Table 1.

Comparison of different MR methods, including whether valid IV assumption A2 or A3 can be violated

Method	A2	A3	Key assumptions	Implementation challenges/performance
cML-MA	✓	✓	plurality valid	controlling type I errors with high power
MR-Mix12	✓	✓	plurality valid; ${\hat{\beta }}_{Yi}-\theta {\hat{\beta }}_{Xi}\sim$ a mixture of normals	biased to the null, thus conservative
MR-ContMix9	✓	✓	plurality valid; ${\hat{\theta }}_i\sim \left(1-q\right)N\left(\theta ,SE{\left({\hat{\theta }}_i\right)}^2\right)+qN\left(0,{\psi }^2+SE{\left({\hat{\theta }}_i\right)}^2\right)$; NOME	difficult to pre-choose a fixed value for tuning parameter ψ
CAUSE13	✓	✓	<50% IVs have correlated pleiotropy; ${\gamma }_i{\phi }_i=0$; ${\beta }_{XU}=1$; ${\beta }_{Xi}={\gamma }_i$ or ${\phi }_i$; ${\hat{\beta }}_{Yi}\sim \left(1-q\right)N\left(\theta {\beta }_{Xi}+r_i,{\sigma }^2\right)+qN\left(\left(\theta +{\beta }_{YU}\right){\beta }_{Xi}+r_i,{\sigma }^2\right)$	difficult to estimate some parameters depending on the hidden confounder U; sensitive to assumption of ${\gamma }_i{\phi }_i=0$
MR-Lasso14	✓	✓	plurality valid;7 some condition on the exposure-association strengths of invalid IVs relative to that of valid IVs to ensure consistency;15 NOME	depending on the heterogeneity criterion for choosing the tuning parameter for the Lasso penalty
MR-Weighted-Mode16	✓	✓	plurality valid	sensitive to the difficult bandwidth selection for mode estimation
MR-Weighted-Median3	✓	✓	majority valid	robust to outliers; low powered; sometimes biased
MR-PRESSO1	✓	x	majority valid; InSIDE; Good delete-1 causal estimates	inflated type I errors; unable to completely remove invalid IVs
MR-Egger17	✓	x	InSIDE: ${\phi }_i=0$; ${\alpha }_i\sim N\left(\mu ,{\tau }^2\right)$ for a small m (but no normality needed for a large m); NOME	often biased and low powered
MR-RAPS6	✓	x	InSIDE: ${\phi }_i=0$; ${\alpha }_i\sim N\left(0,{\tau }^2\right)$ if overdispersion is specified	may be sensitive to directional pleiotropy; robust to outliers with Tukey’s loss
MR-IVW (RE)18,19	✓	x	balanced pleiotropy; NOME	sensitive to directional pleiotropy; low powered
MR-IVW (FE)18,19	x	x	all IVs are valid; NOME	efficient when all IVs are valid; sensitive to invalid IVs

The notations are defined in Figure 1 and Equation 1, and q is the (unknown) proportion of invalid IVs while ${\hat{\theta }}_i={\hat{\beta }}_{Yi}/{\hat{\beta }}_{Xi}$ and $SE\left({\hat{\theta }}_i\right)$ are the Wald ratio estimate of θ based on SNP i and its standard error, respectively. NOME refers to no measurement error assumption: the variance of any IV-exposure association estimate is negligible.²⁰