Statistical Models and Diagnostics Tools for Spatially Correlated Skewed and Heterogeneous Data

在应用研究中经常观察到偏斜和异构的数据，例如，在生态学研究中，与生境适宜性有关的物种的丰度，或在卫生服务研究中，住院人数。当数据由于未测量的区域特征而在地理上聚集或随着时间的推移反复收集时，对此类数据进行建模可能会进一步复杂化。对数或平方根变换通常用于实现正态性，然后可以对转换后的数据应用线性回归。然而，在某些上下文中，数据包含大量的零和高极值，因此通过任何形式的转换都可能不容易实现正态性。此外，转换后的响应变量在不同的尺度上操作，可以掩盖原始响应变量的重要信息。生存数据也经常高度扭曲，可能有治愈率（即，很大一部分受试者从未失败）和复杂事件类型（即多状态或竞争风险事件）。近年来，人们越来越有兴趣捕捉生存时间的空间格局，以确定导致这种变异的可能因素。传统的参数生存模型不能同时考虑治愈率、多种事件类型以及空间相关性。为了填补处理这类不同数据的分析方法的空白，我的研究计划的一个重点是开发空间相关倾斜结果的模型，这些模型不需要对数据进行转换，而是可以应用于原始尺度的数据。所提出的建模方法将提高模型预测和参数估计的精度。模型诊断是保证模型有效性的重要步骤，但由于传统残差具有依赖于模型参数的复杂参考分布，对具有离散或不完整信息的偏斜和异构数据进行模型诊断一直是非常具有挑战性的。为了填补这一空白，我们最近扩展了用于诊断零膨胀混合效应模型和参数生存模型的随机分位数残差。该方法将进一步发展到诊断时空模型和空间生存模型。扩展模型诊断方法将指导研究人员开发更好的模型，并从数据中得出更可靠的结论。提出的理论工作是由实际项目驱动的，适用于应用研究的各个方面。研究成果有望通过提供可行、高效和稳健的方法，为统计理论和实践做出贡献，相关培训将培养出高素质的统计学家。将提供R中的软件包，以协助向广泛的受众传播和实施拟议的模型。

Skewed and heterogeneous data are often observed in applied research, e.g., abundance of a species related to habitat suitability in ecological studies, or number of hospitalizations in health services research. Modelling such data can be further complicated when data are geographically clustered due to unmeasured regional characteristics or repeatedly collected over time. Logarithmic or square-root transformations are often used to achieve normality and then a linear regression can be applied to the transformed data. However, in some contexts, the data contain both an abundance of zeros and high extreme values, so normality may not be easily achieved by any forms of transformation. Moreover, the transformed response variable is operated on a different scale that can mask the important information of the original response variable. Survival data is also often highly skewed, which can have cure fraction (i.e., a substantial portion of subjects never fail) as well as complex event types (i.e. multi-state or competing risk events). In recent years, there has been growing interest to capture spatial patterns in survival times for determining the possible factors that contribute towards such variability. Traditional parametric survival models cannot account for cure fraction, multiple event types as well as spatial correlation at the same time. To fill the gap in analytical methods for handling this sort of disparate data, one key focus of my research program is to develop models for spatially correlated skewed outcomes that do not require transformation of the data, but rather can be applied to the data on their original scale. The proposed modeling methods will improve the model prediction and accuracy of parameter estimates.  Model diagnostics is an essential step to ensure the validity of the model, but it has been very challenging to diagnose models for skewed and heterogeneous data with discreteness or incomplete information due to censoring, partly because the traditional residuals have complicated reference distributions that are dependent on the parameters in the model. To fill this gap, we recently extended randomized quantile residuals for diagnosing zero-inflated mixed-effects models and parametric survival models. This method will be further developed to diagnose spatial and spatial-temporal models and spatial survival models. The extended model diagnosis methods will guide researchers developing better models and drawing more reliable conclusion from their data.  The proposed theoretical work is driven by real-life projects, with applicability to various aspects of applied research. The research outcomes are anticipated to contribute to statistical theory and practice by providing feasible, efficient, and robust approaches, and the associated training will produce highly qualified statisticians. Packages in R will be made available to assist in the dissemination and implementation of the proposed models to a wide audience.