Abstract: Traditional factor analysis models and estimation methods for continuous (i.e., interval or ratio scale) data are not appropriate for item-level data that are categorical in nature. The authors provide a brief review and synthesis of the item factor analysis estimation literature for categorical data (e.g., 0-1 type response scales) under the multidimensional response model. Popular categorical item factor analysis models and estimation methods found in the structural equation modeling and item response theory literatures are presented.
The Monte Carlo simulation studies are conducted and revealed: (1) Similar parameter estimates have been obtained of Modified weighted least squares for categorical data method (WLSMV) from the structural equation model (SEM) framework and adoptive Restricted Maximum Likelihood (MLR) and Markov chain Monte Carlo (MCMC) methods from the multidimensional item response theory (MIRT) framework. Even with a small sample and the item response theory (IRT) estimates converted to SEM parameters, the WLSMV, MLR, and MCMC results are strikingly similar. But in small sample size and long test, weighted least squares for categorical data (WLSc) did not obtain the convergence parameter estimations, although in short test, WLSc estimates have been obtained, the estimates are consistently more discrepant than those produced by the other estimation techniques. (2) The precision of the estimators enhances as the quantity of the sample increases, and the differences between WLSMV and MLR are very trivial, and the precisions of WLSMV and MLR methods are not worse than that of the MCMC method in most conditions. (3) The precision of item factor loading and of item difficulty parameter is influenced by the test length, and the precision of item discrimination and of item difficulty parameter is influenced by the number of test dimension. (4) The precision of the estimators decreases as the number of dimensions measured by the item increases, especially for item discrimination and item factor loading parameter.
Both SEM and IRT can be used for factor analysis of dichotomous item responses. In this case, the measurement models of both approaches are formally equivalent. They were refined within and across different disciplines, and make complementary contributions to central measurement problems encountered in almost all empirical social science research fields. The authors conclude with considerations for categorical item factor analysis and give some advice for applied researchers.

Key words: Multidimensional Item Response Theory, confirmatory factor analysis, parameter estimation, categorical data