Statistical process monitoring and hierarchical Bayesian modeling are two ways to learn more about processes of interest. In this work, we consider two main components: risk-adjusted monitoring and Bayesian hierarchical models for spatial data. Usually, if prior information about a process is known, it is important to incorporate this into the monitoring scheme. For example, when monitoring 30-day mortality rates after surgery, the pre-operative risk of patients based on health characteristics is often an indicator of how likely the surgery is to succeed. In these cases, risk-adjusted monitoring techniques are used. In this work, the practical limitations of the traditional implementation of risk-adjusted monitoring methods are discussed and an improved implementation is proposed. A method to perform spatial risk-adjustment based on exact locations of concurrent observations to account for spatial dependence is also described. Furthermore, the development of objective priors for fully Bayesian hierarchical models for areal data is explored for Gaussian responses. Collectively, these statistical methods serve as analytic tools to better monitor and model spatial processes. / Ph. D. / The purpose of this research was to advance understanding of help-seeking behaviors of lowincome older adults who were deemed ineligible to receive state-funded assistance. I used health services data from two independent state agencies to assess factors associated with service use and health status; follow-up interviews were conducted to explore self-management strategies of rural older adults with unmet needs. Older adults who did not receive help were at increased risk for hospitalization and mortality compared to individuals who received helped. Rural older adults were significantly more likely to not receive help and were at increased risk for mortality, placing them in a vulnerable position. Interviews with rural-dwelling older adults that were not receiving help highlighted the challenges associated with living with unmet needs but demonstrated resilience through their use of physical and psychological coping mechanisms to navigate daily challenges and maintain health and well-being. They had to deal with numerous difficulties performing instrumental activities of daily living (IADL); mobility was an underlying problem that led to subsequent IADL limitations, such as difficulty with household chores and meal preparation. Policymakers need to advocate for services that allow older adults to address preemptively their care needs before they become unmanageable. Ensuring the availability of services for near-risk older adults who are proactive in addressing their functional care needs would benefit individuals and caregivers on whom they rely. Such services not only support older adults’ health, functioning, and well-being but may be cost-effective for public programs. Policies should reduce unmet needs among older adults by increasing service access in rural communities because even if services exist, they may not be available to this near-risk population of older adults.Many current scientific applications involve data collection that has some type of spatial component. Within these applications, the objective could be to monitor incoming data in order to quickly detect any changes in real time. Another objective could be to use statistical models to characterize and understand the underlying features of the data. In this work, we develop statistical methodology to monitor and model data that include a spatial component. Specifically, we develop a monitoring scheme that adjusts for spatial risk and present an objective way to quantify and model spatial dependence when data is measured for areal units. Collectively, the statistical methods developed in this work serve as analytic tools to better monitor and model spatial data.

This dissertation consists of four independent but related parts, each in a Chapter. The first part is an introductory. It serves as the background introduction and offer preparations for later parts. The second part discusses two population multivariate normal distributions with common covariance matrix. The goal for this part is to derive objective/non-informative priors for the parameterizations and use these priors to build up constructive random posteriors of the Kullback-Liebler (KL) divergence of the two multivariate normal populations, which is proportional to the distance between the two means, weighted by the common precision matrix. We use the Cholesky decomposition for re-parameterization of the precision matrix. The KL divergence is a true distance measurement for divergence between the two multivariate normal populations with common covariance matrix. Frequentist properties of the Bayesian procedure using these objective priors are studied through analytical and numerical tools. The third part considers the star-shape Gaussian graphical model, which is a special case of undirected Gaussian graphical models. It is a multivariate normal distribution where the variables are grouped into one "global" group of variable set and several "local" groups of variable set. When conditioned on the global variable set, the local variable sets are independent of each other. We adopt the Cholesky decomposition for re-parametrization of precision matrix and derive Jeffreys' prior, reference prior, and invariant priors for new parameterizations. The frequentist properties of the Bayesian procedure using these objective priors are also studied. The last part concentrates on the discussion of objective Bayesian analysis for partial correlation coefficient and its application to multivariate Gaussian models. / Ph. D.