Ellis and van den Wollenberg (1993) and Junker (1993) recently studied the problem of characterizing monotone unidimensional latent variable models for binary repeated measures. We generalize their work with a de Finetti-like characterization of the distribution of repeated measures X = (X1, X2,...) 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 Xj 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 (1993), if and only if this distribution satisfies conditional association (CA; Holland and Rosenbaum, 1986) 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 (1993), and generalizes methods of Ellis and van den Wollenberg (1993) 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 iid-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.