A Composite Likelihood Approach for Factor Analyzing Ordinal Data

Ordinal variables are widely used in psychological and educational assessment. One popular model for analyzing ordinal variables is based on a modification of the standard factor analysis model. This modified factor analysis model postulates a continuous latent response variable that corresponds to each observed ordinal variable and then specifies a usual factor analysis model on the item-level latent response variables. Various methodologies are available for estimating parameters in the modified factor analysis model and they usually involve two to three computational steps.;This dissertation considers the Underlying Bivariate Normal (UBN) approach for estimating the modified factor analysis model from ordinal data. The UBN method relies on univariate and bivariate margins only, and it estimates all the model parameters in a single step. As a limited-information procedure, the UBN approach has asymptotic properties that are different from full-information Maximum Likelihood Estimation (MLE) procedure and requires theoretical results that are specifically developed for limited-information approaches. The current research applies the Godambe information matrix, instead of the usual Fisher information matrix, to obtain standard error estimates and evaluates the model fit by a residual based quadratic form test statistic.;The UBN approach is illustrated by using both simulation studies and two real data examples. Performance of the UBN approach is evaluated by checking the closeness between the UBN estimates and the corresponding true population values. The UBN estimates are also compared with results given by the Expectation-Maximization (EM) algorithm and the Metropolis-Hastings Robbins-Monro (MH-RM) method within the Item Response Theory (IRT) framework. Simulation results show that: (1) the UBN approach is able to recover the true population values of model parameters; (2) the UBN estimates become more precise when sample size increases; (3) the standard error estimates given by the Godambe information matrix are consistent with the observed empirical standard errors; (4) the residual based test statistic possesses the assumed asymptotic properties and is useable as a goodness-of-fit test statistic. The two real data illustrations demonstrate that the UBN approach is feasible for large scale real world problems.