In many modern applications of statistical models, high-dimensional interdependencies may cause standard likelihood-based inference meets difficulties. High dimensional ordinal data, for instance, will encounter the problem of prohibitively large computational demands. This dissertation develops the statistical theory for a new multistage ordinal methodology in the context of structural equation modeling (SEM), based on a recently developed maximum pairwise likelihood method. Unlike earlier methods, the maximum pairwise likelihood approach maximizes an objective function based on the product of bivariate probabilities from any two different pairs of variables to estimate both thresholds, polychoric and polyserial correlations. The asymptotic distribution of these estimators is used to develop a second stage estimation and testing procedure for SEM based on generalized least squares, and a new goodness-of-fit statistic is obtained that is asymptotically chi-square distributed. Simulation studies to evaluate the performance of the proposed method are described and summarized.

This method is further extended to multiple groups. Additional identification conditions are presented and simulation studies on detection of group difference are also provided.