In many longitudinal studies, several longitudinal processes may be associated. For example, a time-dependent covariate in a longitudinal model may be measured with errors or have missing data so it needs to be modeled together with the response process in order to address the measurement errors and missing data. In such cases, a joint inference is appealing since it can incorporate information of all processes simultaneously. The joint inference is not only more efficient than separate inferences but it may also avoid possible biases. In addition, longitudinal data often contain outliers, so robust methods for the joint models are necessary. In this talk, we discuss joint models for two correlated longitudinal processes with missing data, measurement errors, and outliers. We consider two-step methods and joint likelihood methods for joint inference, and propose robust methods based on M-estimators to address possible outliers for joint models. Simulation studies are conducted to evaluate the performances of the proposed methods, and a real AIDS dataset is analyzed using the proposed methods.
Given the simple setting of point outcome, treatment and confounding variables, all of which are binary, a double-robust estimator for the average causal effect the can be cast as arising from compromises between parametric and non-parametric outcome models. Inspired by this idea, a Bayesian model averaging estimator is introduced as a weighted average between a Bayesian saturated model and a parametric outcome model. However, this new Bayesian estimator cannot scale up well in presence of sparse data. We propose a Bayesian hierarchical framework to address this issue. Several estimation are investigated via simulation studies.