  Mendelian randomization uses genetic variants to make causal inferences about
a modifiable exposure. Subject to a genetic variant satisfying the instrumental
variable assumptions, an association between the variant and outcome implies a
causal effect of the exposure on the outcome. Complications arise with a binary
exposure that is a dichotomization of a continuous risk factor (for example,
hypertension is a dichotomization of blood pressure). This can lead to
violation of the exclusion restriction assumption: the genetic variant can
influence the outcome via the continuous risk factor even if the binary
exposure does not change. Provided the instrumental variable assumptions are
satisfied for the underlying continuous risk factor, causal inferences for the
binary exposure are valid for the continuous risk factor. Causal estimates for
the binary exposure assume the causal effect is a stepwise function at the
point of dichotomization. Even then, estimation requires further parametric
assumptions. Under monotonicity, the causal estimate represents the average
causal effect in `compliers', individuals for whom the binary exposure would be
present if they have the genetic variant and absent otherwise. Unlike in
randomized trials, genetic compliers are unlikely to be a large or
representative subgroup of the population. Under homogeneity, the causal effect
of the exposure on the outcome is assumed constant in all individuals; often an
unrealistic assumption. We here provide methods for causal estimation with a
binary exposure (although subject to all the above caveats). Mendelian
randomization investigations with a dichotomized binary exposure should be
conceptualized in terms of an underlying continuous variable.
