Reduced rank models for contingency tables

In recent years much attention has been given to models for two-way contingency tables that can be formulated in terms of reduced rank of a matrix with probabilities. A well-known reduced rank model is the independence model, where the rank is one. For rank higher than one distinct classes of reduced rank models are possible. Each has the independence model as the special case for rank one. A first class of such models is closely related to what is known under names as canonical analysis or correspondence analysis. Recently much attention has been given to the maximum likelihood estimation of versions of these models by Goodman (1985, 1986, 1987) and Gilula & Haberman (1986, 1988). A second class of models that can be formulated in terms of reduced rank is latent class analysis, LCA, for two-way tables. Latent class analysis was proposed by Lazersfeld (1950a, b). See Clogg (1981) for a more recent review. In this paper we relate these classes of models to each other. The relation has been discussed earlier by Gilula (1979, 1983, 1984), Gilula & Haberman (1986), Goodman (1987), and van der Heijden, Mooijaart & de Leeuw (1989). We summarize existing results in a simple way using new proofs. Gilula (1979) provided conditions that had to hold for rank-2 correspondence analysis to imply rank-2 latent class analysis. We show here that rank-2 correspondence analysis always implies rank-2 latent class analysis. This implies that the theorem and the example given by Gilula (1979) are incorrect.