Non-parametric spatial models for clustered ordered periodontal data

Clinical attachment level is regarded as the most popular measure to assess periodontal disease (PD). These probed tooth site level measures are usually rounded and recorded as whole numbers (in millimetres) producing clustered (site measures within a mouth) error prone ordinal responses representing some ordering of the underlying PD progression. In addition, it is hypothesized that PD progression can be spatially referenced, i.e. proximal tooth sites share similar PD status in comparison with sites that are distantly located. We develop a Bayesian multivariate probit framework for these ordinal responses where the cut point parameters linking the observed ordinal clinical attachment levels to the latent underlying disease process can be fixed in advance. The latent spatial association characterizing conditional independence under Gaussian graphs is introduced via a non-parametric Bayesian approach motivated by the probit stick breaking process, where the components of the stick breaking weights follow a multivariate Gaussian density with the precision matrix distributed as G-Wishart. This yields a computationally simple, yet robust and flexible, framework to capture the latent disease status leading to a natural clustering of tooth sites and subjects with similar PD status (beyond spatial clustering), and improved parameter estimation through sharing of information. Both simulation studies and application to a motivating PD data set reveal the advantages of considering this flexible non-parametric ordinal framework over other alternatives.