Robust Mendelian Randomization Estimation using Weighted Quantile Regression

1. Problem Statement

Mendelian Randomization (MR) utilizes genetic variants as instrumental variables (IVs) to infer causal relationships between exposures and outcomes. The validity of MR relies on three core assumptions:

1. Relevance: The IV is associated with the exposure.
2. Independence from Outcome (Conditional): The IV is independent of the outcome given the exposure and unmeasured confounders.
3. Independence from Confounders: The IV is independent of unmeasured confounders.

The Challenge: In practice, pleiotropy (where a genetic variant affects multiple traits) is pervasive.

• Uncorrelated Pleiotropy: Violates Assumption 2 (direct effect on outcome not mediated by exposure).
• Correlated Pleiotropy: Violates Assumption 3 (affects exposure and outcome via shared confounders).

Existing methods often fail when a large proportion of IVs are invalid:

• Inverse-Variance Weighted (IVW): Fails if any IVs are invalid.
• Weighted Median: Requires at least 50% of IVs to be valid ("Majority Valid" assumption).
• Mode-based estimators (e.g., MR-Weighted-Mode): Require the largest group of IVs to be valid ("Plurality Valid" assumption) but can be unstable with small sample sizes or weak pleiotropic effects.
• Mixture Models (e.g., MR-Mix, MR-CAUSE): Computationally intensive and difficult to fit with few SNPs.
• Selection-based methods (e.g., MR-Lasso, cML): Struggle to distinguish invalid IVs with weak pleiotropic effects from noise, leading to inflated Type I error rates.

2. Methodology: MR-Quantile

The authors propose MR-Quantile, a novel method based on Weighted Quantile Regression (WQR) that is robust to both correlated and uncorrelated pleiotropy, relying only on the Plurality Valid assumption.

Core Statistical Framework

• Ratio Estimates: For each SNP $i$, the Wald ratio is calculated as $\hat{r}_i = \hat{\beta}_{Yi} / \hat{\beta}_{Xi}$.
• Distributional Assumption: The method assumes the ratio estimates follow an Asymmetric Laplace Distribution (ALD). The ALD is chosen because:
  1. It naturally links likelihood-based inference to quantile regression.
  2. Its heavy tails provide robustness against outliers (invalid instruments with strong pleiotropy).
  3. The mode of the ALD corresponds to the quantile parameter $\theta$.
• Likelihood Formulation:
  The log-likelihood function for parameters $(\theta, \tau, \lambda)$ is:
  $$\ell(\theta, \tau, \lambda) = \sum \log w_i + \sum \log(\lambda \tau(1-\tau)) - \lambda \sum w_i \rho_\tau(\hat{r}_i - \theta)$$
  Where $\rho_\tau$ is the check loss function, $w_i$ are inverse-variance weights, and $\tau$ is the skewness parameter.

Key Innovations

1. Data-Driven Quantile Selection ($\hat{\tau}$):
   Unlike traditional WQR where $\tau$ is fixed (e.g., 0.5 for the median), MR-Quantile estimates the optimal quantile level $\tau$ via Maximum Likelihood Estimation (MLE). This allows the method to automatically target the mode of the valid instruments without needing to know the proportion of invalid IVs in advance.
   • The MLE for $\tau$ is derived analytically based on the relationship between the weighted mean of residuals and the location parameter.
2. Iterative Estimation:
   An iterative algorithm (Algorithm 1) is used to jointly estimate the causal effect $\theta$, the scale parameter $\lambda$, and the quantile level $\tau$ until convergence.
3. Inference:
   Standard errors and confidence intervals are derived using a parametric bootstrap approach, resampling the summary statistics to account for uncertainty in $\hat{\tau}$ and $\hat{\lambda}$.

3. Key Contributions

• Robustness to High Invalidity: The method remains consistent even when more than 50% of IVs are invalid, provided the valid IVs constitute the largest group (Plurality Valid assumption).
• Handling Weak Pleiotropy: It specifically addresses the challenge of "weak" invalid IVs where pleiotropic effects are small and difficult to distinguish from noise, a scenario where selection-based methods (like Lasso) often fail.
• Computational Efficiency: The method is significantly faster than mixture models (e.g., MR-Mix) and data-perturbation methods (cML-MA-BIC-DP), making it scalable for large-scale applications like Transcriptome-Wide Association Studies (TWAS).
• Unified Framework: It generalizes the Weighted Median estimator (which is a special case where $\tau=0.5$) to a flexible quantile regression framework.

4. Simulation Results

The authors conducted extensive Monte Carlo simulations comparing MR-Quantile against 11 other MR methods under various scenarios:

• Scenario A: Strong Pleiotropic Effects
  • Performance: MR-Quantile controlled Type I error rates effectively when the InSIDE assumption was violated (correlated pleiotropy).
  • Comparison: It outperformed IVW, MR-Egger, and MR-PRESSO in terms of bias and Root Mean Square Error (RMSE). It performed comparably to cML-MA-BIC and MR-Lasso but with much lower computational cost.
• Scenario B: Weak Invalid IVs
  • Performance: This was the critical test. Methods relying on selecting invalid IVs (cML-MA-BIC, MR-Lasso) showed inflated Type I error rates because they could not distinguish weak pleiotropy from noise.
  • Result: MR-Quantile was the only method (alongside cML-MA-BIC-DP) that maintained Type I error rates close to the nominal 0.05 level across all settings.
  • Power: MR-Quantile demonstrated higher power to detect causal effects than cML-MA-BIC-DP, which was overly conservative.
• Computational Speed: MR-Quantile was nearly 120 times faster than cML-MA-BIC-DP when $p=100$ SNPs, highlighting its suitability for high-throughput analyses.

5. Real-World Application

The method was applied to investigate the causal effect of Resting Heart Rate (RHR) on Atrial Fibrillation (AF).

• Data: 425,748 individuals (VA Million Veteran Program) for RHR and 228,926 AF cases (meta-analysis).
• Instruments: 104 independent SNPs significantly associated with RHR.
• Findings:
  • The ratio estimates were highly heterogeneous, suggesting significant pleiotropy.
  • MR-Quantile estimated a causal log-relative risk of -0.23 (RR = 0.796, 95% CI: 0.728–0.870).
  • Conclusion: Higher RHR is associated with a lower risk of AF (a protective effect), consistent with previous MR studies but contradicting some clinical intuition (though the authors note potential non-linearities and context-specific pleiotropy).
  • PheWAS Analysis: Suggested that invalid IVs might cluster around traits like BMI and pulmonary function, indicating distinct biological pathways.

6. Significance and Limitations

Significance:
MR-Quantile fills a critical gap in the MR toolkit by providing a method that is both statistically robust to high proportions of invalid IVs (including weak ones) and computationally efficient. It offers a reliable alternative for large-scale genetic studies where the validity of instruments is uncertain.

Limitations:

1. Asymptotic Theory: The non-differentiability of the ALD density at the mode prevents standard asymptotic variance derivation; the authors rely on bootstrapping, which may reduce power in small samples compared to asymptotic methods.
2. Single Exposure: Currently limited to single-exposure models; extension to multivariable MR is not yet implemented.
3. Sample Overlap: Assumes independent summary statistics for exposure and outcome; future work is needed to handle overlapping GWAS samples.

In summary, MR-Quantile represents a significant advancement in causal inference, offering a robust, data-driven, and efficient solution to the pervasive problem of pleiotropy in genetic epidemiology.