On meredith's solution for weighted procrustes rotation

Meredith developed a criterion for weighted procrustes rotation. The solution was given using Lagrange multipliers. We show that the solution can be obtained more simply, and that our approach enables us to transform the orthogonalization problem, also considered by Meredith, to that of minimizing a quadratic form on the unit sphere.     

On oblique procrustes rotation

The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years. These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications.In this paper an effective iterative method for obtaining the exact solution is developed. By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown. Newton's method is suggested for solving this equation. Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations.Part of this research was carried out at the National Institute for Personnel Research (South Africa). It was completed while the author was a Visiting Research Psychologist at Educational Testing Service.     

On browne's solution for oblique procrustes rotation

Browne [1967] has given a method of solving the problem (originally stated by Mosier, [1939]) of finding a least squares fit to a specified factor structure. The problem is one of minimizing the sum of squared residuals of —FT with Diag (T'T)=I. Browne's solution involves the eigenvectors and values ofF'F and leads to an iterative solution.This paper gives a form of the solution which does not involve solution of an eigenvalue problem but does require an iteration similar to Browne's. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed which deals with this case. A better starting value for the iteration is also proposed for which convergence is guaranteed using the ordinary Newton iteration.Part of this work was presented at the April 1969 meetings of the Psychometric Society. The anthor is indebted to Dr. Ledyard Tucker for some helpful discussions. This work was supported in part by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health, and Grant No. GM-12868 from the Institute of General Medical Sciences, Public Health Service.     

The varimax criterion for analytic rotation in factor analysis

An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that thenormal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure—although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.Part of the material in this paper is from the writer's Ph.D. thesis. I am indebted to my committee, Professors F. T. Tyler, R. C. Tryon, and H. D. Carter, chairman, for many helpful suggestions and criticisms. Dr. John Caffrey suggested the namevarimax, and wrote the original IBM 602A computer program for this criterion.I am also indebted to the staff of the University of California Computer Center for help in programming the procedures described in the paper for their IBM 701 electronic computer. Since their installation is partially supported by a grant from the National Science Foundation, the assistance of this agency is acknowledged.     

The weighted varimax rotation and the promax rotation

Kaiser's iterative algorithm for the varimax rotation fails when (a) there is a substantial cluster of test vectors near the middle of each bounding hyperplane, leading to non-bounding hyperplanes more heavily overdetermined than those at the boundaries of the configuration of test vectors, and/or (b) there are appreciably more thanm (m factors) tests whose loadings on one of the factors of the initialF-matrix, usually the first, are near-zero, leading to overdetermination of the hyperplane orthogonal to this initialF-axis before rotation. These difficulties are overcome by weighting the test vectors, giving maximum weights to those likely to be near the primary axes, intermediate weights to those likely to be near hyperplanes but not near primary axes, and near-zero weights to those almost collinear with or almost orthogonal to the first initialF-axis. Applications to the Promax rotation are discussed, and it is shown that these procedures solve Thurstone's hitherto intractable “invariant” box problem as well as other more common problems based on real data.     

A solution to the weighted procrustes problem in which the transformation is in agreement with the loss function

This paper provides a generalization of the Procrustes problem in which the errors are weighted from the right, or the left, or both. The solution is achieved by having the orthogonality constraint on the transformation be in agreement with the norm of the least squares criterion. This general principle is discussed and illustrated by the mathematics of the weighted orthogonal Procrustes problem.     

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.     

MMPI factor analytic findings for adolescents: item- and scale-level factor structures

Although numerous studies have examined the factor structures of the Minnesota Multiphasic Personality Inventory (MMPI) in investigations of adults, very little attention has been directed to this issue in adolescent samples. In this study, we investigated the item-level and scale-level factor structures using a sample of 1,762 adolescents receiving psychiatric services at the time of their MMPI assessment. Results identified 17 factors related to item responses and 5 factors based on scale-level data. Findings are discussed in relation to results obtained in adult populations and the potential utility of factor analytically derived content scales for adolescent MMPI interpretation.     

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.     

On antithetic and metathetic words in the analytic situation

Freud's view of primal words emphasized their origins in the vocal acquisition of language and their similarity to primary process behavior in young children. Our English vernacular is rich in word play of reversals and antitheses, some words being truly ancient, others originating in the subversions of slang and in the child's penchant for mirror imaging and other primary process play. In the regression of analytic work the uttered word can combine with dream imagery and behavioral enactment to convey the antitheses abounding in the oral, anal, and early phallic concerns of the individuating child acquiring language.     

On shame,countertransference, and analytic style

Some analysands experience a restricted space in the analytic situation with special counter-transferential consequences. The author discusses how shame is involved in these situations, and projected on to the analyst. This leads to an important choice of direction for the analyst regarding counter-transference acting out or conditions for a real analytic situation. Shame plays a special rôle in these choices of direction. The author illustrates the problem with a clinical vignette and shows how integration of shame is accomplished clinically, and continues with a discussion of the connections between the analyst's analytic style, his own communicative style as a defense against shame and the analytic styles of different analytic “schools”. A discussion of Liberman's concept, of “asymmetrical dialogue” and its connection with countertransference acting out and analytic styles, forms a conclusion to the paper.     

