Reduced Rank Models for Contingency Tables

Abstract

Biomtlrika (1991), 78, 1, pp. 229-32 Printed in Great Britain Reduced rank models for contingency tables BY JAN DE LEEUW Departments of Psychology and Mathematics, University of California, Los Angeles, California 90024-1563, U.S.A. AND PETER G. M. VAN DER HEIJDEN Department of Empirical and Theoretical Sociology, University of Utrecht, 3508 TC Utrecht, The Netherlands SUMMARY Some key words: Canonical analysis; Correspondence analysis; Latent class analysis; Reduced rank models. 1. INTRODUCTION 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. 2. GENERAL REDUCED RANK MODELS The basic model studied in this paper assumes that a n n x m probability matrix II has rank p, where p = min (n,m). We call this model R p . The probability matrix II has all elements nonnegative, while the sum of the ny is equal to one. We suppose, unless indicated otherwise, that II is full, in the sense that its row sums ir l+ and its column sums v +J are all positive. Thus no row or column is equal to zero. We compare this model with the canonical model C p , in which at most p - 1 of the canonical correlations between the row and the column variables of the table are nonzero. These canonical correlations are the stationary values of the product moment correlation coefficient, seen as a function of scores for rows and scores for columns. Downloaded from biomet.oxfordjournals.org at University of California, Los Angeles on May 18, 2011 Reduced rank models for the analysis of two-way contingency tables are introduced. Two classes of reduced rank models are discerned, with well-known exponents canonical analysis and latent class analysis. The relation between these two classes is discussed. Results on the subject mentioned earlier in the literature are shown to be either redundant or inaccurate.