统计与数据科学学院知识库

验证性因子分析作为一种常用的测量模型，通常用于量表的开发。研究者根据已有的先验信息构建验证性因子分析的模型结构，并通过一些严格的假设对模型进行限制。而这些假设的存在，常常导致验证性因子分析不能对数据进行有效的拟合或拟合得到的模型结构不符合理论假设。目前，贝叶斯方法的兴起使得研究人员在使用验证性因子分析时更加灵活。通过对先验的选择，将正则化的方法引入。在放松了原有假设的同时增强了模型在分析实际数据时的实用性与稳健性。

根据前人的研究，在贝叶斯框架下对验证性因子分析模型结构的研究可以被视为一个贝叶斯变量选择问题。因此，为了进一步提升贝叶斯框架下，验证性因子分析的模型选择和参数估计能力，考虑通过引入贝叶斯弹网惩罚实现验证性因子分析模型中的残差协方差矩阵的模型选择，以此达到整体的模型选择的目的。本文主要研究为：首先，在贝叶斯弹网模型的基础上，通过加入分块Gibbs采样，实现对协方差矩阵和精度矩阵的识别，提出贝叶斯协方差弹网模型，并进一步拓展至贝叶斯自适应协方差弹网模型；其次，将其与验证性因子分析结合，通过贝叶斯协方差弹网模型实现残差协方差矩阵的模型选择与参数估计，达到修改验证性因子分析模型结构的目的；最后，通过数值模拟和实例分析评估该方法的有效性和实用性。

通过数值模拟与实例分析，将本文所提出的验证性因子分析方法与已有方法进行对比表明，所提出的方法对于识别验证性因子分析的模型结构具有良好的性能。

As a commonly used measurement model, confirmatory factor analysis is usually used in the development of scales. Researchers construct a confirmatory factor analysis model structure based on the existing prior information, and restrict the model through some strict assumptions. The existence of these assumptions often results in confirmatory factor analysis unable to effectively fit the data or the model structure obtained by the fit does not conform to the theoretical assumptions. Currently, the rise of Bayesian methods makes researchers more flexible when using confirmatory factor analysis. Through the choice of a prior, the regularization method is introduced. While relaxing the original assumptions, it enhances the practicality and robustness of the model when analyzing actual data.

According to previous studies, the study of confirmatory factor analysis model structure under the Bayesian framework can be regarded as a Bayesian variable selection problem. Therefore, in order to further improve the model selection and parameter estimation capabilities of confirmatory factor analysis under the Bayesian framework, considered to implement the model selection of the residual covariance matrix in the confirmatory factor analysis model by introducing the Bayesian elastic net penalty. The main research of this thesis is: First, on the basis of the Bayesian elastic net model, adding block Gibbs sampling to realize the identification of the covariance matrix and the precision matrix, proposes the Bayesian covariance elastic net model, and further expands it to Bayesian adaptive Covariance elastic net model; Secondly, combine it with confirmatory factor analysis, and realize the model selection and parameter estimation of the residual covariance matrix through the Bayesian covariance elastic net model to achieve the purpose of modifying the structure of the confirmatory factor analysis model; Finally, the effectiveness and practicability of the method are evaluated through numerical simulation and empirical analysis.

Through numerical simulation and case analysis, the comparison between the confirmatory factor analysis method proposed in this thesis and the existing methods shows that the proposed method has good performance in identifying the model structure of confirmatory factor analysis.