Symmetry, quasi-symmetry, and marginal homogeneity on the latent level

(Quasi-)symmetry models can be applied to panel and trend data in many ways, some of which are seemingly totally unrelated to the concept of (quasi-)symmetry. Log-linear models are very well suited to estimate the amount of (quasi-)symmetry in a square frequency table. Log-linear models with latent variables make it possible to assess the (quasi-)symmetrical nature of the multivariate distributions of the latent variables, thereby correcting for the unreliability in the measurement of the manifest variables. Latent (quasi-)symmetry models can be applied by introducing a slight extension of Goodman's well-known version of the EM-algorithm for the estimation and testing of latent structure models.