Bayesian informative dropout model for longitudinal binary data with random effects using conditional and joint modeling approaches.

Dropouts are common in longitudinal study. If the dropout probability depends on the missing observations at or after dropout, this type of dropout is called informative (or nonignorable) dropout (ID). Failure to accommodate such dropout mechanism into the model will bias the parameter estimates. We propose a conditional autoregressive model for longitudinal binary data with an ID model such that the probabilities of positive outcomes as well as the drop-out indicator in each occasion are logit linear in some covariates and outcomes. This model adopting a marginal model for outcomes and a conditional model for dropouts is called a selection model. To allow for the heterogeneity and clustering effects, the outcome model is extended to incorporate mixture and random effects. Lastly, the model is further extended to a novel model that models the outcome and dropout jointly such that their dependency is formulated through an odds ratio function. Parameters are estimated by a Bayesian approach implemented using the user-friendly Bayesian software WinBUGS. A methadone clinic dataset is analyzed to illustrate the proposed models. Result shows that the treatment time effect is still significant but weaker after allowing for an ID process in the data. Finally the effect of drop-out on parameter estimates is evaluated through simulation studies.