), we achieve identification of the treatment effect by leveraging heteroscedasticity of the exposure. We then derive the class of influence functions of the treatment effect, based on which, we construct a continuous updating estimator and establish its consistency and asymptotic normality under a many weak invalid instruments asymptotic regime by developing novel semiparametric theory. We also provide a measure of weak identification, an overidentification test, and a graphical diagnostic tool. We demonstrate in simulations that GENIUS-MAWII has clear advantages in the presence of directional or correlated horizontal pleiotropy compared to other methods. We apply our method to study the effect of body mass index on systolic blood pressure using UK Biobank.

GENIUS-MAWII: For Robust Mendelian Randomization with Many Weak Invalid Instruments

TL;DR

A new MR method, G-Estimation under No Interaction with Unmeasured Selection (GENIUS)-MAny Weak Invalid IV, is proposed, which simultaneously addresses the 2 salient challenges in MR: many weak instruments and widespread horizontal pleiotropy.

Abstract

Mendelian randomization (MR) has become a popular approach to study causal effects by using genetic variants as instrumental variables. We propose a new MR method, GENIUS-MAWII, which simultaneously addresses the two salient phenomena that adversely affect MR analyses: many weak instruments and widespread horizontal pleiotropy. Similar to MR GENIUS (Tchetgen Tchetgen et al., 2021), we achieve identification of the treatment effect by leveraging heteroscedasticity of the exposure. We then derive the class of influence functions of the treatment effect, based on which, we construct a continuous updating estimator and establish its consistency and asymptotic normality under a many weak invalid instruments asymptotic regime by developing novel semiparametric theory. We also provide a measure of weak identification, an overidentification test, and a graphical diagnostic tool. We demonstrate in simulations that GENIUS-MAWII has clear advantages in the presence of directional or correlated horizontal pleiotropy compared to other methods. We apply our method to study the effect of body mass index on systolic blood pressure using UK Biobank.

Paper Structure

This paper contains 36 sections, 13 theorems, 188 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Challenges in Mendelian randomization
  3. Prior work
  4. Our contributions
  5. Review of the GENIUS identification strategy
  6. Semiparametric theory
  7. Estimation and inference with many invalid IVs
  8. Measure of weak identification and diagnosis
  9. Measure of weak identification
  10. Overidentification test and graphical diagnosis
  11. Simulations
  12. A simulation when assumptions for GENIUS-MAWII hold
  13. A simulation under assumption violation and weak identification
  14. Application to the UK Biobank data
  15. Discussion
  16. ...and 21 more sections

Key Result

Theorem 1

(a) Under the conditional moment restriction $E(R_A (Y- \beta_0 A )\mid \bm{Z},\bm{X})=E(R_A (Y- \beta_0 A )\mid \bm{X})$, let $h(\bm Z, \bm X)$ be any scalar-valued function, the class of influence functions of $\beta_0$ is where $\Delta = R_A R_Y- \beta R_A^2$, $R_A= A- E(A\mid \bm{Z}, \bm{X}),$ and $R_Y= Y- E(Y\mid \bm{Z}, \bm{X})$. (b) The efficient influence function of $\beta_0$ is obtained

Figures (4)

  • Figure 1: Illustration of three types of SNPs: $\mathcal{S}_1$ consists of valid IVs, $\mathcal{S}_2$ consists of invalid IVs with uncorrelated pleiotropic effects (i.e., InSIDE is satisfied), and $\mathcal{S}_3$ consists of invalid IVs with correlated pleiotropic effects.
  • Figure 2: Residual plot for GENIUS-MAWII. The blue line is the estimated conditional mean using the smoothing splines, with gray point-wise confidence band (almost invisible in this plot). We see that the blue line is close to a straight horizontal line through zero, indicating that the errors are centered at zero, so there is no evidence of assumption violation.
  • Figure 3: A diagram of how to choose $\bm X$ in order to satisfy our identifying assumptions.
  • Figure 4: Organization of the proof.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma S1
  • proof
  • Lemma S2
  • proof
  • Lemma S3
  • proof
  • Lemma S4
  • ...and 15 more