An empirical evaluation of three procedures of confirmatory factor analysis with ordinal data

Confirmatory factor analysis (CFA) is routinely applied to ordinal variables like questionnaire items. Treating ordinal manifest variables as if they were continuous in CFA leads to less-than-optimal results. Three procedures have been proposed to properly fit CFA models with ordinal manifest variables. These three procedures are the polychoric correlation approach (Joreskog, 1990, 1994; Muthen, 1984), the underlying bivariate normal approach (Joreskog and Moustaki, 2001; Xi, 2011), and the item response theory approach (Bock et al., 1988; Cai, 2010a,b). Statistical properties of the polychoric correlation approach under correctly specied models have been well studied. The properties of the underlying bivariate normal approach and the item response theory approach have not been studied and contrasted. The goal of the present dissertation is to empirically evaluate the performances of three procedures under a variety of conditions. To serve the purpose of comparison, the normal theory maximum likelihood approach which ignoring the characteristics of ordinal manifest variables is also examined in the current study. Of particular interests are their respective performances under misspecified CFA models. To examine the performances of different approaches under misspecified CFA models, different levels of model error are introduced into population covariance matrices. The levels of model error are controlled to have RMSEA index with 0, .05, and .08, which falls in the range from perfect fit, close fit and fair fit. The model errors are introduced to the population covariance matrices by the procedure described in Cudeck and Browne (1992). The independent variables manipulated in the simulation studies include the level of model misspecification, sample size, model size, number of response categories, and the distribution of ordinal variables. The outcome variables include point estimates, standard error estimates, and condence intervals. The chi-square test statistics and fit measures are also examined to understand the influences of the factors on model fit. Guidelines for substantive researchers are provided. The results indicate that across the manipulated conditions, the composite likelihood function approach provides satisfactory performances in terms of the relative bias of point and standard error estimates, empirical coverage rates of confidence intervals, and empirical rejection rates of overall model fit test statistic. The item response theory approach provides similar results as the composite likelihood function approach. The polychoric correlation approach can also provides unbiased point estimates for model parameters. And the normal theory maximum likelihood approach is not recommended to apply with the ordinal manifest variables.