In Figure 2 (left side), although the invalid IVs do not contribute directly to the median estimate, they do influence it. The simple median estimate in this case is an average of the fifth and sixth ratio estimates. If the invalid IVs were not present, the median estimate would be the average of the third and fourth ratio estimates. The presence of invalid instruments does not affect the median estimate asymptotically, but in this example it will bias the estimate in finite samples, as the fifth and sixth ratio estimates will always be larger than the third and fourth ratio estimates. This is likely to be a problem when the estimates from invalid IVs are not balanced about the true causal effect (as in this example, all four invalid estimates are greater than the true causal effect).
