<P> where Y (\ displaystyle Y) is the p x 1 vector of observed random variables, ξ (\ displaystyle \ xi) is the unobserved latent variables, or variables in the multidimensional case, and Λ (\ displaystyle \ Lambda) is a p x k matrix with k equal to the number of latent variables . Since, Y (\ displaystyle Y) are imperfect measures of ξ (\ displaystyle \ xi), the model also consists of error, ε (\ displaystyle \ epsilon). Estimates in the maximum likelihood (ML) case generated by iteratively minimizing the fit function, </P> <P> F M L = l n Λ Ω Λ ′ + I − d i a g (Λ Ω Λ ′) + t r (R (Λ Ω Λ ′ + I − d i a g (Λ Ω Λ ′) − 1) − l n (R) − p (\ displaystyle F_ (ML) = ln \ Lambda \ Omega \ Lambda (') + I - diag (\ Lambda \ Omega \ Lambda (')) + tr (R (\ Lambda \ Omega \ Lambda (') + I - diag (\ Lambda \ Omega \ Lambda (')) ^ (- 1)) - ln (R) - p) </P> <P> where Λ Ω Λ ′ + I − d i a g (Λ Ω Λ ′) (\ displaystyle \ Lambda \ Omega \ Lambda (') + I - diag (\ Lambda \ Omega \ Lambda ('))) is the variance - covariance matrix implied by the proposed factor analysis model and R (\ displaystyle R) is the observed variance - covariance matrix . That is, values are found for freed model parameters that minimizes the difference between the mode - implied variance - covariance matrix and observed variance - covariance matrix . </P> <P> Although numerous algorithms have been used to estimate CFA models, ML remains the primary estimation procedure . That being said, CFA models are often applied to data conditions that deviate from the normal theory requirements for valid ML estimation . For example, social scientists often estimate CFA models with non-normal data and indicators scaled using discrete ordered categories . Accordingly, alternative algorithms have been developed that attend to the diverse data conditions applied researchers encounter . The alternative estimators have been characterized into two general type: (1) robust and (2) limited information estimator . </P>

Goodness of fit indices in confirmatory factor analysis