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Mendelian randomization with fine-mapped genetic data: choosing from large numbers of correlated instrumental variables
Stephen Burgess, Verena Zuber, Elsa Valdes-Marquez, Benjamin B Sun, Jemma C Hopewell
TL;DR
Fine-mapped regions can contain many correlated variants, creating a trade-off between unstable estimates from including too many variants and sensitive estimates from selecting too few. The paper uses PCA on summarized genetic associations and correlations to form instruments, finding estimates that are less precise than variable-selection methods but less variable to variant-selection choices. Its main limitation is that including too many variants can still lead to numerical instabilities and inflated Type 1 error rates under problematic data conditions.
Problem
Choosing among many correlated variants is difficult because including too many can produce unstable estimates, while selecting few variants can make results sensitive to arbitrary choices.
Method
The paper applies principal components analysis to summarized genetic associations and correlation estimates to form instruments using the totality of variant data.
Results
PCA estimates are less variable with respect to variant-selection choices than estimates from GCTA and pruning approaches.
Takeaways & Limitations
The approach provides a more robust alternative to seemingly arbitrary variable-selection choices in fine-mapped Mendelian randomization analyses.
Takeaways & Limitations
Including too many variants can still cause numerical instabilities and inflated Type 1 error rates.
Abstract
from arXiv · showhide
Mendelian randomization uses genetic variants to make causal inferences about the effect of a risk factor on an outcome. With fine-mapped genetic data, there may be hundreds of genetic variants in a single gene region any of which could be used to assess this causal relationship. However, using too many genetic variants in the analysis can lead to spurious estimates and inflated Type 1 error rates. But if only a few genetic variants are used, then the majority of the data is ignored and estimates are highly sensitive to the particular choice of variants. We propose an approach based on summarized data only (genetic association and correlation estimates) that uses principal components analysis to form instruments. This approach has desirable theoretical properties: it takes the totality of data into account and does not suffer from numerical instabilities. It also has good properties in simulation studies: it is not particularly sensitive to varying the genetic variants included in the analysis or the genetic correlation matrix, and it does not have greatly inflated Type 1 error rates. Overall, the method gives estimates that are not so precise as those from variable selection approaches (such as using a conditional analysis or pruning approach to select variants), but are more robust to seemingly arbitrary choices in the variable selection step. Methods are illustrated by an example using genetic associations with testosterone for 320 genetic variants to assess the effect of sex hormone-related pathways on coronary artery disease risk, in which variable selection approaches give inconsistent inferences.
Background
The paper addresses how to choose correlated genetic variants for Mendelian randomization while retaining efficiency without numerical instability. It motivates a principal-components approach that uses summarized associations and correlation estimates.
- When variants are uncorrelated, including all instrumental variables maximizes power, but correlated variants make the appropriate selection unclear.
- Using many highly correlated variants can cause numerical instability, whereas using few variants may discard data and produce choice-sensitive estimates.
- For correlated variants, inverse-variance weighted estimation can be implemented with summarized genetic associations and a correlation-adjusted weighting matrix.
- The paper illustrates variant-selection guidance using hundreds of correlated variants in a fine-mapped gene region and provides implementation code.
- The proposed method uses principal components to ensure all variants contribute without introducing numerical instabilities.
Motivating example: serum testosterone and coro-
The motivating example examines testosterone and coronary artery disease using hundreds of variants in the SHBG region. Variable-selection methods produced substantially different and sometimes unreliable causal estimates.
- The analysis uses 320 variants in the SHBG gene region, where variants are associated with both testosterone and SHBG.
- A positive Mendelian randomization finding would indicate a causal role for sex hormone-related mechanisms, but would not distinguish testosterone from SHBG.
- The study compares conditional selection, stepwise conditional GCTA selection, and stepwise pruning across correlation thresholds.
- Causal estimates from similarly sized conditional and pruning analyses differed by over two standard errors and yielded opposing substantive conclusions.
- At a pruning threshold of 0.9 the standard error was undefined, while at 0.95 the estimate was clearly spurious.
- The authors conclude that it is unclear whether the data support testosterone as a causal risk factor for coronary heart disease.
Choosing the right number of variants
The paper explains why using too many or too few correlated variants can fail and proposes PCA-IVW to retain information while avoiding unstable matrix inversion. PCA forms weighted orthogonal instruments and applies IVW to the transformed data.
- Too many variants: near-singular genetic correlation matrix: In the motivating example, inverse correlation-matrix elements exceeded 10 million at high correlation thresholds.
- Too many variants: near-singular genetic correlation matrix: Near-singular correlation matrices can silently produce overly precise and highly misleading estimates when small data discrepancies are magnified by matrix inversion.
- Too few variants: Conditional analyses may retain fewer than 10 variants and ignore the remaining over 300 variants, making estimates sensitive to variant choice.
- Principal components analysis: PCA is applied to a weighted genetic correlation matrix whose diagonal elements represent inverse-variance weights for variant-specific causal estimates.
- Principal components analysis: The first principal component explains the largest proportion of risk-factor variance, with later components explaining successively smaller orthogonal proportions.
- Principal components analysis: The weighted matrix is intended to improve on PCA using an unweighted genetic-correlation matrix by preferentially selecting more precisely estimated variants.
- Principal components analysis: Selected components transform the risk-factor associations, outcome associations, and correlation matrix before inverse-variance weighted estimation.
- Results: In the testosterone example, PCA-based estimates were similar in precision to conditional analysis but less precise than GCTA estimates.
Simulation study
The simulations assessed how variant selection, correlation-matrix uncertainty, and rounded associations affect Mendelian randomization estimates. PCA estimates were less precise than pruning estimates but were substantially more stable across these choices, while aggressive pruning and rounding could inflate error rates.
