Unidimensional linear latent variable models

Abstract: "Linear structural equation models with latent (unmeasured) variables are used widely in sociology, psychometrics, and political science. When such models have a unidimensional (pure) measurement model (Gerbing and Anderson 82, 88; Scheines 92) they imply constraints on the measured covariances which can be used to either confirm unidimensionality or find submodels which are unidimensional. Assuming unidimensionality, the causal relations among the latent variables can be partially determined by examining other (related) constraints on the measured covariances.In this paper I prove first that unidimensionality is detectible from constraints on only the measured covariances no matter what the structure among latent variables, and second that in a structural equation model with a unidimensional measurement model, for any three latents T[subscript i], T[subscript j], and T[subscript k], [rho]T[subscript i],T[subscript j].T[subscript k] = 0 only if certain constraints hold on only the measured covariances."