  Causal discovery algorithms estimate causal graphs from observational data.
This can provide a valuable complement to analyses focussing on the causal
relation between individual treatment-outcome pairs. Constraint-based causal
discovery algorithms rely on conditional independence testing when building the
graph. Until recently, these algorithms have been unable to handle missing
values. In this paper, we investigate two alternative solutions: Test-wise
deletion and multiple imputation. We establish necessary and sufficient
conditions for the recoverability of causal structures under test-wise
deletion, and argue that multiple imputation is more challenging in the context
of causal discovery than for estimation. We conduct an extensive comparison by
simulating from benchmark causal graphs: As one might expect, we find that
test-wise deletion and multiple imputation both clearly outperform list-wise
deletion and single imputation. Crucially, our results further suggest that
multiple imputation is especially useful in settings with a small number of
either Gaussian or discrete variables, but when the dataset contains a mix of
both neither method is uniformly best. The methods we compare include random
forest imputation and a hybrid procedure combining test-wise deletion and
multiple imputation. An application to data from the IDEFICS cohort study on
diet- and lifestyle-related diseases in European children serves as an
illustrating example.
