A longitudinal Bayesian mixed effects model with hurdle Conway‐Maxwell‐Poisson distribution

Abstract

Dental caries (i.e., cavities) is one of the most common chronic childhood diseases and may continue to progress throughout a person's lifetime. The Iowa Fluoride Study (IFS) was designed to investigate the effects of various fluoride, dietary and nondietary factors on the progression of dental caries among a cohort of Iowa school children. We develop a mixed effects model to perform a comprehensive analysis of the longitudinal clustered data of IFS at ages 5, 9, 13, and 17. We combine a Bayesian hurdle framework with the Conway‐Maxwell‐Poisson regression model, which can account for both excessive zeros and various levels of dispersion. A hierarchical shrinkage prior distribution is used to share the temporal information for predictors in the fixed‐effects model. The dependence among teeth of each individual child is modeled through a sparse covariance structure of the random effects across time. Moreover, we obtain the parameter estimates and credible intervals from a Gibbs sampler. Simulation studies are conducted to assess the accuracy and effectiveness of our statistical methodology. The results of this article provide novel tools to statistical practitioners and offer fresh insights to dental researchers on effects of various risk and protective factors on caries progression.

Analyzing dental fluorosis data using a novel Bayesian model for clustered longitudinal ordinal outcomes with an inflated category

“…This choice is intuitively reasonable as we expect that if the latent factors increase the probability of fluorosis, they should also increase the severity of fluorosis. A similar choice has made in the context of other hurdle model, 16,20 sometimes allowing different scalings of the random effects in the different model components to account for differences in link function. As both models (1) and ( 2) are based on a logit link function, different scaling coefficients are not needed.…”

“…While we wish to consider the differences in the regression coefficient vectors $$$ {\boldsymbol{\alpha}}_{0t} $$$ and $$$ {\boldsymbol{\alpha}}_{1t} $$$ across time, we typically believe that there will be similarities in these coefficients across time. To that end, we aim to flexibly share information about the coefficients across time through hierarchical shrinkage, as in Reference 16. We let $$$ {\alpha}_{0 ut} $$$ and $$$ {\alpha}_{1 ut} $$$ be the $$$ u $$$‐th elements from $$$ {\boldsymbol{\alpha}}_{0t} $$$ and $$$ {\boldsymbol{\alpha}}_{1t} $$$, respectively.…”

“…The design vector $$$ {\mathbf{w}}_{1, itjk} $$$ specifies the form of the temporal dependence, and we focus on the case where $$$ {\boldsymbol{\beta}}_i $$$ is a $$$ T $$$‐dimensional vector such that $$$ {\beta}_{it}={\mathbf{w}}_{1, itjk}^{\prime }{\boldsymbol{\beta}}_i $$$ provides the temporal random effect at time $$$ t $$$. We let $$$ {\boldsymbol{\beta}}_i\sim {N}_T\left({\mathbf{0}}_T,{\boldsymbol{\Sigma}}_{\beta}\right) $$$, and one may wish to seek structure on the temporal random effects covariance $$$ {\boldsymbol{\Sigma}}_{\beta } $$$ either by imposing a parametric structure (such as equi‐ or auto‐correlation) or by using prior distributions that pull $$$ {\boldsymbol{\Sigma}}_{\beta } $$$ toward more structured choices (as was considered in Reference 16). Here, we are typically interested in cases where $$$ T $$$ is small ($$$ T=3 $$$ in the IFS example), so there is little practical gain in more complex priors.…”

“…The goal of the IFS was to study the associations between various dietary and non-dietary factors and two outcomes of interest: dental fluorosis (spots on teeth) and dental caries (cavities) for a cohort of Iowa school children. Previous studies have focused on the dental caries outcome, [14][15][16][17] and here, we focus on methodology for the fluorosis outcome. The fluorisis data are a longitudinal and multilevel clustered ordinal outcome.…”