Table 2.

Summary of select sensitivity analyses for exploring bias due to horizontal pleiotropy in Mendelian randomization (MR)

Name	Brief description	MR assumptions	Other issues
For one-sample MR
sisVIVE [67]	It is an extension to two-stage least squares, which incorporates LASSO penalization	(1) relevance; (2) independence; (3) the alternative to exclusion restriction: at least 50% of proposed IVs are valid; (4) monotonicitya	It works for continuous outcomes only, is computationally intensive, and the current implementation do not provide 95% CIs
MR-GENIUS [68]	It is a version of G-estimation which is robust to time-varying SNP-exposure associations, unmeasured confounding and violation of IV assumptions	(1) the alternative to relevance: proposed genetic IVs should strongly affect the variance rather than the mean of the exposure; both (2) independence and (3) exclusion restriction can be relaxed; (4) the alternative to homogeneityb: no additive interaction with unmeasured selection	Estimates on binary exposures have ambiguous units. Proposed genetic IVs often explain a small variance of the exposure
MR-MiSTERI [69]	It is another version of G-estimation for estimating the causal effect among compliersa, which is robust to time-varying SNP-exposure associations, unmeasured confounding and violation of IV assumptions	(1) relevance, (2) independence, and (3) exclusion restriction all can be relaxed; (4) the alternative to monotonicitya: exposure-outcome effect does not vary with proposed invalid IV on additive scale; selection bias due to confounding does not vary with proposed invalid IV on multiplicative scale; residual variance for outcome is heteroscedastic and thus varies with proposed invalid IV	Its R package can only be used for continuous exposure and outcome at present
For two-sample MR
MR-Egger [70]	It allows a non-zero intercept to test unbalanced horizontal pleiotropy	(1) relevance; (2) independence; (3) the alternative to exclusion restriction: InSIDEc; (4) homogeneityb	It is sensible to outliers and tends to suffer from low statistical power
Weighted median [71]	It is defined as the median of a weighted empirical density function of the Wald ratio estimates	(1) relevance; (2) independence; (3) the alternative to exclusion restriction: at least 50% of weight comes from valid IVs; (4) homogeneityb	Nil
Weighted mode [72]	It calculates the weighted mode of the Wald ratio estimates	(1) relevance; (2) independence; (3) the alternative to exclusion restriction: zero modal pleiotropy assumption d; (4) homogeneitya or monotonicity b	Researchers need to choose a bandwidth to obtain the clustering effect, and different bandwidths might provide inconsistent estimates [10]
MR-PRESSO [73]	It assesses horizontal pleiotropy based on the contribution of each SNP to heterogeneity and provides adjusted MR estimates by removing outlier SNPs	(1) relevance; (2) independence; (3) the alternative to exclusion restriction: InSIDEc and outliers (identified via MR-PRESSO global test) are due to potential horizontal pleiotropy; (4) homogeneitya or monotonicityb	After removing outlier SNPs, the standard errors would decrease. Therefore, it would be more likely to reject the null
MR-TRYX [74]	It assesses horizontal pleiotropy based on the contribution of each SNP to heterogeneity and attempts to adjust for their horizontal pleiotropic effects using extra publicly available GWAS from MR-Base	(1) relevance; (2) independence; (3) the alternative to exclusion restriction: outliers (identified via RadialMR [75]) are due to potential horizontal pleiotropy; (4) homogeneitya or monotonicityb	GWAS from MR-Base may not cover the whole genome or conducted in the target population (e.g. only female participants)

ACE average causal effect, CI confidence interval, GWAS genome-wide association studies, InSIDE instrument strength independent of direct effect, IV instrumental variable, SNPs single nucleotide polymorphisms

^aMonotonicity means proposed IV cannot increase exposure level in some participants while decrease it in others. Thus, MR quantifies the magnitude of ACE among the unknown subpopulation of compliers who are monotonically affected by the proposed IV (i.e. local ACE)[2, 4]

^bHomogeneity means the exposure-outcome effect is homogeneous across population and does not depend on proposed IV. Thus, MR quantifies the magnitude of ACE for the reference population (i.e. global ACE)[2, 4]

^cThe strength of genetic IV – exposure association should not be correlated with the strength of the pleiotropic effects across proposed IVs[70]

^dThe majority of SNPs could be invalid providing that the set of SNPs which form the largest homogeneous cluster are valid[72]