Bayesian inference for generalized linear mixed models with predictors subject to detection limits: an approach that leverages information from auxiliary variables.

This paper is motivated from a retrospective study of the impact of vitamin D deficiency on the clinical outcomes for critically ill patients in multi-center critical care units. The primary predictors of interest, vitamin D2 and D3 levels, are censored at a known detection limit. Within the context of generalized linear mixed models, we investigate statistical methods to handle multiple censored predictors in the presence of auxiliary variables. A Bayesian joint modeling approach is proposed to fit the complex heterogeneous multi-center data, in which the data information is fully used to estimate parameters of interest. Efficient Monte Carlo Markov chain algorithms are specifically developed depending on the nature of the response. Simulation studies demonstrate the outperformance of the proposed Bayesian approach over other existing methods. An application to the data set from the vitamin D deficiency study is presented. Possible extensions of the method regarding the absence of auxiliary variables, semiparametric models, as well as the type of censoring are also discussed.