带有复杂多元数据的非参数非线性结构方程模型：贝叶斯分析

There is often the need to assess interrelationships among latent variables in behavioral, social and psychological research. Structural equation models (SEMs) comprise a flexible class of models for modeling multivariate correlated data to analyze the interrelationships among latent variables. Basically, SEMs are formulated by a measurement model, which is a confirmatory factor analysis model for grouping correlated manifest variables to "measure" their corresponding latent variables, and a regression type structural model with latent variables for examining the effects of exogenous latent variables on endogenous latent variables of interest. In real data analysis, it is common to encounter multivariate data with complex data structure, which include continuous data, categorical data, heterogeneous data and missing data (especially non-ignorable missing data). However, it is difficult to analyze them simultaneously with the existing models and software. Moreover, because the major objective of SEMs is the analysis of latent variables, the structural model plays the most important role. Although the existing nonlinear SEM has been found to be useful, its structural model is parametric and hence may be too restrictive to many research settings. It is thus necessary to consider more general structural models for revealing the true functional relationships between endogenous and exogenous latent variables and fixed covariates. Motivated by the development of nonparametric modeling, we will develop a novel Bayesian nonparametric nonlinear SEM with complex multivariate data, in which the important structural model is formulated via unspecified smooth functions of latent variables or covariates. Hence, it can be regarded as a generalization of the ordinary nonparametric regression model with the new inclusion of latent variables. The objectives of our project are listed as follows: (1) To establish a novel nonparametric nonlinear SEM with complex multivariate data, in which the crucial structural model is formulated by unspecified smooth functions of latent variables or covariates. (2) To develop statistical methods for estimation and model selection to analyze the proposed model. Given the complexity of the nonparametric modeling and the data structure, we will derive novel methods to solve the involved difficulties. Bayesian P-splines coupled with data augmentation and Markov chain Monte Carlo techniques will be used to achieve the results. (3) To develop related computer programs that will obtain the desired results, and to put these programs onto a freely accessible website for practitioners. (4) To achieve novel applications by applying the newly developed methodologies to real-world researches. As both nonparametric modeling and SEMs have extremely wide applicability, we expect that our newly developed methodologies will be very useful in many fields. And the achieved results will open a new research frontier in the study of SEMs.

结构方程模型是公认的研究多元相关数据的重要方法。实际研究中的多元数据集往往具有复杂的数据结构，经常同时包含连续数据、分类数据、异构数据和缺失数据（特别是不可忽略缺失数据），现有的模型和软件都难以处理。本项目拟建立崭新的带有复杂多元数据的非参数非线性结构方程模型，除了能够有效处理上述的复杂多元数据之外，更重要的是，该模型创新地将非参数回归和结构方程模型相结合，对涉及潜变量的未知光滑函数进行非参数建模，以求更准确细致地刻画潜变量之间的关系。在贝叶斯方法的框架下，结合P-样条技术，数据增广和马尔科夫链蒙特卡洛方法，本项目将集中研究和探讨有效的统计分析方法，包括未知参数估计、未知光滑函数估计以及模型选择等。另外，本项目还将致力于把提出的模型应用到实际研究问题当中，以更好地解决其他应用学科中复杂多元数据分析的问题。本项目完成后，将把所编写的相关电脑程序放在互联网上供其他有需要的研究者参考和使用。

本项目建立了带有复杂多元数据的非参数非线性结构方程模型。该模型除了能够有效地处理复杂多元数据外，还创新地将非参数建模和结构方程模型相结合，对结构模型部分涉及潜变量的未知光滑函数进行非参数建模。另外，为了保证所提出的模型能更准确地刻画潜变量间的关系，本项目在执行过程中增加了对不同类型数据的局部项目依赖性建模这一部分。 因此本项目的主要研究内容包括：（1）带复杂多元数据的非参数非线性结构方程模型的贝叶斯分析；（2）带局部项目依赖的结构方程模型的贝叶斯分析。. 首先，受LASSO算法估计稀疏化的方差协方差矩阵的启发，本项目把LASSO算法和验证性因素分析模型结合，建立了带局部项目依赖的验证性因素分析模型，将结构方程模型中测量模型的误差项，从传统的需要两两相互独立的假设推广到更一般的情况，成功解决了实际数据分析中不满足“局部独立性”假设时应如何建模的问题。其次，基于变换模型的思想并结合P-样条技术对结构方程模型中的未知函数进行非参数建模和逼近，所提出的模型能够处理多组别情况下的连续数据、有序分类数据、删失数据和随机缺失数据。这项工作主要解决了含潜变量和复杂多元数据的半参数结构方程模型的建模和统计分析的问题。另外，通过对测量误差的方差协方差矩阵进行贝叶斯LASSO建模，本项目建立了多层异构模型，用于同时处理密集追踪数据、异构数据以及可观察变量间的局部依赖性。该分析方法在保持因子结构（包括因子数目和因子负荷情况）稳定性的同时保证了模型的简约性。在贝叶斯方法的框架下，本项目分别针对上述模型提出了相应的统计分析方法，无论是计算机模拟研究还是实际数据分析都验证了所提出的方法是有效和有实际应用价值的。作为本项目的一个特色，我们将把与项目研究成果相关的电脑程序放在互联网上以供其他有需要的研究者免费参考和使用。. 非参数方法和结构方程模型在不同研究领域已经有着广泛的应用，本项目所提出的新模型则是基于实证研究的需求，在考虑了复杂多元数据的情况下，创新地将两者结合起来。该新模型有较好的理论价值和实际应用价值，不但能在学术研究上进一步丰富和完善结构方程模型的相关研究，进一步拓展其应用空间，而且能对以结构方程模型作为数据分析工具的相关学科的研究起到积极的推动作用。.

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