Confirmatory Factor Analysis Of Ordered Categorical Variables

Ordered categorical variable is widely used in the fields of Psychology and Social Science,like,the Likert scales(“1” represents extremely disagree and “5” represents extremely agree)andtrue-false questions in cognitive tests.In the practical research,researchers often treat the ordered categorical variable as a continuous variable for data analysis.However,the ordered categorical variable is different from the continuous variable because of its discrete property.Therefore,many statistical researchers suggest that model with the ordered categorical variable needs to be analyzed with special techniques.In addition,the Confirmatory Factor Analysis(CFA)assumed that an indicator is loaded on a target factor while zero cross-loadings are specified on the other factors.The cross-loading problem often results in poor fitting model and biased parameter's estimates.Therefore,it becomes an important research topic to find appropriate approach to analyze theCFA with order categorical data when there are potentially unknown cross-loadings.Traditional Frequentist approach in analyzing CFA often encounters obstacles of small sample size and non-normal distributed data,which often results in difficulty in convergence and unbiased estimates.Previous research has already proved the strengths of the Bayesian method.First,Bayesian approach could provide an accurate parameter estimate when the sample size is relatively small.Second,Bayesian estimation,which is based on posterior distribution of the unknown parameters,may be less affected by the non-normal data.Finally,Bayesian approach could deal with more complex model.Nevertheless,it is still not clear whether the Bayesian approach could perform well in the parameter estimation of the CFA model with ordered categorical data.Two simulation studies were set 300 conditions(Study 1)and 810 conditions(Study2)to compare the Frequentist and Bayesian approaches in analyzing the CFA model with ordered categorical data.Results support the approach of treating ordered categorical variables as continuous variables in analyzing of the CFA model with ordered categorical data when the number of category is four or above.It is proved that the performance Bayesian approach is better under the conditions of small number of category,small sample size,and asymmetric data distribution.Moreover,Bayesian approach could provide more accurate credibility interval estimate for the parameter.Compared with the Bayesian approach,Exploratory Structural Equation Modeling(ESEM)does not rely on prior information and could be more applicable.In the real study,a CFA model of the “dark triad” was established to illustrate the analyzing processes of the ESEM and Bayesian approaches.This study also provided some guidelines for the applied researchers.