The Annals of Statistics

Conditional Association and Unidimensionality in Monotone Latent Variable Models

Paul W. Holland and Paul R. Rosenbaum

Full-text: Open access


Latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables, usually assuming the conditional independence of the observable variables given the latent variables. These models play an important role in educational measurement and psychometrics, in sociology and in population genetics, and are implicit in some work on systems reliability. We study a broad class of latent variable models, namely the monotone unidimensional models, in which the latent variable is a scalar, the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable. All models in this class imply a new strong form of positive dependence among the observable variables, namely conditional (positive) association. This positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data. Various applications, generalizations and a numerical example are discussed.

Article information

Ann. Statist., Volume 14, Number 4 (1986), 1523-1543.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62P15: Applications to psychology 62H99: None of the above, but in this section

Latent variable models item response theory population genetics latent class models factor analysis systems reliability conditional association associated random variables $TP_2$ $MTP_2$ SPOD


Holland, Paul W.; Rosenbaum, Paul R. Conditional Association and Unidimensionality in Monotone Latent Variable Models. Ann. Statist. 14 (1986), no. 4, 1523--1543. doi:10.1214/aos/1176350174.

Export citation