The weighted oblimin rotation

Cureton & Mulaik (1975) proposed the Weighted Varimax rotation so that Varimax (Kaiser, 1958) could reach simple solutions when the complexities of the variables in the solution are larger than one. In the present paper the weighting procedure proposed by Cureton & Mulaik (1975) is applied to Direct Oblimin (Clarkson & Jennrich, 1988), and the rotation method obtained is called Weighted Oblimin. It has been tested on artificial complex data and real data, and the results seem to indicate that, even though Direct Oblimin rotation fails when applied to complex data, Weighted Oblimin gives good results if a variable with complexity one can be found for each factor in the pattern. Although the weighting procedure proposed by Cureton & Mulaik is based on Landahl's (1938) expression for orthogonal factors, Weighted Oblimin seems to be adequate even with highly oblique factors. The new rotation method was compared to other rotation methods based on the same weighting procedure and, whenever a variable with complexity one could be found for each factor in the pattern, Weighted Oblimin gave the best results. When rotating a simple empirical loading matrix, Weighted Oblimin seemed to slightly increase the performance of Direct Oblimin.The author is obliged to Henk A. L. Kiers and three anonymous reviewers for helpful comments on an earlier version of this paper.     

A simple general method for oblique rotation

A simple and very general algorithm for oblique rotation is identified. While motivated by the rotation problem in factor analysis, it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length. The algorithm has two steps. The first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns. For this reason it is called a gradient projection algorithm. Because the projection step is very simple, implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple. It is proven that the algorithm is strictly monotone, that is as long as it is not already at a stationary point, each step will decrease the value of the criterion. Examples from a variety of areas are used to demonstrate the algorithm, including oblimin rotation, target rotation, simplimax rotation, and rotation to similarity and simplicity simultaneously. While it may be, the algorithm is not intended for use as a standard algorithm for well established problems, but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort.The author would like to thank the review team for their insights and recommendations.     

Techniques for rotating two or more loading matrices to optimal agreement and simple structure: A comparison and some technical details

Matrices of factor loadings are often rotated to simple structure. When more than one loading matrix is available for the same variables, the loading matrices can be compared after rotating them all (separately) to simple structure. An alternative procedure is to rotate them to optimal agreement, and then compare them. In the present paper techniques are described that combine these two procedures. Specifically, five techniques that combine the ideals of rotation to optimal agreement and rotation to simple structure are compared on the basis of contrived and empirical data. For the contrived data, it is assessed to what extent the rotations recover the underlying common structure. For both the contrived and the empirical data it is studied to what extent the techniques give well matching rotated matrices, to what extent these have a simple structure, and to what extent the most prominent parts of the different loading matrices agree. It was found that the simple procedure of combining a Generalized Procrustes Analysis (GPA) with Varimax on the mean of the matched loading matrices performs very well on all criteria, and, for most purposes, offers an attractive compromise of rotation to agreement and simple structure. In addition to this comparison, some technical improvements are proposed for Bloxom's rotation to simple structure and maximum similarity.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to René van der Heijden for assistance in programming the procedures in the simulation study reported in this paper, and to Jos ten Berge, three anonymous reviewers and an associate editor for helpful comments on an earlier version of this paper.     

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.     

Orthogonal rotations to maximal agreement for two or more matrices of different column orders

Methods for orthogonal Procrustes rotation and orthogonal rotation to a maximal sum of inner products are examined for the case when the matrices involved have different numbers of columns. An inner product solution offered by Cliff is generalized to the case of more than two matrices. A nonrandom start for a Procrustes solution suggested by Green and Gower is shown to give better results than a random start. The Green-Gower Procrustes solution (with nonrandom start) is generalized to the case of more than two matrices. Simulation studies indicate that both the generalized inner product solution and the generalized Procrustes solution tend to attain their global optima within acceptable computation times. A simple procedure is offered for approximating simple structure for the rotated matrices without affecting either the Procrustes or the inner product criterion.The authors are obliged to Charles Lewis for helpful comments on a previous draft of this paper and to Frank Brokken for preparing a computer program that was used in this study.     

