The IVW estimate is an efficient analysis method when all genetic variants are valid IVs. Unfortunately, it will be biased even if only one genetic variant is invalid. For this reason, the IVW estimate can be said to have a 0% breakdown level. However, an estimator exists that enjoys a 50% breakdown level; that is, it provides a consistent estimate of the causal effect when up to (but not including) 50% of genetic variants are invalid. This simple estimator is the median ratio estimate (Han, 2008). Specifically, let β^j denote the jth ordered ratio estimate (arranged from smallest to largest). If the total number of genetic variants is odd (J=2k+1), the simple median estimator is the middle ratio estimate β^k+1. If it is even (J=2k), the median is interpolated between the two middle estimates 12(β^k+β^k+1). In terms of notation, we assume in this manuscript that genetic variants are ordered according to their ratio estimates.
