Problems in avoiding the singularity problem in analyzing matrices for optimal scaling are addressed. Conditions are given under which the stationary points and values of a ratio of quadratic forms in two singular matrices can be obtained by a series of simple matrix operations. (Author/JKS)

The problem of demonstrating invariance of factor structures across criterion groups is addressed. Procedures are outlined which combine the replication of factor structures across sex-race groups with use of the coefficient of invariance to demonstrate the level of invariance associated with factors identified in a self concept measure.…

Derived formulas for the asymptotic standard errors of component loading estimates to cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. Used the formulas with a real correlation matrix of 355 subjects who took 12 psychological tests. (SLD)

An alternative to the PROMAX exploratory method is presented for constructing a target matrix in Procrustean rotation in factor analysis. A technique is proposed based on vector majorization. The approach is illustrated with several standard numerical examples. (SLD)

Two Statistical Analysis System (SAS) macros are presented that perform the modified principal components approach of L. R. Tucker (1966) to modeling generalized learning curves analysis up to a rotation of the components. Three SAS macros are described that rotate the factor patterns to have characteristics Tucker considered desirable. (SLD)

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…

Various analytic choices in principal components and common factor analysis are discussed. Differences and similarities among these extraction methods are explained, and aids in interpreting the origin of detected effects are explored. Specifically, the nature and use of structure and pattern coefficients are examined. Communalities and methods…

Data reduction techniques seek to combine variables that account for patterns of variation in observed dependent variables in such a way that a simpler model is available for analysis. Factor analysis is a data reduction technique that attempts to model or explain a set of variables in terms of their associations. To understand why this technique…

Most researchers using factor analysis extract factors from a matrix of Pearson product-moment correlation coefficients. A method is presented for extracting factors in a non-parametric way, by extracting factors from a matrix of Spearman rho (rank correlation) coefficients. It is possible to factor analyze a matrix of association such that…

A relatively simple technique for assessing the convergence of sets of variables across method domains is presented. The technique, two-step principal components analysis, empirically orthogonalizes each method domain into sets of components, and then analyzes convergence among components across domains. (Author)

A method is proposed and illustrated for estimating the degree to which a factor rotation to a hypothesized target represents an improvement over rotation to a random target. (Author)

The problem of demonstrating replicability of factor structure across random variables is addressed. Procedures are outlined which combine the use of random subsample replication strategies with the correlations between factor score estimates across replicate pairs to generate a coefficient of replicability and confidence intervals associated with…

Results showed that the canonical correlations are very stable upon replication. The results also indicated that there is no solid evidence for concluding that components are superior to the coefficients, at least not in terms of being more reliable. (Author/BJG)

The distributional characteristics of the Kaiser-Rice measure of sampling adequacy (MSA) were investigated with sample correlation matrices from multivariate normal populations where the level of correlation (LC) was systematically varied. Two additional variables were manipulated--sample size (SS) and number of variables (NV). Ten matrices were…