The simple median estimator is inefficient, especially when the precision of the individual estimates varies considerably. In order to account for this, a weighted median can be defined as follows. Let wj be the weight given to the jth ordered ratio estimate, and let sj=∑k=1jwk be the sum of weights up to and including the weight of the jth ordered ratio estimate. Weights are standardized, so that the sum of the weights sJ is 1. The weighted median estimator is the median of a distribution having estimate β^j as its pj=100(sj−wj2)th percentile. For all other percentile values, we extrapolate linearly between the neighboring ratio estimates. The contribution of the jth genetic variant to the empirical distribution is proportional to its weight wj. The simple median estimator can be thought of as a weighted median estimator with equal weights. Although the simple median provides a consistent estimate of causal effect if at least 50% of IVs are valid, the weighted median will provide a consistent estimate if at least 50% of the weight comes from valid IVs. We assume that no