- Varying variants: 10 000 random subsets of 180 variants were used to assess sensitivity to measuring only part of the available genetic data.The simulations repeatedly selected subsets from the motivating example.
- Varying the correlation matrix: 10 000 bootstrap correlation matrices were generated by resampling the reference data to assess sensitivity to correlation-matrix variation.Each bootstrap sample matched the original reference-data sample size.
- Compared approaches: Pruning used correlation thresholds of 0.2, 0.4, 0.6, and 0.8, whereas PCA retained components explaining 99% or 99.9% of variance.These six approaches were compared in both simulation studies.
- Varying the correlation matrix: For ρ = 0.8, pruning-based causal estimates were unstable and especially sensitive to changes in the correlation matrix.The mean causal estimate also changed substantially, while the standard deviation decreased as pruning thresholds increased.
- Robustness of PCA: PCA estimates were less precise but far less variable between analyses than estimates from the pruning approaches.The simulations varied both the included variants and the correlation matrix.
- Rounding and error rates: Type 1 error rates exceeded nominal levels for ρ = 0.8, and rounding association estimates further exacerbated false-positive findings across methods.Mean estimates remained close to the true causal effect, but rounding was associated with inflated Type 1 error rates in all methods.
Discussion
Fine-mapped Mendelian randomization must balance instability from including many correlated variants against sensitivity and information loss from selecting only a few. The proposed principal-components approach uses summarized genetic associations and correlations to incorporate all variants while remaining more robust to analytical choices, at a cost in precision.
- Theoretical basis: Theoretical arguments support including variants associated with the risk factor in a conditional analysis and using conditional associations as allele-score weights.This connects the choice of included variants and their weighting to the objective of obtaining the most efficient allele score.
- Motivation: Fine-mapped regions can contain many correlated variants, making it unclear which instruments to select and creating a trade-off between information and stability.Including too few variants produces less precise, highly variable estimates, whereas including too many can cause numerical instabilities and inflated Type 1 error rates.
- Method: The proposed method forms orthogonal instruments as linear combinations of variants using principal components of a weighted genetic correlation matrix.Weighting emphasizes components explaining the most variance in the risk factor, while orthogonality reduces numerical instability and summarized association and correlation data suffice for implementation.
- Robustness: The approach incorporates data from all genetic variants, producing causal estimates that are less variable with respect to choices of variants and other implementation details.The method is advocated for fine-mapped genetic data in Mendelian randomization, where summarized and two-sample settings can introduce discrepancies in genetic associations.
- Trade-offs: Principal-components estimates are less precise than estimates from variable-selection approaches such as GCTA and pruning, but they are less sensitive to how the analysis is implemented.The reported trade-off is robustness to variant-selection choices in exchange for reduced precision.
Figure legends
Figure 1 compares genetic associations with testosterone and coronary artery disease risk for variant sets selected using correlation thresholds of 0.95 and 0.8. The heavy dashed line represents the inverse-variance weighted estimate accounting for variant correlations.
- Figure 1 plots estimated genetic associations and 95% confidence intervals with testosterone and coronary artery disease risk.Testosterone associations are shown in nmol/L after log transformation, and coronary artery disease associations are shown as log odds ratios.
- 104 genetic variants are shown for threshold correlation 0.95, corresponding to r2 = 0.9025.
- 62 genetic variants are shown for threshold correlation 0.8, corresponding to r2 = 0.64.
- The heavy dashed line is the inverse-variance weighted estimate accounting for correlation between variants.
A.1 Software code
The software appendix provides implementations for inverse-variance weighted and two-stage least squares analyses, including correlated variants and allele scores. It also states conditions under which these methods are equivalent and discusses heterogeneity and principal-component choices.
- Genetic variants can be collapsed into an allele score and used as an instrument in the two-stage least squares method.
- 30: When variants are perfectly uncorrelated and weights are univariable regression coefficients, inverse-variance weighted and two-stage least squares analyses are equivalent.This equivalence requires the variants to be perfectly uncorrelated.
- Random-effects analysis is preferred when heterogeneity between variant-specific causal estimates is expected and enough variants are available to estimate heterogeneity.Heterogeneity is generally expected across multiple gene regions but not necessarily among variants in the same gene region with similar mechanisms.
- The inverse-variance weighted method accounting for correlation can be implemented through weighted linear regression.For continuous outcomes, associations are usually estimated by linear regression; for binary outcomes, logistic regression is used.
- The code uses summarized genetic associations with the risk factor and outcome, together with standard errors and correlations between genetic variants.Risk-factor associations are denoted betaXG with standard errors sebetaXG, while outcome associations are denoted betaYG with standard errors sebetaYG.
- The principal-component implementation includes enough components to explain 99% of the variance in the risk factor.
A.2 Proof of equality of 2SLS and inverse-variance weighted estimates
This appendix proves the relationship between two-stage least squares and inverse-variance weighted estimation. With correlated variants, the weighted regression uses a correlation-matrix-based weighting structure, whereas uncorrelated variants yield simpler equivalence conditions.
- When variants are uncorrelated, regression coefficients from univariate regressions equal those from multivariable regression.The assumption of uncorrelated instrumental variables underlies this equality.
- If instrumental variables are perfectly uncorrelated, the off-diagonal elements of (Z^T Z)^-1σ^2 are zero.
- The two-stage least squares standard errors are obtained from the square roots of the diagonal elements of (Z^T Z)^-1σ^2.Here, σ is the residual standard error.
- Two-stage least squares estimates causal effects by regressing the risk factor on instrumental variables and then the outcome on fitted risk-factor values.
- When variants are correlated, weighted regression of outcome associations on risk-factor associations uses the variant correlation matrix in the weighting structure.