Neighbor relation modeling and variance component identification in hierarchical areal data models with application to periodontology, cancer epidemiology, and sports

Attachment loss, the extent of a tooth's root (in millimeters) that is no longer attached to surrounding bone by periodontal ligament, is often used to measure the current state of a patient's periodontal disease and monitor disease progression. Attachment loss data can be analyzed using a conditionally autoregressive (CAR) prior distribution, which smooths fitted values toward neighboring values. However, in this setting it is desirable to have more than one class of neighbor relation in the spatial structure, so the different classes of neighbor relations can induce different degrees of smoothing. For example, we may wish to allow smoothing of neighbor pairs bridging the gap between teeth to be different from smoothing of pairs that do not bridge such gaps. This dissertation develops a two-neighbor-relation CAR (2NRCAR) model to handle this situation.;The 2NRCAR model can also be applied to non-periodontal data. For example, basketball coaches at all levels use shot charts to study shot locations and outcomes for their own teams as well as upcoming opponents. This dissertation develops hierarchical spatial models for shot-chart data, which allow for spatially-varying effects of covariates. The 2NRCAR spatial model permits differential smoothing of fitted surfaces in two spatial directions, which naturally correspond to polar coordinates: distance to the basket and angle from the line connecting the two baskets. We illustrate our approach using the 2003--2004 shot chart data for Minnesota Timberwolves guard Sam Cassell.;Our analyses using the 2NRCAR model were initially hampered by computational problems caused by weakly-identified variance parameters. This dissertation introduces the class of two-variance hierarchical linear models and characterizes the aspects of these models that lead to well-identified or poorly-identified variances. These ideas are illustrated with the periodontal data set and examined in some generality for specific two-variance models including the CAR, one-way random effects, and multiple membership models. We also connect this theory with other constrained regression methods and suggest a diagnostic that can be used to search for missing spatially-varying fixed effects in the CAR model.;Spatial models often include fixed effects to measure the effect of spatially-varying covariates and CAR random effects to account for spatial clustering of the outcomes. Based on our analysis of variance identification, diagnostics are proposed to measure the collinearity between the fixed effect covariates and the CAR random effects and to measure each region's influence on the change in the fixed effect's estimates due to the addition of the CAR random effects. A new model that alleviates the collinearity between the fixed-effect covariates and the CAR random effects is developed and extensions of these methods to point-referenced data models are discussed.