Structural Equation Modeling with a Small Number of Observations (N) and a Large Number of Variables (p)

该研究项目将开发结构方程建模（SEM）方法，该方法可应用于具有少量观测值（N）和大量变量（p）的数据。 新方法将扩大扫描电镜的适用性，重要的研究领域，观察的数量可能是相当小的，如数据从难以达到的人口。 数据和数据分析方法是推动科学发展的关键。 SEM被许多学科使用，包括心理学，教育学，社会学，管理学，健康科学和医学。 现有的SEM方法要求观测值的数量与变量数量的平方成比例。 然而，可能难以收集SEM所需的大量观察结果以得出可靠的结论。 当N大于30或变量数的两倍时，新方法有望产生可靠的结果。 该研究有可能改善数据分析并降低数据收集成本。 在小N和/或大P的情况下，现有的SEM方法面临四个主要的实际问题：（1）近奇异样本协方差矩阵和相关的参数估计不收敛问题;（2）非正态数据的参数估计效率低下;（3）不可靠的模型检验统计量;和（4）不准确的标准误差。 该项目将通过在样本协方差矩阵或岭方法中添加适当的对角矩阵来解决第一个问题，这种方法已被证明是有效的。 其他问题将通过一种称为经验建模的新方法来解决，其中在各种条件下对检验统计量/标准误差的经验行为进行建模。 通过结合经验建模和岭方法，该项目预计将产生有效的参数估计和可靠的模型推断。

This research project will develop structural equation modeling (SEM) methodology that can be applied to data with a small number of observations (N) and a large number of variables (p). The new methods will broaden the applicability of SEM to important areas of research where the number of observations may be quite small, such as data from hard-to-reach populations. Data and methods for data analysis are key to advancing science. SEM is used by many disciplines, including psychology, education, sociology, management, health sciences, and medicine. Existing SEM methodology requires the number of observations to be proportional to the square of the number of variables. However, it may be difficult to collect the large number of observations required by SEM to reach reliable conclusions. The new methodology is expected to yield reliable results when N is either larger than 30 or two times the number of variables. The research has the potential to both improve data analyses and reduce data collection costs. The methods will be implemented in freely available, user-friendly software.With small N and/or large p, existing methods for SEM face four major practical issues: (1) near singular sample covariance matrices and the related issue of nonconvergence in parameter estimation; (2) inefficient parameter estimates with non-normal data; (3) unreliable model test statistics; and (4) inaccurate standard errors. The project will address the first issue by adding a proper diagonal matrix to the sample covariance matrix or the ridge method, an approach that has been shown to be effective. The other issues will be addressed by a new method termed empirical modeling, in which the empirical behavior of a test statistic/standard error is modeled under a variety of conditions. By combining empirical modeling and the ridge method, the project is expected to produce efficient parameter estimates and reliable model inference.