Modal-based estimation via heterogeneity-penalized weighting: model averaging for consistent and efficient estimation in Mendelian randomization when a plurality of candidate instruments are valid

Abstract

BackgroundA robust method for Mendelian randomization does not require all genetic variants to be valid instruments to give consistent estimates of a causal parameter. Several such methods have been developed, including a mode-based estimation method giving consistent estimates if a plurality of genetic variants are valid instruments; i.e. there is no larger subset of invalid instruments estimating the same causal parameter than the subset of valid instruments.MethodsWe here develop a model-averaging method that gives consistent estimates under the same ‘plurality of valid instruments’ assumption. The method considers a mixture distribution of estimates derived from each subset of genetic variants. The estimates are weighted such that subsets with more genetic variants receive more weight, unless variants in the subset have heterogeneous causal estimates, in which case that subset is severely down-weighted. The mode of this mixture distribution is the causal estimate. This heterogeneity-penalized model-averaging method has several technical advantages over the previously proposed mode-based estimation method.ResultsThe heterogeneity-penalized model-averaging method outperformed the mode-based estimation in terms of efficiency and outperformed other robust methods in terms of Type 1 error rate in an extensive simulation analysis. The proposed method suggests two distinct mechanisms by which inflammation affects coronary heart disease risk, with subsets of variants suggesting both positive and negative causal effects.ConclusionsThe heterogeneity-penalized model-averaging method is an additional robust method for Mendelian randomization with excellent theoretical and practical properties, and can reveal features in the data such as the presence of multiple causal mechanisms.

Bayesian variable selection with a pleiotropic loss function in Mendelian randomization

“…The list of methods that we considered in our simulation study is not exhaustive. For example, we did not implement the heterogeneity penalization approach presented in Burgess et al (2018). Similar to JAM-MR, this approach relies on model averaging, but instead of our algorithm's stochastic search it uses an exhaustive search over possible models, and is therefore only applicable for small numbers of genetic variants.…”

“…Since we are using heterogeneity as a proxy for pleiotropic behaviour, JAM-MR implicitly makes a plurality assumption: the algorithm targets the largest set of genetic variants with homogeneous univariate causal effect estimates, and it is assumed that this set corresponds to the valid instruments. This assumption is similar to those made in Hartwig et al (2017) and Burgess et al (2018). If smaller sets of genetic variants with homogeneous ratio estimates exist, the stochastic search will sometimes identify them; in practice, this can be checked by inspecting the list of posterior model probabilities.…”

“…We now discuss how to specify the loss function (D, γ). It is common in the Mendelian randomization literature to use some measure of heterogeneity between univariate causal effect estimates as a proxy for pleiotropic behaviour (Hartwig et al, 2017;Burgess et al, 2018). The intuition is that genetic variants which are valid instruments yield the same univariate causal effect estimates, up to some random variation.…”

“…This second loss function has the advantage that it also models uncertainty in the univariate ratio estimatesθ j through its dependence on the standard errorsŝ 2 θj . A similar function was used in Burgess et al (2018) to obtain a pleiotropy-robust exhaustive-search model averaging procedure for Mendelian randomization. In the simulations and the real-data application of this paper, we have used the weighted loss function (18).…”

“…In practical applications, we typically do not know which genetic variants exhibit pleiotropic effects and there is need for methods to perform Mendelian randomization in the presence of pleiotropic variants. Traditional approaches include MR-Egger regression (Bowden et al, 2015) and median estimation , while several algorithms for pleiotropy-robust Mendelian randomization have been developed recently (Hartwig et al, 2017;Kang et al, 2016;Rees et al, 2019;Burgess et al, 2018;Verbanck et al, 2018;Zhao et al, 2018;Qi and Chatterjee, 2018;Bowden et al, 2018;Burgess et al, 2019). Recent reviews and comparison of the various methods can be found in Slob and Burgess (2019) and Qi and Chatterjee (2019).…”