Linear Logistic Latent Class Analysis and the Rasch Model

Abstract

As a special case of linear logistic latent class analysis, the paper introduces the latent class/Rasch model (LC/RM). While its item latent probabilities axe restricted according to the RM, its unconstrained class sizes describe an unknown discrete ability distribution. It is shown that when the number of classes is (at least) half the number of items plus one, the LC/RM perfectly fits the raw score distribution in most cases and thus results in the same item parameter estimates as obtained under the conditional maximum likelihood method in the RM. Two reasons account for this equivalence. Firstly, the maximum likelihood (ML) method for the LC/RM can be seen to be a version of the Kiefer-Wolfowitz approach, so that the structural item parameters as well as the unknown ability distribution are estimated consistently. Secondly, through its moment structure the observed raw score distribution constrains the ability distribution. Irrespective of whether the ability distribution is assumed to be continuous or discrete, it has to be replaced by a step function with the number of steps (=classes) being half the number of items plus one. An empirical example (Stouffer-Toby data on role conflict) illustrates these findings. Finally, conclusions are drawn concerning the properties of the ML item parameter estimates in the LC/RM, and some relations to other methods of parameter estimation in the RM are mentioned.