A simulation study to compare competing estimators in structural equation models with ordinal variables

Structural equation models have been around for now a long time. They are intensively used to analyze data from di.erent fields such as psychology, social sciences, economics, management, etc. Their estimation can be performed using standard statistical packages such as LISREL. However, these implementations su.er from an important drawback: they are not suited for cases in which the variables are far from the normal distribution. This happens in particular with ordinal data that have a non symmetric distribution, a situation often encountered in practice. An alternative approach would be to use generalized linear latent variable models (GLLVM) as defined for example in Bartholomew and Knott 1999 and Moustaki and Knott (2000). These models consider the data as they are, i.e. binary or ordinal but the loglikelihood function is intractable and needs numerical approximations to compute it. Several approaches exist such as Gauss-Hermite quadratures or simulation based methods, as well as the Laplace approximation, i.e. the Laplace approximated maximum likelihood estimator (LAMLE) proposed by Huber, Ronchetti, and Victoria-Feser (2004) for these models. The advantage of the later is that it is very fast and hence can cope with relatively complicated models. In this paper, we perform a simulation study to compare the parameters' estimators provided by LISREL which is taken as a benchmark, and the LAMLE when the data are generated from a confirmatory factor analysis model with normal variables which are then transformed into ordinal ones. We will show that while the LISREL estimators can provide seriously biased estimators, the LAMLE not only is unbiased, but one can also recover an unbiased estimator of the correlation matrix of the original normal variables.