  Data analyses typically rely upon assumptions about missingness mechanisms
that lead to observed versus missing data. When the data are missing not at
random, direct assumptions about the missingness mechanism, and indirect
assumptions about the distributions of observed and missing data, are typically
untestable. We explore an approach, where the joint distribution of observed
data and missing data is specified through non-standard conditional
distributions. In this formulation, which traces back to a factorization of the
joint distribution, apparently proposed by J.W. Tukey, the modeling assumptions
about the conditional factors are either testable or are designed to allow the
incorporation of substantive knowledge about the problem at hand, thereby
offering a possibly realistic portrayal of the data, both missing and observed.
We apply Tukey's conditional representation to exponential family models, and
we propose a computationally tractable inferential strategy for this class of
models. We illustrate the utility of this approach using high-throughput
biological data with missing data that are not missing at random.
