Maximally correlated orthogonal composites and oblique factor analytic solutions

Kaiser presented a method for finding a set of derived orthogonal variables which correlate maximally with a set of original variables. A simpler, more complete derivation of Kaiser's result is given and compared to related types of transformations. The transformation derived here suggests a direct method for finding the orthogonal factor solution which is maximally similar to a given oblique solution.     

Hierarchical factor solutions without rotation

A method is presented for securing a hierarchical factor solution which achieves simple structure at each hierarchical level without rotation or even preliminary arbitrary orthogonal or oblique solutions. The method is based upon the assumption that if overlap is removed from clusters the remaining specifics will achieve simple structure automatically. The problem presented earlier by Schmid and Leiman, using oblique simple structural rotation as a basis, is reworked by this new approach.     

A general approach to confirmatory maximum likelihood factor analysis

We describe a general procedure by which any number of parameters of the factor analytic model can be held fixed at any values and the remaining free parameters estimated by the maximum likelihood method. The generality of the approach makes it possible to deal with all kinds of solutions: orthogonal, oblique and various mixtures of these. By choosing the fixed parameters appropriately, factors can be defined to have desired properties and make subsequent rotation unnecessary. The goodness of fit of the maximum likelihood solution under the hypothesis represented by the fixed parameters is tested by a large samplex ² test based on the likelihood ratio technique. A by-product of the procedure is an estimate of the variance-covariance matrix of the estimated parameters. From this, approximate confidence intervals for the parameters can be obtained. Several examples illustrating the usefulness of the procedure are given.This work was supported by a grant (NSF-GB 1985) from the National Science Foundation to Educational Testing Service.     

Two-matrix orthogonal rotation procedures

Examples are presented in which it is either desirable or necessary to transform two sets of orthogonal axes to simple structure positions by means of the same transformation matrix. A solution is then outlined which represents a two-matrix extension of the general orthomax orthogonal rotation criterion. In certain circumstances, oblique two-matrix solutions are possible using the procedure outlined and the Harris-Kaiser [1964] logic. Finally, an illustrative example is presented in which the preceding technique is applied in the context of an inter-battery factor analysis.The work reported herein was supported by Grant S72-1886 from the Canada Council. The author acknowledges the helpful contributions of Nancy Reid and Lawrence Ward to parts of this paper.     

General resolution of correlation matrices into components and its utilization in multiple and partial regression

The derivation of multiple and partial regression statistics from uniqueness-augmented factor loadings, presented in the literature for orthogonal factor solutions, is generalized to oblique solutions. A mathematical rationale for the general case, without restriction to uncorrelated factors, is presented. Use of the general formulation is illustrated with a two-factor, seven-variable example.This report is based on work done under ARDC Project 7702, in support of the research and development program of the Air Force Personnel and Training Research Center, Lackland Air Force Base, Texas. Permission is granted for reproduction, translation, publication, use, and disposal in whole and in part by or for the United States Government.     

A General Approach for Fitting Pure Exploratory Bifactor Models

This article proposes a procedure for fitting a pure exploratory bifactor solution in which the general factor is orthogonal to the group factors, but the loadings on the group factors can satisfy any orthogonal or oblique rotation criterion. The proposal combines orthogonal Procrustes rotations with analytical rotations and consists of a sequence of four steps. The basic input is a semispecified target matrix that can be (a) defined by the user, (b) obtained by using Schmid-Leiman orthogonalization, or (c) automatically built from a conventional unrestricted solution based on a prescribed number of factors. The relevance of the proposal and its advantages over existing procedures is discussed and assessed via simulation. Its feasibility in practice is illustrated with two empirical examples in the personality domain.     

NFACOM: A new program for relating solutions in exploratory factor analysis

A new version of the program FACOM (comparison of exploratory factor analysis solutions) is presented. The new version (NFACOM) greatly extends the possibilities of the previous versions. It is now possible to use congruence as well as least squares comparisons in both orthogonal and oblique situations. What is more, n solutions, not just two, can be compared. Recently developed algorithms have been implemented; others, which have proved not to be theoretically well founded, have been omitted.     

