Constrained maximum likelihood-based Mendelian randomization robust to both correlated and uncorrelated pleiotropic effects

Abstract

With the increasing availability of large-scale GWAS summary data on various complex traits and diseases, there have been tremendous interests in applications of Mendelian randomization (MR) to investigate causal relationships between pairs of traits using SNPs as instrumental variables (IVs) based on observational data. In spite of the potential significance of such applications, the validity of their causal conclusions critically depends on some strong modeling assumptions required by MR, which may be violated due to the widespread (horizontal) pleiotropy. Although many MR methods have been proposed recently to relax the assumptions by mainly dealing with uncorrelated pleiotropy, only a few can handle correlated pleiotropy, in which some SNPs/IVs may be associated with hidden confounders, such as some heritable factors shared by both traits. Here we propose a simple and effective approach based on constrained maximum likelihood and model averaging, called cML-MA, applicable to GWAS summary data. To deal with more challenging situations with many invalid IVs with only weak pleiotropic effects, we modify and improve it with data perturbation. Extensive simulations demonstrated that the proposed methods could control the type I error rate better while achieving higher power than other competitors. Applications to 48 risk factor-disease pairs based on large-scale GWAS summary data of 3 cardio-metabolic diseases (coronary artery disease, stroke, and type 2 diabetes), asthma, and 12 risk factors confirmed its superior performance.

Keywords: GWAS; causal inference; data perturbation; goodness-of-fit test; instrumental variable.

Figure 1

Causal model with exposure X and outcome Y (A) Three IV assumptions. (B) A general causal model.

Figure 2

Main simulations: Empirical type I error rates at the nominal level of 0.05 with sample size n = 50,000 and with m = 10 or 100 SNPs, among which 0 to 60% were invalid IVs with the InSIDE assumption either holding or violated

Figure 3

Main simulations: Empirical type I error (for $\theta =0$) and power (for $\theta \ne 0$) curves with sample size n = 50,000

Figure 4

Main simulations: Empirical distributions of the estimates of the causal effect θ by the methods with n = 50,000 and $\theta =0$ The numbers below each panel are the mean$\left(\hat{\theta }\right)$, SD$\left(\hat{\theta }\right)$, mean squared error (MSE) of $\hat{\theta }$ from top to bottom.

Figure 5

Main simulations: Empirical distributions of the estimates of the causal effect θ by the methods with n = 50,000 and $\theta =0.1$ The numbers below each panel are the mean$\left(\hat{\theta }\right)$, SD$\left(\hat{\theta }\right)$, mean squared error (MSE) of $\hat{\theta }$ from top to bottom.

Figure 6

Secondary simulations: Empirical type I error rates (for $\theta =0$) and power (for $\theta \ne 0$) with sample size n = 50,000 or 100,000, and with m = 10 or 100 exposure-associated SNPs

Figure 7

Simulation results with many invalid IVs having weak pleiotropic effects (A) Empirical type I error (for $\theta =0$) and power (for $\theta \ne 0$) curves with h_y = 0.2 and h_u = 0. (B) Relative frequencies of the goodness-of-fit tests rejecting the null hypothesis.

Figure 8

Results of cML-MA-BIC, MR-CAUSE, MR-Mix, and MR-IVW to detect causal relationships among 48 risk factor-disease pairs

Figure 9

The numbers of the significant risk factor-disease pairs detected by various methods at the significance cutoff of p value < 0.001 MR-RAPS refers to RAPS2 (with the Tukey loss and overdispersion).

Figure 10

BIC and scatter plots for four pairs For four risk factor-disease pairs, the left panels show the numbers of invalid IVs versus BIC values, while the right panels show ${\hat{\beta }}_{Xi}$ versus ${\hat{\beta }}_{Yi}$ (with their errors bars indicating ${\hat{\sigma }}_{Xi}$ and ${\hat{\sigma }}_{Yi}$). In the right panels, those for invalid IVs detected by BIC are blue, the red solid lines give the causal estimates (after removing the detected invalid IVs), and the black dashed lines are for the estimates based on all IVs.

Figure 11

Q-Q plots for 53 (likely) null trait-pairs in the secondary real data examples