Search

Methods of multivariate analysis for continuous data have been applied extensively in epidemiology, economics, engineering, and other fields. However, multivariate models for count data have not been applied to a similar extent, and there is relatively little work published on models, especially when accounting for both spatial and temporal dependence. In the first part of the thesis, we propose a hierarchical multivariate Poisson (MVP) model that simultaneously models spatial and temporal correlation of observed data counts in a fairly general setting. In particular, MVP allows for modeling the spatially/temporally dependent counts as a function of location-specific and time-varying covariates. To characterize temporal trends, we propose a broken-line regression model within MVP and apply it to joinpoint detection. Bayesian inference is conducted using Markov chain Monte Carlo (MCMC) methods to approximate required posterior summaries. We use a backward selection algorithm coupled with the shaver method to obtain joinpoint coefficients via Bayesian Lasso. We apply the proposed model to spatial temporal pertussis incidence data from 2000 to 2015 from several midwestern states in the U.S. To evaluate the appropriateness of the model, in the second part of this thesis we develop a goodness-of-fit (GOF) statistic for fitting discrete generalized linear models (GLMs) based on the sum of standardized residuals (SSRs). This work is an extension of our earlier work (Chen L. et al. 2017) which proposed a GOF test for binary responses. We derive the asymptotic distribution of the test statistic and show how it can be applied to popular count regression models such as Poisson regression, Negative Binomial regression and Binomial regression using different link functions. Using numeric examples we show that the proposed test is substantially more powerful than most of the currently available GOF tests under various model misspecification scenarios applied to various discrete GLMs like logistic, Poisson, and negative binomial.