The issue of measurement invariance commonly arises in factor-analytic contexts, with methods for assessment including likelihood ratio tests, Lagrange multiplier tests, and Wald tests. These tests all require advance definition of the number of groups, group membership, and offending model parameters. In this paper, we study tests of measurement…

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…

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 method proposed by Harman and Fukuda to treat the so-called Heywood case in the minres method in factor analysis (i.e., the case where the resulting communalities are greater than one) involves the frequent solution of eigenvalue problems. A simple method to treat this problem is presented. (Author/JKS)

Some mathematical aspects of minimum trace factor analysis (MTFA) are discussed. The uniqueness of an optimal point of MTFA is proved, and necessary and sufficient conditions for any particular point to be optimal are given. The connection between MTFA and classical minimum rank factor analysis is discussed. (Author/JKS)

A model for four-mode component analysis is developed and presented. The developed model, which is an extension of Tucker's three-mode factor analytic model, allows for the simultaneous analysis of all modes of a four-mode data matrix and the consideration of relationships among the modes. (Author/JKS)

It is proved for the common factor model that, under certain conditions maintaining the distinctiveness of each factor, a given factor will be determinate if there exists an unlimited number of variables in the model, each having an absolute correlation with the factor greater than some arbitrarily small quantity. (Author/JKS)

Factor analysis programs in SAS, BMDP, and SPSS are discussed and compared in terms of documentation, methods and options available, internal logic, computational accuracy, and results provided. Based on these comparisons, preferences for SAS and against SPSS factor analysis programs are recommended. (Author/JKS)

The extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries is discussed. Extension of this work to minimum trace factor analysis is presented. (Author/JKS)

Canonical correlation and redundancy analysis are two approaches to analyzing the interrelationships between two sets of measurements made on the same variables. A component method is presented which uses aspects of both approaches. An empirical example is also presented. (Author/JKS)

A rigorous definition for a factor analysis model and a complete solution of the factor score indeterminacy problem are presented in this technical paper. The meaning and application of these results are discussed. (Author/JKS)

A simple roof is presented: that the squared multiple correlation of a variable with the remaining variables in the set of variables is a lower bound to the communality of that variable. (Author/JKS)

A new method is proposed for a simultaneous factor analysis of dichotomous responses from several groups of individuals. The method makes it possible to compare factor loading pattern, factor variances and covariances, and factor means over groups. Generalized least squares is used as the estimation procedure. (Author/JKS)

A jackknife-like procedure is developed for producing standard errors of estimate in maximum likelihood factor analysis. Unlike earlier methods based on information theory, the procedure developed is computationally feasible on larger problems. Examples are given to demonstrate the feasibility of the method. (Author/JKS)

This note proposes an alternative implementation of the regression method which should be slightly faster than the principal components methods for estimating missing data. (RC)