Paradoxes in instrumental variable studies with missing data and one-sided noncompliance

Edward H. Kennedy, Dylan S. Small

It is common in instrumental variable studies for instrument values to be missing, for example when the instrument is a genetic test in Mendelian randomization studies. In this paper we discuss two apparent paradoxes that arise in so-called single consent designs where there is one-sided noncompliance, i.e., where unencouraged units cannot access treatment. The first paradox is that, even under a missing completely at random assumption, a complete-case analysis is biased when knowledge of one-sided noncompliance is taken into account; this is not the case when such information is disregarded. This is reminiscent of the surprising fact that estimating known propensity scores can improve efficiency of inverse-probability-weighted estimators; in our setting, discarding compliance information can avoid bias, rather than improve efficiency. The second paradox is that, although incorporating information about one-sided noncompliance does not lead to efficiency gains without missing data, the story is different when instrument values are missing: there, incorporating such information changes the efficiency bound, allowing possible efficiency gains.