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

在应用研究中经常观察到不对称和不同种类的数据，例如，生态学研究中与栖息地适宜性有关的物种丰富度，或卫生服务研究中的住院次数。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而重复收集时，对此类数据进行建模可能会更加复杂。*通常使用对数或平方根变换来实现正态分布，然后对变换后的数据进行线性回归。然而，在某些情况下，数据既包含大量的零，也包含大量的极值，因此通过任何形式的转换可能都不容易实现常态。此外，变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。生存数据通常也是高度倾斜的，这可能有治愈分数(即相当一部分受试者从未失败)以及复杂的事件类型(即多状态或竞争性风险事件)。近年来，人们越来越有兴趣捕捉生存时间的空间模式，以确定导致这种变异性的可能因素。传统的参数生存模型不能同时考虑治愈分数、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关键重点是为空间相关的倾斜结果开发模型，这些模型不需要转换数据，而是可以应用于原始规模的数据。所提出的建模方法将提高模型预测和参数估计的精度。*模型诊断是确保模型有效性的重要步骤，但对于具有离散性或不完全信息的偏态和异质数据，由于截尾的影响，模型的诊断一直是非常困难的，部分原因是传统的残差具有复杂的参考分布，这依赖于模型中的参数。为了填补这一空白，我们最近扩展了随机分位数残差，用于诊断零膨胀混合效应模型和参数生存模型。该方法将被进一步发展用于诊断空间和时空模型以及空间生存模型。扩展的模型诊断方法将指导研究人员开发更好的模型，并从他们的数据中得出更可靠的结论。*建议的理论工作由实际项目驱动，适用于应用研究的各个方面。预计研究成果将通过提供可行、有效和可靠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人员。将提供R中的套餐，以协助向广大受众传播和实施拟议的模式。**

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.**