Constrained estimation of the polychoric correlation matrix

Full Maximum Likelihood (ML) estimation of polychoric correlation matrices is too computationally expensive. As an alternative, other approaches use pairwise bivariate normal densities of the categorical distributions as an estimation method. These methods, however, usually result in matrices that are not positive semidefinite. In this presentation, we introduce a new approach to estimate polychoric correlation matrices that use pairwise bivariate normal densities, but the form of the correlation matrix is restricted to guarantee that it is positive semidefinite. Our approach is based on using a transformation of the lower diagonal values of the Cholesky (or LDL) decomposition matrix as the parameters of the models. This restriction is monotonic in regards to the estimated correlations and, therefore, are easy to include in an optimization routine and should result in a well-behaved objective function. We also show that extending the original approach to a regularized Bayesian approach—i.e., using zero-centered symmetric distribution as priors to the lower diagonal values of the Cholesky (or LDL) decomposition matrix—helps in guaranteeing a better convergence of the estimates. Results of a pilot simulation study are presented and discussed in regards to computation efficiency (i.e., computation time) and the precision of the estimates. Suggestions for future studies are also presented, focusing mainly on what aspects of the simulation these studies should focus on.