Bi-factor analysis is a form of confirmatory factor analysis originally introduced by Holzinger and Swineford ("Psychometrika" 47:41-54, 1937). The bi-factor model has a general factor, a number of group factors, and an explicit bi-factor structure. Jennrich and Bentler ("Psychometrika" 76:537-549, 2011) introduced an exploratory form of bi-factor…

A Metropolis-Hastings Robbins-Monro (MH-RM) algorithm for high-dimensional maximum marginal likelihood exploratory item factor analysis is proposed. The sequence of estimates from the MH-RM algorithm converges with probability one to the maximum likelihood solution. Details on the computer implementation of this algorithm are provided. The…

The present paper shows that the usual factor analytic structured data dispersion matrix lambda psi lambda' + delta can readily arise from a set of scores y = lambda eta + epsilon, shere the "common" (eta) and "unique" (epsilon) factors have nonzero covariance: gamma = Cov epsilon,eta) is not equal to 0. Implications of this finding are discussed…

Three-mode factor analysis (3MFA) and PARAFAC are methods that describe three-way data. A class of 3MFA models is introduced that falls between 3MFA and PARAFAC and contains the good properties of both approaches, including the unique axes property that has distinguished the PARAFAC model. (SLD)

Proposes an index for assessing the degree of factor simplicity in the context of principal components and exploratory factor analysis. The index does not depend on the scale of the factors, and its maximum and minimum are related only to the degree of simplicity in the loading matrix. (SLD)

Investigated the conditions under which the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. Extends the results to the image factor analysis model. Discusses implications for practice. (SLD)

Gives sufficient and necessary conditions for the observability of factors in terms of the parameter matrices and a finite number of variables. Outlines five conditions that rigorously define indeterminacy and shows that (un)observable factors are (in)determinate, and extends L. Guttman's (1955) proof of indeterminacy to Heywood (H. Heywood, 1931)…

Provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously. Computationally the approach relies on repeated orthomax rotation applied to supermatrices containing the frontal, lateral, or horizontal slabs, respectively. Exemplary analyses illustrate the procedure. (Author/SLD)

A form of Browne's (1967) solution of finding a least squares fit to a specified factor structure is given which does not involve solution of an eigenvalue problem. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed. (Author/RC)

An index of factorial simplicity, employing a quartimax transformational criteria, is developed. This index is both for each row separately and for a factor pattern matrix as a whole. The index varies between zero and one. The problem of calibrating the index is discussed. (Author/RC)

Gives a set of minimally sufficient axioms to define and distinguish common factor theory, image theory, and component theory and analyzes claims that have been made for image theory as a device for improving factor theory. (Author/RC)

Presents a construction method for all factors that satisfy the assumptions of the model for factor analysis, including partially determined factors where certain error variances are zero. Illustrates that variable elimination can have a large effect on the seriousness of factor indeterminacy. (SLD)

Explores the relationship between the higher-order factor model and the hierarchical factor model and shows that the Schmid-Leiman transformation process (J. Schmid and J. Leiman, 1957) produces constrained hierarchical factor solutions. Shows that the two models are not mathematically equivalent unless appropriate direct effects are added. (SLD)

A counterexample is produced to a conjecture by K. G. Joreskog concerning sufficiency conditions for uniqueness of a restricted factor matrix. A substitute condition is stated and proved for the most common situation where the restricted elements are specified to be zero. (Author)

A class of oblique rotation procedures is proposed to rotate a pattern matrix so that it optimally resembles a matrix that has an exact simple pattern. It is demonstrated that the method can recover relatively complex simple structures where other simple structure rotation techniques fail. (SLD)