Exploratory Bi-factor Analysis: The Oblique Case

Bi-factor analysis is a form of confirmatory factor analysis originally introduced by Holzinger and Swineford (Psychometrika 47:41?C54, 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?C549, 2011) introduced an exploratory form of bi-factor analysis that does not require one to provide an explicit bi-factor structure a priori. They use exploratory factor analysis and a bifactor rotation criterion designed to produce a rotated loading matrix that has an approximate bi-factor structure. Among other things this can be used as an aid in finding an explicit bi-factor structure for use in a confirmatory bi-factor analysis. They considered only orthogonal rotation. The purpose of this paper is to consider oblique rotation and to compare it to orthogonal rotation. Because there are many more oblique rotations of an initial loading matrix than orthogonal rotations, one expects the oblique results to approximate a bi-factor structure better than orthogonal rotations and this is indeed the case. A surprising result arises when oblique bi-factor rotation methods are applied to ideal data.     

On the least-squares orthogonalization of an oblique transformation

After proving a special case of a theorem stated by Eckart and Young, namely, that an oblique transformationG is the product of two different orthogonal transformations and an intervening diagonal, this note shows that the best fitting orthogonal approximation toG is obtained simply by replacing the intervening diagonal by the identity matrix. This result is shown to be identical with two earlier orthogonalizing procedures whenG is of full rank. A multiplicity of solutions is shown for the case of a singularG.I am grateful to J. J. Mellinger for pointing out a flaw in a previous version of this paper.Opinions expressed herein are those of the author, not necessarily those of the Army.     

Confirmatory factor analysis of the Kaufman Adolescent and Adult Intelligence Test with young adolescents

This study examined the factor structure of the Kaufman Adolescent and Adult Intelligence Test (KAIT) in 375 11- to 14-year-olds (normative sample) and 60 sixth- and eighth-graders (cross-validation sample). Previous exploratory and confirmatory factor analyses of the KAIT have included optional subtests, which are often not administered in practice, and have not tested oblique versus orthogonal two-factor models. In this study, three factor models were tested via confirmatory factor analysis for the six core subtests of the KAIT: a one-factor general intelligence (g) model, an orthogonal fluid intelligence (Gf)-crystallized intelligence (Gc) model, and an oblique Gf-Gc model. The orthogonal Gf-Gc model fit poorly in both samples. The g model fit only the cross-validation sample. The oblique Gf-Gc model fit both samples and fit significantly better than the g model in both samples. Additional tests of factor structure, path coefficients, and covariance between Gf and Gc indicated that the data from both samples fit even the most restrictive oblique Gf-Gc model tested. KAIT users can be confident that the Fluid and Crystallized IQs from the KAIT are adequate representations of a robust and interpretable factor structure when only the core subtests are administered to young adolescents.     

The development of hierarchical factor solutions

Although simple structure has proved to be a valuable principle for rotation of axes in factor analysis, an oblique factor solution often tends to confound the resulting interpretation. A model is presented here which transforms the oblique factor solution so as to preserve simple structure and, in addition, to provide orthogonal reference axes. Furthermore, this model makes explicit the hierarchical ordering of factors above the first-order domain.Grateful acknowledgment is given to Dr. Lloyd G. Humphreys for his encouragement and valuable suggestions in the development of this task. This investigation was carried out under the Air Force Personnel and Training Research Center program in support of Project Nos. 7702 and 7950. Permission is granted for reproduction, translation, publication, and use or disposal in whole or in part by or for the United States Government.     

Factoring test scores and implications for the method of averages

The general procedure and detailed steps for attaining complete factor analyses of scores are presented. Both orthogonal and oblique factors are considered. It is shown that a single average by conventional procedure gives an incomplete summarization of the data when the rank exceeds one. There should be as many averages as there are common factors.     

Standard errors for procrustes solutions

The asymptotic standard errors for the procrustes solutions are derived for orthogonal rotation, direct oblique rotation and indirect oblique rotation. The standard errors for the first two rotations are obtained using the augmented information matrices. For the indirect oblique solution, the standard errors of rotated parameters are derived from the information matrix of unrotated loadings using the chain rule for information matrices. For all three types of rotation, the standard errors of rotated parameters are presented for unstandardized and standardized manifest variables. Numerical examples show the similarity of theoretical and simulated values.     

FACOM: A library for relating solutions obtained in exploratory factor analysis

A library for comparing factorial solutions obtained in exploratory analyses is presented. The procedure allows both orthogonal and oblique solutions to be compared by computing several indices. The library is controlled with interactive menus or programs.     

