  We develop a fully non-parametric, easy-to-use, and powerful test for the
missing completely at random (MCAR) assumption on the missingness mechanism of
a dataset. The test compares distributions of different missing patterns on
random projections in the variable space of the data. The distributional
differences are measured with the Kullback-Leibler Divergence, using
probability Random Forests. We thus refer to it as "Projected Kullback-Leibler
MCAR" (PKLM) test. The use of random projections makes it applicable even if
very few or no fully observed observations are available or if the number of
dimensions is large. An efficient permutation approach guarantees the level for
any finite sample size, resolving a major shortcoming of most other available
tests. Moreover, the test can be used on both discrete and continuous data. We
show empirically on a range of simulated data distributions and real datasets
that our test has consistently high power and is able to avoid inflated type-I
errors. Finally, we provide an R-package PKLMtest with an implementation of our
test.
