  Models for analyzing multivariate data sets with missing values require
strong, often unassessable, assumptions. The most common of these is that the
mechanism that created the missing data is ignorable - a twofold assumption
dependent on the mode of inference. The first part, which is the focus here,
under the Bayesian and direct-likelihood paradigms, requires that the missing
data are missing at random; in contrast, the frequentist-likelihood paradigm
demands that the missing data mechanism always produces missing at random data,
a condition known as missing always at random. Under certain regularity
conditions, assuming missing always at random leads to an assumption that can
be tested using the observed data alone namely, the missing data indicators
only depend on fully observed variables. Here, we propose three different
diagnostic tests that not only indicate when this assumption is incorrect but
also suggest which variables are the most likely culprits. Although missing
always at random is not a necessary condition to ensure validity under the
Bayesian and direct-likelihood paradigms, it is sufficient, and evidence for
its violation should encourage the careful statistician to conduct targeted
sensitivity analyses.
