Insights into the use of Bayesian models for informative missing data

Many studies are affected by missing data, which complicates subsequent analyses for re-searchers. Here, we are concerned with missing outcomes generated by a missingness mechanism that is informative. In this case, ad hoc approaches are not suitable and if we wish to adequately model this type of missing data, we need to use ‘statistically principled ’ methods. We investigate one of these methods, Bayesian full probability modelling, in which a joint model consisting of a model of interest and a model for the informative missing data mechanism is specified. Using simulated data, we explore the performance of Bayesian methods, finding that the addition of a model of missingness generally improves the overall fit of the model of interest leading to better prediction, but that the estimates of parameters of interest can be adversely affected by skewness in the response variable. The effective number of parameters, pD, is a measure of the ratio of the information in the likelihood to that of the posterior. We consider the use of the scaled pD of the model of missingness as a diagnostic that indicates the amount of informativeness in the missing data given our assumptions. We find that it is useful for indicating how far our missing data departs from missing at random, but that it should not be used for choosing the ‘best ’ model of missingness. These points are illustrated with two real examples, which analyse test score data from the 1958 British birth cohort study and data from a clinical trial.