Table 1.

Overview of sensitivity analyses available to examine evidence of violations of Mendelian randomization assumptions

Method	Purpose	What it does	Assumptions	Strengths	Limitations
MR-Egger regression and intercept test	Examines invalidation of the third MR assumption (i.e., horizontal pleiotropy). Specifically, this method tests for the presence of directional pleiotropy (MR-Egger intercept test) and the robustness of findings to directional pleiotropy (MR-Egger regression)	Performs a weighted generalized linear regression of the SNP-outcome effect estimates on the SNP-exposure effect estimates with an unconstrainted intercept term. If the InSIDE and NOME assumptions are met, the intercept term can provide a formal statistical test for directional pleiotropy and the slope generated from MR-Egger regression can provide an effect estimate that is adjusted for directional pleiotropy	InSIDE, NOME	Permits unbiased causal effects to be estimated even when all variants are invalid IVs	Sensitive to outliers; requires the InSIDE assumption to hold; low statistical power in the presence of no invalid instruments
Weighted median [61]	Examines invalidation and robustness of findings of the third MR assumption (i.e., horizontal pleiotropy)	Individual SNP effect estimates are ordered and weighted by the inverse of their variance. Providing at least 50% of the instruments are valid, the weighted median of this distribution is taken as an unbiased estimate of the causal effect	The median estimate (weighted by precision of SNPs) is unaffected by horizontal pleiotropy	Greater statistical power than MR-Egger; does not require the InSIDE assumption	Requires at least 50% of the information from variants to come from valid IVs
Weighted mode [62]	Examines invalidation and robustness of findings of the third MR assumption (i.e., horizontal pleiotropy)	Individual SNP effect estimates are ordered and weighted by the inverse of their variance. Providing the ZEMPA assumption is satisfied, the weighted mode generates a causal estimate using the mode of a smoothed empirical density function of the distribution of weighted SNP effect estimates	ZEMPA	Can generate unbiased causal estimates even when many SNPs in an instrument are invalid	Lower statistical power to detect causal effects than weighted median, under the condition of no invalid instruments; sensitive to bandwidth parameter
MR-CAUSE [63]	Examines invalidation and robustness of findings of the third MR assumption (i.e., horizontal pleiotropy)	Compares the expected log pointwise posterior density (i.e., estimate of how well the posterior distribution of a model is expected to predict a new set of data) under three models: a “sharing model” (i.e., permitting horizontal pleiotropy but no causal effect between traits), a “causal model” (i.e., permitting horizontal pleiotropy and assuming a causal effect), and a “null model” (i.e., neither a causal nor shared factor)	Assumes a single unobserved shared factor between two traits of interest	Can account for both correlated and non-correlated horizontal pleiotropy. Greater statistical power than MR-Egger and weighted mode when there is a true causal effect and no correlated horizontal pleiotropy	Inferior control of false positive rate in the presence of no causal effect and 0 to 50% of variants acting through a shared factor, as compared to MR-Egger and the weighted mode. Has somewhat lower statistical power than the weighted median approach when there is a true causal effect and no correlated horizontal pleiotropy
Multivariable MR [64]	Examines invalidation and robustness of findings of the third MR assumption (i.e., horizontal pleiotropy)	Performs a weighted generalized linear regression with adjustment for measured horizontal pleiotropy between instruments and outcomes	Requires that there are at least as many genetic instruments available as there are exposures	Can adjust estimates for the presence of measured horizontal pleiotropy	There can still be horizontal pleiotropy through variants having effects on unmeasured outcomes that are independent to the exposure of interest
Outlier detection tests (e.g., MR-PRESSO [65], Radial MR [66])	Remove or down-weight genetic variants that are outliers in an MR analysis	Differing methods	Perform better when a large proportion of variants are not horizontally pleiotropic	Explicitly remove or down-weight contributions of outliers that may be indicative of IV assumption violations; can improve statistical efficiency of models	Residual directional pleiotropy can remain after removing or down-weighting outlying SNPs; methods are underpowered when few SNPs are available; interpretation of an “outlying” variant may be ambiguous when there are few SNPs available in a multi-SNP instrument
Colocalization	Examines whether an association of a SNP with two or more traits represents both traits sharing a single causal variant or distinct causal variants in linkage disequilibrium	Differing methods	Some methods assume at most a single causal variant within the region for two traits examined	Can rule out findings being driven by two traits having distinct causal variants in high linkage disequilibrium	Can be underpowered for disease outcomes as compared to molecular traits
Steiger filtering/reverse direction MR [67]	Examines whether the association of a variant with two traits (e.g., A and B) represents a proximal effect of the variant on trait A which then influences trait B or vice versa. Reverse direction MR attempts to understand direction of effect between two traits	Steiger filtering compares the proportion of the variance explained in the exposure and outcome by SNPs used as instruments to help establish directionality between associations	No horizontal pleiotropy	Can help to elucidate the direction of association between two traits	Steiger filtering is sensitive to differences in measurement error and sample size across traits examined

InSIDE INstrument Strength Independent of Direct Effect, NOME NO Measurement Error, ZEMPA ZEro Modal Pleiotropy Assumption