Two-pronged Strategy for Using DIC to Compare Selection Models with Non-Ignorable Missing Responses

Data with missing responses generated by a non-ignorable missing- ness 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 di-cult because the assumptions about the missingness model are never veriflable from the data at hand. For complete data, the Deviance Information Criterion (DIC) is routinely used for Bayesian model comparison. However, when an anal- ysis includes missing data, DIC can be constructed in difierent 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 difierent constructions of the DIC. A DIC based on the observed data likeli- hood is used to compare joint models with difierent models of interest but the same model of missingness, and a comparison of models with the same model of interest but difierent 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 difierent assumptions about the two parts of the model, and is illustrated by examples with simulated missingness and an appli- cation 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.

[1]  Sylvia Richardson,et al.  Improving ecological inference using individual‐level data , 2006, Statistics in medicine.

[2]  L. Joseph.  4. Bayesian data analysis (2nd edn). Andrew Gelman, John B. Carlin, Hal S. Stern and Donald B. Rubin (eds), Chapman & Hall/CRC, Boca Raton, 2003. No. of pages: xxv + 668. Price: $59.95. ISBN 1‐58488‐388‐X , 2004 .

[3]  Geert Molenberghs,et al.  Selection models and pattern‐mixture models to analyse longitudinal quality of life data subject to drop‐out , 2002, Statistics in medicine.

[4]  Michael G. Kenward,et al.  Using Hierarchical Likelihood for Missing Data Problems , 2007 .

[5]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[6]  Jon Wakefield,et al.  Ecological inference for 2 × 2 tables , 2004 .

[7]  Garrett M. Fitzmaurice,et al.  Methods for Handling Dropouts in Longitudinal Clinical Trials , 2003 .

[8]  M. Plummer.  Penalized loss functions for Bayesian model comparison. , 2008, Biostatistics.

[9]  A. Rotnitzky,et al.  Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis by DANIELS, M. J. and HOGAN, J. W , 2009 .

[10]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[11]  D. Rubin,et al.  On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood , 1983 .

[12]  F. Vaida,et al.  Conditional Akaike information for mixed-effects models , 2005 .

[13]  Arthur P. Dempster,et al.  The direct use of likelihood for significance testing , 1997, Stat. Comput..

[14]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[15]  Mario Peruggia,et al.  On the variability of case-deletion importance sampling weights in the Bayesian linear model , 1997 .

[16]  M. Kenward,et al.  Informative Drop‐Out in Longitudinal Data Analysis , 1994 .

[17]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[18]  C. Robert,et al.  Deviance information criteria for missing data models , 2006 .

[19]  G Molenberghs,et al.  Parametric models for incomplete continuous and categorical longitudinal data , 1999, Statistical methods in medical research.

[20]  A. Gelman.  Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper) , 2004 .

[21]  Jerry Nedelman,et al.  Book review: “Bayesian Data Analysis,” Second Edition by A. Gelman, J.B. Carlin, H.S. Stern, and D.B. Rubin Chapman & Hall/CRC, 2004 , 2005, Comput. Stat..

[22]  J. Schafer,et al.  Missing data: our view of the state of the art. , 2002, Psychological methods.