  Many real datasets contain values missing not at random (MNAR). In this
scenario, investigators often perform list-wise deletion, or delete samples
with any missing values, before applying causal discovery algorithms. List-wise
deletion is a sound and general strategy when paired with algorithms such as
FCI and RFCI, but the deletion procedure also eliminates otherwise good samples
that contain only a few missing values. In this report, we show that we can
more efficiently utilize the observed values with test-wise deletion while
still maintaining algorithmic soundness. Here, test-wise deletion refers to the
process of list-wise deleting samples only among the variables required for
each conditional independence (CI) test used in constraint-based searches.
Test-wise deletion therefore often saves more samples than list-wise deletion
for each CI test, especially when we have a sparse underlying graph. Our
theoretical results show that test-wise deletion is sound under the justifiable
assumption that none of the missingness mechanisms causally affect each other
in the underlying causal graph. We also find that FCI and RFCI with test-wise
deletion outperform their list-wise deletion and imputation counterparts on
average when MNAR holds in both synthetic and real data.
