Using DIC to compare selection models with non-ignorable missing responses

Data with missing responses generated by a non-ignorable missingness mechanism can be analysed
by jointly modelling the response and a binary variable indicating whether the response is
observed or missing. Using a selection model factorisation, the resulting joint model consists of
a model of interest and a model of missingness. In the case of non-ignorable missingness, model
choice is difficult because the assumptions about the missingness model are never verifiable from
the data at hand. For complete data, the Deviance Information Criterion (DIC) is routinely used
for Bayesian model comparison. However, when an analysis includes missing data, DIC can be
constructed in different ways and its use and interpretation are not straightforward. In this paper,
we present a strategy for comparing selection models by combining information from two measures
taken from different constructions of the DIC. A DIC based on the observed data likelihood is used
to compare joint models with different models of interest but the same model of missingness, and a
comparison of models with the same model of interest but different models of missingness is carried
out using the model of missingness part of a conditional DIC. This strategy is intended for use
within a sensitivity analysis that explores the impact of different assumptions about the two parts
of the model, and is illustrated by examples with simulated missingness and an application which
compares three treatments for depression using data from a clinical trial. We also examine issues
relating to the calculation of the DIC based on the observed data likelihood.