  The National Health and Nutrition Examination Survey (NHANES) is a major
program of the National Center for Health Statistics, designed to assess the
health and nutritional status of adults and children in the United States. The
analysis of NHANES dental caries data faces several challenges, including (1)
the data were collected using a complex, multistage, stratified,
unequal-probability sampling design; (2) the sample size of some primary
sampling units (PSU), e.g., counties, is very small; (3) the measures of dental
caries have complicated structure and correlation, and (4) there is a
substantial percentage of nonresponses, for which the missing data are expected
to be not missing at random or non-ignorable. We propose a Bayesian
hierarchical spatial model to address these analysis challenges. We develop a
two-level Potts model that closely resembles the caries evolution process and
captures complicated spatial correlations between teeth and surfaces of the
teeth. By adding Bayesian hierarchies to the Potts model, we account for the
multistage survey sampling design and also enable information borrowing across
PSUs for small area estimation. We incorporate sampling weights by including
them as a covariate in the model and adopt flexible B-splines to achieve robust
inference. We account for non-ignorable missing outcomes and covariates using
the selection model. We use data augmentation coupled with the noisy exchange
sampler to obtain the posterior of model parameters that involve
doubly-intractable normalizing constants. Our analysis results show strong
spatial associations between teeth and tooth surfaces and that dental hygienic
factors, fluorosis and sealant reduce the risks of having dental diseases.
