Key points from chapter

• In this Chapter, we introduced four extensions to robust methods for Mendelian randomization with summarized data: 1) robust regression (MM-estimation); 2) penalized weights; 3) Lasso selection; and 4) least trimmed squares selection.

3.6 Discussion 87

These methods can be used to assess the robustness of findings from Mendelian randomization investigations with multiple genetic variants.

• The methods have been considered in two applied examples: one where there is evidence of over-dispersion in the causal estimates (the causal effect of body mass index on schizophrenia risk), and the other containing outliers (the causal effect of low-density lipoprotein on Alzheimer’s disease risk).

• Although the performances of the proposed methods in the simulation study were not significantly better than the robust methods that are already in the literature, the applied example suggested that the methods, particularly robust regression with penalized weights, may be a worthwhile addition to a Mendelian randomization study when there is a small proportion of heterogeneous causal estimates.

Chapter 4

Extending the MR-Egger method

for multivariable Mendelian

randomization to correct for both

measured and unmeasured

pleiotropy

4.1

Introduction

For some sets of risk factors, including lipid fractions, several risk factors have common genetic predictors. Although such genetic variants are pleiotropic, they can be used to estimate causal effects in a multivariable Mendelian randomization framework [28]. In multivariable Mendelian randomization, the IV assumptions are extended to allow a genetic variant to be associated with multiple risk factors, provided all associated risk factors are included in the analysis. Alternatively, when genetic variants are suspected to violate the IV assumptions through unknown pleiotropic pathways, methods have been developed to estimate consistent causal effects under weaker assumptions. These include the weighted median [52] and MR-Egger methods [29].

The extension of MR-Egger to a multivariable setting has been implemented by Helgadottir et al. [114] as part of a sensitivity analysis in their applied work investigating the effect of lipid fractions on coronary heart disease (CHD) risk. In the supplementary material, Helgadottir et al. [114] suggest that multiple covariates can be included in the MR-Egger method proposed by Bowden et al. [29]. The authors then go on to include

summary level data on high-density lipoprotein cholesterol, low-density lipoprotein cholesterol and triglycerides in the same MR-Egger model, and refer to this model as ‘multiple Egger regression’. The supplementary material contains the estimates for each lipid fraction from the ‘multiple Egger regression’ model.

The model we propose in this Chapter for extending MR-Egger to the multivariable setting is essentially the same as the ‘multiple Egger regression’ model considered by Helgadottir et al. [114] in their applied investigation. Since Helgadottir et al. [114] only considered the extension of MR-Egger within the context of an applied investigation, there remains several methodological issues relating to the implementation of the method, and the assumptions required. In particular, Helgadottir et al. [114] did not address the issues of residual pleiotropic effects, orientation of the genetic variants, or the assumptions required to obtain consistent causal effects.

In this Chapter, we extend the MR-Egger method to the multivariable setting through theoretical arguments (Section 4.2). In Section 4.3, we provide an example analysis using published summary level data on lipid fractions and CHD risk. We also perform two simulation studies to compare the performance of the methods: one where the risk factors do not have causal effects on each other (Section 4.4); and another where this assumption is relaxed (Section 4.5). Finally, in Section 4.6 we discuss the results from Sections 4.2 to 4.5, and consider the implications of extending the MR-Egger method to the multivariable setting on future research.

4.2

Methods

In this Section, we use theoretical arguments to expand the MR-Egger method to the multivariable setting. Initially, we consider the causal effect of a risk factor X on an outcome Y using J genetic variants Gj (j = 1, . . . , J) that are assumed to be

uncorrelated (not in linkage disequilibrium). Then, we expand to consider multiple risk factors X1, X2, . . . , XK, and from the MR-Egger and multivariable IVW models we

outline the regression model for multivariable MR-Egger. We provide the assumptions required for the causal estimates from multivariable MR-Egger to be consistent, and compare the precision of the causal estimates from univariable MR-Egger and multi- variable MR-Egger. The advantages of using multivariable MR-Egger over univariable MR-Egger are discussed in detail. Finally, we provide a recommendation on how the reference alleles of the genetic variants should be orientated in multivariable MR-Egger.

4.2 Methods 91

We assume that summarized data are available on the associations of each genetic variant with the risk factor and with the outcome: the beta-coefficients ( ˆβXj, ˆβYj) and

their standard errors (se( ˆβXj), se( ˆβYj)) from univariable regression on each variant Gj

in turn. For the multivariable setting, we assume that there is summary level data for the genetic variants on each risk factor, and these genetic variants are associated with at least one of the risk factors, and the risk factors are associated with at least one of the genetic variants. Finally, we assume that the associations of genetic variants with the risk factor and the outcome, and the causal effect of the risk factor on the outcome, are linear and homogeneous across the population. To distinguish between the parameters from the different methods considered, we use the following subscript notation: UI (‘univariable inverse variance weighted (IVW)’); UE (‘univariable MR-Egger’); MI (multivariable IVW); and ME (‘multivariable MR-Egger’).