Varisim: A new machine method for orthogonal rotation

The derivative of Kaiser's Varimax criterion, if set to zero, yields a set of equations which are quite similar to those obtained for a least-squares problem of the Procrustes type. This similarity suggested an iterative technique for orthogonal rotation, dubbed Varisim, which was programmed for the IBM 7094 in FORTRAN. An empirical comparison between Varimax and Varisim, which was based on a number of data sets taken from the literature yielded three major results so far: (i) Varisim is slower than Varimax, roughly by a factor of 3, (ii) Varisim yields factors which in general contribute more evenly to the common test variance than Varimax factors, and which (iii) line up more closely with oblique configurations obtained with Binormamin than Varimax factors.This paper is based on parts of a thesis submitted to the Graduate College of the University of Illinois in partial fulfillment of the requirements for a degree in Psychology. Some of its contents were read at the joint meeting of the Psychonomic and the Psychometric Society, Niagara Falls, Ontario, 1964. Most of the computer-bound work was carried out while the author was employed by the Statistical Service Unit, University of Illinois; it was written up while the author held a Postdoctoral Fellowship at the Psychometric Laboratory of the University of North Carolina. Although space limitations forbid more complete acknowledgments, I am bound to record my deep gratitude to my advisor, Prof. H. F. Kaiser.     

A simple graphical method for orthogonal rotation of axes

A simple method of making orthogonal rotations by projecting coordinates from one plot to another is presented. An easily constructed apparatus for rapid and accurate work is described in detail. By the technique proposed 12 to 15 rotations of pairs of axes involving 20 test variables can be completed per hour by a practiced technician.     

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 comparative study in multiple-factor analysis of neurotic tendency

The original aim of this investigation was to discover what relations exist between race attitudes and certain personality traits. For that purpose, a standard neurotic inventory and an attitude scale were applied to three dissimilar groups of subjects, and the results, obtained by multiple-factor analysis carried out independently for each of the three groups, compared with one another. The components of neurotic tendency, as displayed by each of the three groups, show a high measure of agreement and appear to be reconcilable with other interpretations of neurotic personality. So far as the original aim of the study is concerned, there appears to be a slight but significant tendency for the neurotic factor of hypersensitiveness to affect the race attitudes of Europeans towards the native in South Africa in a negative or unfavorable direction.     

A general rotation criterion and its use in orthogonal rotation

Measures of test parsimony and factor parsimony are defined. Minimizing their weighted sum produces a general rotation criterion for either oblique or orthogonal rotation. The quartimax, varimax and equamax criteria are special cases of the expression. Two new criteria are developed. One of these, the parsimax criterion, apparently gives excellent results. It is argued that one of the most important factors bearing on the choice of a rotation criterion for a particular problem is the amount of information available on the number of factors that should be rotated. This research was supported by the National Research Council of Canada research grant 291-13 to Dr. G. A. Ferguson.     

The rigid orthogonal Procrustes rotation problem

The problem of rotating a matrix orthogonally to a best least squares fit with another matrix of the same order has a closed-form solution based on a singular value decomposition. The optimal rotation matrix is not necessarily rigid, but may also involve a reflection. In some applications, only rigid rotations are permitted. Gower (1976) has proposed a method for suppressing reflections in cases where that is necessary. This paper proves that Gower’s solution does indeed give the best least squares fit over rigid rotation when the unconstrained solution is not rigid. Also, special cases that have multiple solutions are discussed. The author is obliged to Henk Kiers and Alwin Stegeman for helpful comments on a previous draft.     

On the rank of the reduced correlational matrix in multiple-factor analysis

Since the factor problem is reduced mathematically to the expression of the obtained correlational matrix in terms of a matrix of lower rank, criteria for the determination of this lower rank are of first importance. These criteria are investigated by means of certain mathematical deductions, and brought into relation with Spearman's and Thurstone's factor theories.     

Powered vector method of factor analysis

The powered vector method of factor analysis yields directly and without rotation factors satisfying objectives of parsimony, orthogonality, and meaningfulness. The method is objective, computationally efficient, and easily programmed for digital computers. The computational procedure is described. Illustrative analyses are presented. Results of applications of the powered vector method are compared with results obtained using the principal axes solution followed by orthogonal rotation.     

