On the oblique rotation of a factor matrix to a specified pattern

This paper presents a procedure for rotating an arbitrary factor matrix to maximum similarity with a specified factor pattern. The sum of squared distances between specified vectors and rotated vectors in oblique Euclidian space is minimized. An example of the application of the procedure is given.This research was supported in part by the National Institute of Child Health and Human Development, Research Grant 1 PO1 HDO1762.The names of the authors are given in alphabetical order. Their contributions to the paper are equal.     

Properties and applications of gramian factoring

The Gramian factorizationG of a GramianR is square and symmetric and has no negative characteristic roots. It is shown to be that square factorization that is, in the least-squares sense, most isomorphic toR, most like a scalarK, and most highly traced, and to be the necessary and sufficient relation between the oblique vectors of an oblique transformation and the orthogonal vectors of the least-squares orthogonal counterpart. A slightly modified Gramian factorization is shown to be the factorization that is most isomorphic to a specified diagonalD, and to be the main part of an iterative procedure for obtaining simplimax, a square factor matrix with simple structure maximized in the sense of having the largest sum of squared diagonal loadings. Several published applications of Gramian factoring are cited.     

Oblique factor analytic solutions by orthogonal transformations

A general framework for obtaining all possible factor analytic solutions, orthogonal and oblique, for a given common factor space is developed in detail. Interestingly, and seemingly paradoxically, any one of these solutions may be obtained by orthogonal transformations of selected matrices; thus an oblique solution may be determined by orthogonal transformations. Within the possible oblique solutions, two distinct categories of solutions emerge, a special case of the simpler of which apparently provides a definitive solution to the problem of independent, but correlated, clusters. Possible further specializations of the general approach to specific problems are discussed.     

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.     

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).     

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.     

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.     

Standard errors for obliquely rotated factor loadings

In a manner similar to that used in the orthogonal case, formulas for the aymptotic standard errors of analytically rotated oblique factor loading estimates are obtained. This is done by finding expressions for the partial derivatives of an oblique rotation algorithm and using previously derived results for unrotated loadings. These include the results of Lawley for maximum likelihood factor analysis and those of Girshick for principal components analysis. Details are given in cases including direct oblimin and direct Crawford-Ferguson rotation. Numerical results for an example involving maximum likelihood estimation with direct quartimin rotation are presented. They include simultaneous tests for significant loading estimates.This research was supported in part by NIH Grant RR-3. The author is indebted to Dorothy Thayer who implemented the algorithms required for the example and to Gunnar Gruvaeus and Allen Yates for reviewing an earlier version of this paper. Special thanks are extended to Michael Browne for many conversations devoted to clarifying the thoughts of the author.     

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.     

The MacQuarrie test for mechanical ability

Data originally analyzed by Charles H. Goodman on the MacQuarrie Test for Mechanical Ability are subjected to the principal axes factoring method. The maximum variance was extracted with three factors. Rotation to an oblique simple structure yielded a factor pattern which satisfies the simple structure concept more adequately than the orthogonal factor matrix, thus leading to greater clarity of interpretation of the factors.     

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.     

A counterexample to two-dimensional varimax-rotation

Usually, an iterative procedure based on two-dimensional rotations is employed to find the varimax solution in factor analysis. A matrix is given where this procedure does not yield the maximum value of the varimax criterion. However, random orthogonal transformations of some matrices and subsequent varimax-rotation using the iterative procedure seem to indicate that usually no local maxima exist.     

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.     

On the symmetric treatment of an asymmetric approach to factor analysis

On the assumption that a partitioning can be found such that three mutually exclusive test vector configurations span the same factor space, a procedure is derived whereby symmetric parts of the correlation matrix are estimated from functions of asymmetric parts treated symmetrically. This yields an explicit matrix formula for communality estimation which generalizes earlier work by Albert. Conventional factoring methods, with all their computational and fitting advantages, can be applied once the symmetric portions of the correlation matrix have been estimated. Extension to four subgroups of test vectors allows for a matrix generalization of the old tetrad difference criterion to the multiple-factor case.     

Asymptotic sampling variances for factor analytic models identified by specified zero parameters

Formulas are derived for the asymptotic variances and covariances of the maximum likelihood estimators for oblique simple structure models which are identified by prior specification of zero elements in the factor loading matrix. The formulas are expressed in terms of the various submatrices of the inverse of the required variance-covariance matrix. A numerical example using artificial data is given and problems in the application of the formulas discussed.Now at The Pennsylvania State University.     

A matrix formulation of Kaiser's varimax criterion

Kaiser has given the varimax criterion for the solution of the rotation problem in factor analysis as well as a practical computational procedure for maximizing this criterion. In the present paper, the maximization condition is shown as a matrix equation involving only the unknown orthogonal rotation matrix. This matrix equation can be solved iteratively as a sequence of symmetric eigenproblems.This investigation was supported by Public Health Service grant number MH 07285-03 from the National Institute of Mental Health.     

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.     

A feasible method for standard errors of estimate in maximum likelihood factor analysis

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. Unlike earlier methods based on the jackknife, the present procedure is not plagued by the factor alignment problem, the Heywood case problem, or the necessity to jackknife by groups. Standard errors may be produced for rotated and unrotated loading estimates using either orthogonal or oblique rotation as well as for estimates of unique factor variances and common factor correlations. The total cost for larger problems is a small multiple of the square of the number of variables times the number of observations used in the analysis. Examples are given to demonstrate the feasibility of the method.The research done by R. I. Jennrich was supported in part by NSF Grant MCS 77-02121. The research done by D. B. Clarkson was supported in part by NSERC Grant A3109.     

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.     

Some relationships between factors and components

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.The author is indebted to anonymous referees for their comments, corrections and suggestions which have led to the improvement of this article.