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Conditional Association and Unidimensionality in Monotone Latent Variable Models IRT

Holland, Paul W.; Rosenbaum, Paul R.
Publication Year:
Report Number:
RR-85-47, PSRTR-85-65
ETS Research Report
Document Type:
Page Count:
Subject/Key Words:
Factor Analysis, Item Response Theory (IRT), Mathematical Models, Multivariate Analysis, Psychometrics, Statistical Analysis


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 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. (42pp.)

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