3. Log-Multiplicative Association Models as Latent Variable Models for Nominal and/or Ordinal Data

Abstract

Associations between multiple discrete measures are often due to collapsing over other variables. When the variables collapsed over are unobserved and continuous, log-multiplicative association models, including log-linear models with linear-by-linear interactions for ordinal categorical data and extensions of Goodman's (1979, 1985) RC(M) association model for multiple nominal and/or ordinal categorical variables, can be used to study the relationship between the observed discrete variables and the unobserved continuous ones, and to study the unobserved variables. The derivation and use of log-multiplicative association models as latent variable models for discrete variables are presented in this paper. The models are based on graphical models for discrete and continuous variables where the variables follow a conditional Gaussian distribution. The models have many desirable properties, including having schematic or graphical representations of the system of observed and unobserved variables, the log-multiplicative models can be read from the graphs, and estimates of the means, variances, and covariances of the latent variables given values on the observed variables are a function of the log-multiplicative model parameters. To illustrate some of the advantageous aspects of these models, two examples are presented. In one example, responses to items from the General Social Survey (Davis and Smith 1996) are modeled, and in the other example, panel data from two groups (Coleman 1964) are analyzed.