A characterization of monotone unidimensional latent variable models

Abstract

Recently, the problem of characterizing monotone unidimensional latent variable models for binary repeated measures was studied by Ellis and van den Wollenberg and by Junker. We generalize their work with a de Finetti-like characterization of the distribution of repeated measures $\rm X = (X_1, X_2, \dots)$ that can be represented with mixtures of likelihoods of independent but not identically distributed random variables, where the data satisfy a stochastic ordering property with respect to the mixing variable. The random variables $X_j$ may be arbitrary real-valued random variables. We show that the distribution of X can be given a monotone unidimensional latent variable representation that is useful in the sense of Junker if and only if this distribution satisfies conditional association (CA) and a vanishing conditional dependence (VCD) condition, which asserts that finite subsets of the variables in X become independent as we condition on a larger and larger segment of the remaining variables in X. It is also interesting that the mixture representation is in a certain ordinal sense unique, when CA and VCD hold. The characterization theorem extends and simplifies the main result of Junker and generalizes methods of Ellis and van den Wollenberg to a much broader class of models.

Exchangeable sequences of binary random variables also satisfy both CA and VCD, as do exchangeable sequences arising as location mixtures. In the same way that de Finetti's theorem provides a path toward justifying standard i.i.d.-mixture components in hierarchical models on the basis of our intuitions about the exchangeability of observations, this theorem justifies one-dimensional latent variable components in hierarchical models, in terms of our intuitions about positive association and redundancy between observations. Because these conditions are on the joint distribution of the observable data X, they may also be used to construct asymptotically power-1 tests for unidimensional latent variable models.