A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data.

Pattern mixture modeling is a popular approach for handling incomplete longitudinal data. Such models are not identifiable by construction. Identifying restrictions is one approach to mixture model identification (Little, 1995, Journal of the American Statistical Association 90, 1112-1121; Little and Wang, 1996, Biometrics 52, 98-111; Thijs et al., 2002, Biostatistics 3, 245-265; Kenward, Molenberghs, and Thijs, 2003, Biometrika 90, 53-71; Daniels and Hogan, 2008, in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis) and is a natural starting point for missing not at random sensitivity analysis (Thijs et al., 2002, Biostatistics 3, 245-265; Daniels and Hogan, 2008, in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis). However, when the pattern specific models are multivariate normal, identifying restrictions corresponding to missing at random (MAR) may not exist. Furthermore, identification strategies can be problematic in models with covariates (e.g., baseline covariates with time-invariant coefficients). In this article, we explore conditions necessary for identifying restrictions that result in MAR to exist under a multivariate normality assumption and strategies for identifying sensitivity parameters for sensitivity analysis or for a fully Bayesian analysis with informative priors. In addition, we propose alternative modeling and sensitivity analysis strategies under a less restrictive assumption for the distribution of the observed response data. We adopt the deviance information criterion for model comparison and perform a simulation study to evaluate the performances of the different modeling approaches. We also apply the methods to a longitudinal clinical trial. Problems caused by baseline covariates with time-invariant coefficients are investigated and an alternative identifying restriction based on residuals is proposed as a solution.