Orthogonal rotation to congruence

Two problems are considered. The first is that of rotating two factor solutions orthogonally to a position where corresponding factors are as similar as possible. A least-squares solution for transformations of the two factor matrices is developed. The second problem is that of rotating a factor matrix orthogonally to a specified target matrix. The solution to the second problem is related to the first. Applications are discussed.This research was supported in part by contract Nonr 228 (22) between the office of Naval Research and the University of Southern California. Portions of this paper were presented at the American Psychological Association Convention, Los Angeles, September, 1964.     

Quartic rotation criteria and algorithms

Most of the currently used analytic rotation criteria for simple structure in factor analysis are summarized and identified as members of a general symmetric family of quartic criteria. A unified development of algorithms for orthogonal and direct oblique rotation using arbitrary criteria from this family is given. These algorithms represent fairly straightforward extensions of present methodology, and appear to be the best methods currently available.The research done by R. I. Jennrich was supported by NSF Grant MCS-8301587.     

The orthogonal approximation of an oblique structure in factor analysis

A procedure is derived for obtaining an orthogonal transformation which most nearly transforms one given matrix into another given matrix, according to some least-squares criterion of fit. From this procedure, three analytic methods are derived for obtaining an orthogonal factor matrix which closely approximates a given oblique factor matrix. The case is considered of approximating a specified subset of oblique vectors by orthogonal vectors.Part of this research was carried out while the author was a psychometric fellow at the Educational Testing Service, Princeton, New Jersey.     

Cognitive constraints on ordering operations: the case of geometric analogies

Many tasks (e.g., solving algebraic equations and running errands) require the execution of several component processes in an unconstrained order. The research reported here uses the geometric analogy task as a paradigm case for studying the ordering of component processes in this type of task. In solving geometric analogies by applying mental transformations such as rotate, change size, and add a part, the order of performing the transformations is unconstrained and does not in principle affect solution accuracy. Nevertheless, solvers may bring cognitive constraints with them to the analogy task that influence the ordering of the transformations. First, we demonstrate that solvers have a preferred order for performing mental transformations during analogy solution. We then investigate three classes of explanations for the preferred order, one based on general information processing considerations, another based on task-specific considerations, and a third based on individual differences in analogy ability. In the first and third experiments, college students solved geometric analogies requiring two or three transformations and indicated the order in which they performed the transformations. There was close agreement on nearly the same order for both types of analogies. In the second experiment, subjects were directed to perform pairs of transformations in the preferred or unpreferred order. Both speed and accuracy were greater for the preferred orders, thus validating subjects' reported orders. Ability differences were observed for only the more difficult three-transformation problems: High- and middle-ability subjects agreed on an overall performance order, but the highs were more consistent in their use of this order. Low-ability subjects did not consistently order the transformations for these difficult problems. The general information processing factor examined was working-memory load. A number of task factors have been shown to affect working-memory load during the solution of inductive reasoning problems. Of these, we chose to examine process difficulty. Because analogies are solved in working memory, performing more difficult transformations earlier may reduce working-memory load and facilitate problem solution. However, the observed performance order was not correlated with transformation difficulty. The first task-specific factor considered was that some transformations may be identified earlier, possibly because of perceptual salience, and that the performance order follows the identification order.(ABSTRACT TRUNCATED AT 400 WORDS)     

Higher-order exploratory factor analysis of the Reynolds Intellectual Assessment Scales with a referred sample

The factor structure of the Reynolds Intellectual Assessment Scales (RIAS; [Reynolds, C.R., & Kamphaus, R.W. (2003). Reynolds Intellectual Assessment Scales. Lutz, FL: Psychological Assessment Resources, Inc.]) was investigated with a large (N = 1163) independent sample of referred students (ages 6-18). More rigorous factor extraction criteria (viz., Horn's parallel analysis (HPA); [Horn, J.L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.], and Minimum Average Partial (MAP) analysis; [Velicer, W.F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327.]), in addition to those used in RIAS development, were investigated. Exploratory factor analyses using both orthogonal and oblique rotations and higher-order exploratory factor analyses using the Schmid and Leiman [Schmid, J., and Leiman, J.M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53-61.] procedure were conducted. All factor extraction criteria indicated extraction of only one factor. Oblique rotations resulted in different results than orthogonal rotations, and higher-order factor analysis indicated the largest amount of variance was accounted for by the general intelligence factor. The proposed three-factor solution was not supported. Implications for the use of the RIAS with similarly referred students are discussed.     

Orthogonal inter-battery factor analysis

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.This investigation has been supported by the U.S. Office of Naval Research under Contract Nonr-2752(00).