Table 5.

Summary of methods considered in this paper

Method	Breakdown	IV2	IV3	Comments
Two‐stage least squares	0%	✗	✗	Requires individual‐level data. Biased when at least one genetic variant is an invalid IV.
Inverse‐variance weighted (IVW)	0%	✗	✗	Equivalent to two‐stage least squares method with summary data. Also biased when at least one genetic variant is an invalid IV.
Simple median	50%	✓	✓	Consistent when 50% of genetic variants are valid IVs. Inefficient compared with IVW and weighted median methods.
Weighted median	50%	✓	✓	Consistent when 50% of weight contributed by genetic variants is valid. Efficiency is similar to that of IVW method.
Penalized weighted median	50%	✓	✓	Equivalent to weighted median when there is no causal effect heterogeneity. Downweights the contribution of heterogeneous variants, so may have better finite sample properties, particularly if there is directional pleiotropy.
MR‐Egger regression	100%	✗	✓	Consistent when 100% of genetic variants are invalid, but requires variants to satisfy a weaker assumption (the InSIDE assumption). This assumption is not automatically violated by an association between a genetic variant and a confounder, but it would be violated if several variants were associated with the same confounder. Substantially less efficient than IVW and median‐based methods, and more susceptible to weak instrument bias in a one‐sample setting.

Breakdown refers to the breakdown level, the proportion of information that can come from invalid instrumental variables (IVs) before the method gives biased estimates. IV2 and IV3 refer to whether violations of the second (no association with confounders) and third (no direct effect on the outcome) instrumental variable assumptions are allowed (✓) or not allowed (✗).