(though this so-called “simple structure” restriction may be relaxed). In a confirmatory factor analysis (CFA) model, only the measurement model is specified, and the set of latent variables are allowed to freely covary. Thus, the model-implied covariance matrix of a CFA is Σ(θ) = ΛΨΛ′+Θ, where Ψ is an m × m latent variable covariance matrix and Θ is a k × k matrix of covariances among the residuals, ε. Typically, Θ is diagonal, which implies that indicators are mutually independent conditional on the set of latent variables. That constraint may be relaxed such that select pairs of indicators are allowed to covary over and above their associations via the latent variable structure (i.e., residual covariances are allowed). CFA models are typically used to assess the strength of relations between sets of indicators and their respective underlying latent variables, as well as to assess the fit of a measurement model to data. A well-fitting CFA model implies that the latent variable structure is able to account for the observed covariances among a set of indicator variables.