A factor analytic investigation of children's nighttime fear and coping responses

Checklists of children's nighttime fears and nighttime coping responses, completed by 178 8- to 13-year-old children and one of their parents, were factor analyzed. The resulting factors from each checklist were comparable for children and parents. The nighttime fear categories consisted of content around security either for one's personal life, loss, or safety or for others' safety and continued presence; imaginal-numinous concerns; and characteristics inherent in a nighttime situation. The coping categories consisted of responses related to internal self-control, social support, support from inanimate objects, prayer, and avoidance or escape either by controlling the inanimate environment or by controlling others.     

The structure of temperament in preschoolers: a two-stage factor analytic approach

The structure of temperament traits in young children has been the subject of extensive debate, with separate models proposing different trait dimensions. This research has relied almost exclusively on parent-report measures. The present study used an alternative approach, a laboratory observational measure, to explore the structure of temperament in preschoolers. A 2-stage factor analytic approach, exploratory factor analyses (n = 274) followed by confirmatory factor analyses (n = 276), was used. We retrieved an adequately fitting model that consisted of 5 dimensions: Sociability, Positive Affect/Interest, Dysphoria, Fear/Inhibition, and Constraint versus Impulsivity. This solution overlaps with, but is also distinct from, the major models derived from parent-report measures.     

On the measurement of preferences in the analytic hierarchy process

In this paper we apply multiattribute value theory as a framework for examining the use of pairwise comparisons in the analytic hierarchy process (AHP). On one hand our analysis indicates that pairwise comparisons should be understood in terms of preference differences between pairs of alternatives. On the other hand it points out undesirable effects caused by the upper bound and the discretization of any given ratio scale. Both these observations apply equally well to the SMART procedure which also uses estimates of weight ratios. Furthermore, we demonstrate that the AHP can be modified so as to produce results similar to those of multiattribute value measurement; we also propose new balanced scales to improve the sensitivity of the AHP ratio scales. Finally we show that the so-called supermatrix technique does not eliminate the rank reversal phenomenon which can be attributed to the normalizations in the AHP. © 1997 John Wiley & Sons, Ltd.     

Personality differences between intrinsically religious and nonreligious students: a factor analytic study

By means of Allport's Religious Orientation Inventory (ROI) 145 students were classified as intrinsically religious and 133 as nonreligious. Personality differences between the two groups were explored on the basis of their scores on EPPS, the CPI, and 16 PF Questionnaire. Factor analysis of the combined 49 subscales of these three instruments produced eight factors: Achievement Potential, Self-Control, Social Ascendency, Affiliation, Personal and Social Adequacy, Egocentric Sexuality, Restlessness, and Stereotyped Femininity. Analyses of variance of the factor scores of the two groups of subjects showed significant differences on five of the eight factors, accounting for a total of 25% of the total variance. The intrinsically religious scored significantly higher on Self-Control, Personal and Social Adequacy, and Stereotyped Femininity; the nonreligious scored higher on Egocentric Sexuality and Restlessness. These results were discussed in relation to some of Allport's ideas on religion as a dimension of personality.     

A factor analytic and reliability study of the Devereux Child Behavior Rating Scale

The factor structure of the Devereux Child Behavior (DCB) Rating Scale was found to be different in the present study from those reported by the authors of the scale. Five of the 17 DCB factors were too unreliable to include in the present factor analysis. The findings of the present study were strikingly similar with other factor analytic studies of behavior rating scales in that only three main factors were found: Conduct Disorder, Personality Disorder, and Inadequacy-Immaturity.     

Sex differences and the factor of time in solving Vandenberg and Kuse mental rotation problems

In accounting for the well-established sex differences on mental rotation tasks that involve cube stimuli of the Shepard and Metzler (Shepard & Metzler, 1971) kind, performance factors are frequently invoked. Three studies are presented that examine performance factors. In Study 1, analyses of the performance of a large number of subjects (n=1765) that performed the Vandenberg and Kuse (1976) mental rotation test (MRT) under standard conditions showed that the magnitude of the sex differences increases as subjects proceed through the sequence of problems, and that fewer females than males reach the last problems in a problem set. This supports the role of time constraints in sex differences on the MRT. Study 2 compared the magnitude of sex differences for subjects (n=212) who did the MRT under standard conditions (T1), and with double the time (T2) allowed for the test. No significant reduction in the magnitude of sex differences was observed-even though the overall scores under T2 increased markedly for both sexes. Study 3 examined the suggestion by that mental rotation differences arise because females spend more time than males in assuring that stimuli that do not match do indeed not match, with no sex differences for matching stimuli. This hypothesis was not supported for a sample of 20 males and 26 females. We conclude that performance factors may play a role in sex difference on mental rotation tasks, but do not account for all of the differences.