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…

When the factor analysis model holds, component loadings are linear combinations of factor loadings, and vice versa. This interrelation permits us to define new optimization criteria and estimation methods for exploratory factor analysis. Although this article is primarily conceptual in nature, an illustrative example and a small simulation show…

A multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the Dutch Identity (Holland in "Psychometrika" 55:5-18, 1990). In the model, latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item…

OpenMx is free, full-featured, open source, structural equation modeling (SEM) software. OpenMx runs within the "R" statistical programming environment on Windows, Mac OS-X, and Linux computers. The rationale for developing OpenMx is discussed along with the philosophy behind the user interface. The OpenMx data structures are…

This paper studies changes of standard errors (SE) of the normal-distribution-based maximum likelihood estimates (MLE) for confirmatory factor models as model parameters vary. Using logical analysis, simplified formulas and numerical verification, monotonic relationships between SEs and factor loadings as well as unique variances are found.…

Identifying price sensitive consumers is an important problem in marketing. We develop a Bayesian multi-level factor analytic model of the covariation among household-level price sensitivities across product categories that are substitutes. Based on a multivariate probit model of category incidence, this framework also allows the researcher to…

This paper proposes a semiparametric Bayesian framework for the analysis of associations among multivariate longitudinal categorical variables in high-dimensional data settings. This type of data is frequent, especially in the social and behavioral sciences. A semiparametric hierarchical factor analysis model is developed in which the…

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of "R" factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing…

The association structure between manifest variables arising from the single-factor model is investigated using partial correlations. The additional insights to the practitioner provided by partial correlations for detecting a single-factor model are discussed. The parameter space for the partial correlations is presented, as are the patterns of…

Motivated by Gibbons et al.'s (Appl. Psychol. Meas. 31:4-19, "2007") full-information maximum marginal likelihood item bifactor analysis for polytomous data, and Rijmen, Vansteelandt, and De Boeck's (Psychometrika 73:167-182, "2008") work on constructing computationally efficient estimation algorithms for latent variable…

It is shown that, under very general conditions, uniqueness estimates proposed independently by Guttman (1957) and by Harris (1963) provide tighter upper bounds on the unknown uniqueness values of factor analysis than do existing estimates. (Editor)

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled…

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)

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)

Some three-way factor analysis and multidimensional scaling models incorporate the principle of parallel proportional profiles of R. B. Cattell. Proof is presented for a unique axis orientation for a more general parallel profiles model that incorporates interacting dimensions. Special cases of PARAFAC2 and CANDECOMP models are discussed. (SLD)