*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 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 may be arbitrary real-valued random variables. We
show that the distribution of 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 become
independent as we condition on a larger and larger segment of the
remaining variables in . 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.