In a Bayesian framework, prior distributions on a space of nonparametric continuous distributions may be defined using Polya trees. This dissertation addresses statistical problems for which the Polya tree idea can be utilized to provide efficient and practical methodological solutions. One problem considered is the estimation of risks, odds ratios, or other similar measures that are derived by specifying a threshold for an observed continuous variable. It has been previously shown that fitting a linear model to the continuous outcome under the assumption of a logistic error distribution leads to more efficient odds ratio estimates. We will show that deviations from the assumption of logistic error can result in great bias in odds ratio estimates. A one-step approximation to the Savage-Dickey ratio will be presented as a Bayesian test for distributional assumptions in the traditional logistic regression model. The approximation utilizes least-squares estimates in the place of a full Bayesian Markov Chain simulation, and the equivalence of inferences based on the two implementations will be shown. A framework for flexible, semiparametric estimation of risks in the case that the assumption of logistic error is rejected will be proposed. A second application deals with regression scenarios in which residuals are correlated and their distribution evolves over an ordinal covariate such as time. In the context of prediction, such complex error distributions need to be modeled carefully and flexibly. The proposed model introduces dependent, but separate Polya tree priors for each time point, thus pooling information across time points to model gradual changes in distributional shapes. Theoretical properties of the proposed model will be outlined, and its potential predictive advantages in simulated scenarios and real data will be demonstrated.

近來地理資訊系統(GIS)之資料庫受到不同領域的統計學家廣泛的研究，以期建立及分析可描述空間聚集效應及變異之模型，而描述空間相關存活資料之統計模式為公共衛生及流行病學上新興的研究議題。本文擬建立多維度半參數的貝氏階層模型，並結合空間及非空間隨機效應以描述存活資料中的空間變異。此模式將利用多變量條件自回歸(MCAR)模型以檢驗在不同地理區域中是否存有空間聚集效應。而基準風險函數之生成為分析貝氏半參數階層模型的重要步驟，本研究將利用混合Polya樹之方式生成基準風險函數。美國國家癌症研究院之「流行病監測及最終結果」(Surveillance Epidemiology and End Results, SEER)資料庫為目前美國最完整的癌症病人長期追蹤資料，包含癌症病人存活狀況、多重癌症史、居住地區及其他分析所需之個人資料。本文將自此資料庫擷取美國愛荷華州之癌症病人資料為例作實證分析，並以貝氏統計分析中常用之模型比較標準如條件預測指標(CPO)、平均對數擬邊際概似函數值(ALMPL)、離差訊息準則(DIC)分別測試其可靠度。 / The databases of Geographic Information System (GIS) have gained attention among different fields of statisticians to develop and analyze models which account for spatial clustering and variation. There is an emerging interest in modeling spatially correlated survival data in public health and epidemiologic studies. In this article, we develop Bayesian multivariate semiparametric hierarchical models to incorporate both spatially correlated and uncorrelated frailties to answer the question of spatial variation in the survival patterns, and we use multivariate conditionally autoregressive (MCAR) model to detect that whether there exists the spatial cluster across different areas. The baseline hazard function will be modeled semiparametrically using mixtures of finite Polya trees. The SEER (Surveillance Epidemiology and End Results) database from the National Cancer Institute (NCI) provides comprehensive cancer data about patient’s survival time, regional information, and others demographic information. We implement our Bayesian hierarchical spatial models on Iowa cancer data extracted from SEER database. We illustrate how to compute the conditional predictive ordinate (CPO), the average log-marginal pseudo-likelihood (ALMPL), and deviance information criterion (DIC), which are Bayesian criterions for model checking and comparison among competing models.