A monotonically convergent algorithm for orthogonal congruence rotation

Brokken has proposed a method for orthogonal rotation of one matrix such that its columns have a maximal sum of congruences with the columns of a target matrix. This method employs an algorithm for which convergence from every starting point is not guaranteed. In the present paper, an iterative majorization algorithm is proposed which is guaranteed to converge from every starting point. Specifically, it is proven that the function value converges monotonically, and that the difference between subsequent iterates converges to zero. In addition to the better convergence properties, another advantage of the present algorithm over Brokken's one is that it is easier to program. The algorithms are compared on 80 simulated data sets, and it turned out that the new algorithm performed well in all cases, whereas Brokken's algorithm failed in almost half the cases. The derivation of the algorithm is given in full detail because it involves a series of inequalities that can be of use to derive similar algorithms in different contexts.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the first author. The authors are obliged to Willem J. Heiser and Jos M. F. ten Berge for useful comments on an earlier version of this paper.     

Some comparisons of the multiple-factor and the bi-factor methods of analysis

Bi-factor and multiple-factor analyses of the same data are compared in two respects. First, two criteria are suggested for determining when the factorization is adequate. This problem being more acute for the centroid method than for the bi-factor method, the latter is used primarily for comparison only. It is shown also that the omission from the simple structure of entries smaller than .10 yields a pattern which is a poorer fit to the original correlations than is the bi-factor pattern. Second, the second-order general factor obtained from the intercorrelations of the primaries is found to be highly correlated with the general factor of the bi-factor pattern.     

Kernel principal component analysis combining rotation forest method for linearly inseparable data

Rotation forest (RoF) is an ensemble classifier combining linear analysis theories and decision tree algorithms. In recent existing works, RoF was widely applied to various fields with outstanding performance compared to traditional machine learning techniques, given that a reasonable number of base classifiers is provided. However, the conventional RoF algorithm suffers from classifying linearly inseparable datasets. In this study, a hybrid algorithm integrating kernel principal component analysis (KPCA) and the conventional RoF algorithm is proposed to overcome the classification difficulty for linearly inseparable datasets. The radial basis function (RBF) is selected as the kernel for the KPCA method to establish the nonlinear mapping for linearly inseparable data. Moreover, we evaluate various kernel parameters for better performance. Experimental results show that our algorithm improves the performance of RoF with linearly inseparable datasets, and therefore provides higher classification accuracy rates compared with other ensemble machine learning methods.     

On the equivalence of two oblique congruence rotation methods,and orthogonal approximations

Tucker's method of oblique congruence rotation is shown to be equivalent to a procedure by Meredith. This implies that Monte Carlo studies on congruence by Nesselroade, Baltes and Labouvie and by Korth and Tucker are highly comparable. The problem of rotating two matrices orthogonally to maximal congruence has not yet been solved. An approximate solution to this problem can be derived from Tucker's method. Even better results can be obtained from a Procrustes rotation followed by rotation to simple structure.     

Confirmatory factor analysis of single- and multiple-factor competing models of the dissociative experiences scale in a nonclinical sample

Previous research on the Dissociative Experiences Scale (DES) has demonstrated that (a) dissociation is quantifiable in both clinical and nonclinical samples and (b) a three-factor structure (amnesia, depersonalization, and absorption) is tenable for clinical samples. The factor structurefor nonclinical samples is less clear, with one- and multiple-factor solutions proposed. To clarify the DESfactor structure in nonclinical samples, confirmatory factor analyses were conducted on (a) one-, two-, three-, and four-factorfirst-order models and (b) two bifactor (hierarchical) models of DES scoresfor two samples of nonclinical university students. Results of delta(chi2) and goodness-of-fit indices support the three-factor (first-order) model as bestfitting of the datafor these samples. The utility of this DES model for screening both dissociative pathology and elevated normal dissociative behavior in clinical and nonclinical populations is discussed.     

The simultaneous maximization of congruence for two or more matrices under orthogonal rotation

While a rotation procedure currently exists to maximize simultaneously Tucker's coefficient of congruence between corresponding factors of two factor matrices under orthogonal rotation of one factor matrix, only approximate solutions are known for the generalized case where two or more matrices are rotated. A generalization and modification of the existing rotation procedure to simultaneously maximize the congruence is described. An example using four data matrices, comparing the generalized congruence maximization procedure with alternative rotation procedures, is presented. The results show a marked improvement of the obtained congruence using the generalized congruence maximization procedure compared to other procedures, without a significant loss of success with respect to the least squares criterion. A computer program written by the author to perform the rotations is briefly discussed.     

A single plane method of rotation

The method of rotation described here is applied to one hyperplane at a time. The method seems to be simple and quite effective and it can be applied by a relatively inexperienced computer. The method does not postulate a positive manifold, and hence it is applicable also to bipolar factors.The author wishes to express his indebtedness to the Social Science Research Committee of The University of Chicago for support of the Psychometric Laboratory and these factorial investigations.