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It is common in instrumental variable studies for instrument values to bemissing, for example when the instrument is a genetic test in Mendelianrandomization studies. In this paper we discuss two apparent paradoxes thatarise in so-called single consent designs where there is one-sidednoncompliance, i.e., where unencouraged units cannot access treatment. Thefirst paradox is that, even under a missing completely at random assumption, acomplete-case analysis is biased when knowledge of one-sided noncompliance istaken into account; this is not the case when such information is disregarded.This occurs because incorporating information about one-sided noncomplianceinduces a dependence between the missingness and treatment. The second paradoxis that, although incorporating such information does not lead to efficiencygains without missing data, the story is different when instrument values aremissing: there, incorporating such information changes the efficiency bound,allowing possible efficiency gains. This is because some of the missing valuescan be filled in, based on the fact that anyone who received treatment musthave been encouraged by the instrument (since the unencouraged cannot accesstreatment).

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