Point-Adjusted Gaussian Mixture Model, With Its’ Application In Structural Equation Model

Structural equation model (SEM) is an important new statistical analysis methods, and has been widely applied in social science research. Meanwhile, point inflation is common in questionnaire data. In this paper, we discuss point inflation data in Confirmatory Factor Analysis (CFA). Since a point is excess, both removing the inflated point and without any processing can make model inaccurate. We make a new model, called Point-Adjusted Gaussian mixture model for handling this kind of point inflated data. It assumes that the data distribution is mixed by a distribution degenerate at point y0 and multivariable Gaussian distribution. Using this model, we can get the covariance for fitting structural equation model. The details about the research are described as follows.1)We build Point-Adjusted Gaussian(PAG) mixture model, and present the model theory and EM algorithm for maximum likelihood estimation(MLE). The maximum likelihood estimates (MLE’s) are approximately normal in large samples, and variance and confidence intervals can be constructed by using approximate normality of MLE’s and the observed information matrix corresponding to the PAG Log-likelihood. In addition, we discuss the convergence properties of the EM algorithm for PAG mixture model. Simulations suggest that when there exist point inflated data, the structural equation models based on PAG mixture model are better.2)We apply PAG mixture model to data on classroom experience in SiJiQing Middle School’s students and test the six-dimensional teaching structure with CFA. The model fitting results are compared with SEM based on raw data covariance. And it shows that the Log-likelihood corresponding to SEM with PAG mixture model is bigger, both the values of AIC and the BIC are smaller; That is, PAG mixture model can improve the goodness of fit in CFA when there exist point inflated data.