Constrained latent class models: Theory and applications

Three types of restricted latent class models for binary data are discussed: models assuming equalities of certain latent parameters, linearly constrained latent class analysis, and linear logistic latent class analysis. The basic equations of the latter model state the decomposition of the log odds of the item latent probabilities and of the class sizes into weighted sums of basic parameters representing the effects of predictor variables which are hypothesized to be relevant indicators for the original parameters. The maximum likelihood equations for these effect parameters and a criterion for their local identifiability are given. Further, statistical tests for goodness of fit are sketched, and the practical application of linear logistic latent class analysis is demonstrated by several examples. They comprise some known scaling models as well as a simple model with located classes and items, a model which relates item difficulty to item structure, and a model for measuring latent changes. Finally, the generalization of linear logistic latent class analysis to polytomous manifest variables is outlined.