Adaptation to visuomotor rotations in younger and older adults

Adaptation to a visuomotor rotation is known to be impaired at older adult age. The authors examined whether the impairment is present already at preretirement age and whether it depends on the difficulty of the adaptation task. Moreover, the authors tested predictions of the hypothesis that the age-related impairment pertains primarily to strategic corrections and the explicit knowledge on which they are based but not to the acquisition of an (implicit) internal model of the novel visuomotor transformation. The authors found an age-related impairment of adaptation and explicit knowledge already at preretirement age but no age-related change of aftereffects. With an incremental simplification of the adaptation task, age-related changes were able to be eliminated. Individual differences of the quality of explicit knowledge were associated with differences of adaptation, but not of aftereffects. When age groups were matched by explicit knowledge, age-related impairments of adaptation largely disappeared. However, a reliable difference remained in one of the experiments, suggesting that other processes of adjustment to a visuomotor rotation might be affected by aging as well.     

Continuity of personality patterning from 3 to 15 years in a longitudinal sample

A longitudinal sample of 97 subjects was rated at 3, 7, 11 and 15 years on variables selected as indices of Extraversion-Introversion (EI) and Neuroticism (N) or emotionality. Principal factor analysis and Varimax rotation were performed at each age. Continuity of patterning of the variables across the four ages was determined before, and after, rotation of the items to maximal contiguity by Kaiser's method. Two principal factors, clearly identifiable as EI and N, displayed very high continuity, as did 45 rotated factors, according to Gray's model. Four Varimax factors, which initially appeared dissimilar at different ages, were shown to display high continuity, in terms of loadings on standard reference axes, after alignment of the item vectors. Thus the inadequacy of simply comparing factors as they happen to emerge, was clearly demonstrated.     

A direct solution for pairwise rotations in Kaiser's varimax method

The present note contains a completing-the-squares type approach to the varimax rotation problem. This approach yields a direct proof of global optimality of a solution for optimal rotation in a plane. Because varimax rotation can be interpreted as diagonalization of a set of symmetric matrices, the present solution also applies to the diagonalization problem.The author is obliged to Jos M. F. ten Berge for helpful comments.     

A factorial study of a new measure of authoritarianism

In the present study scores on Ray's Directiveness scale were subjected to Varimax rotation. Three factors explaining more than 50% of the variance of authoritarianism were interpreted and labelled Submissiveness, Dominance and Aggressiveness, respectively. Although these are in line with Ray's earlier conception of the nature of authoritarianism and do, to some extent, support other findings, it is suggested that the Directiveness scale be expanded to accommodate more characteristics of authoritarianism such as rigidity, hostility and superstitiousness.     

Joint factors in self-reports and ratings: Neuroticism,extraversion and openness to experience

Although most dimensional theories of personality assume that the same traits can be assessed in either ratings or self-reports, joint factor analyses of data from these two methods have seldom provided clear evidence in support of this position. Previous analyses have found a preponderance of within-method factors, or have had to transform variables or use unorthodox rotational procedures in order to control the effects of method variance. The present study argues that recent conceptual and technical advances should now make it possible to show joint factors at the second-order level using standard factor techniques. The NEO Inventory and NEO Rating Form, which measure 18 traits in the domains of Neuroticism, Extraversion and Openness to Experience, were administered to a sample of 281 men and women. Varimax rotation of three principal components clearly showed the hypothesized structure within and across self-reports and spouse ratings. Convergent and discriminant validity of the joint factors with the EPI scales was also shown. The results suggest that the effects of method variance can be minimized if well-qualified raters use psychometrically adequate instruments to provide ratings of clearly conceptualized traits. In addition, they provide strong evidence for the validity of the proposed three-domain model of personality.     

Three-mode orthomax rotation

Factor analysis and principal components analysis (PCA) are often followed by an orthomax rotation to rotate a loading matrix to simple structure. The simple structure is usually defined in terms of the simplicity of the columns of the loading matrix. In Three-mode PCA, rotational freedom of the so called core (a three-way array relating components for the three different modes) can be used similarly to find a simple structure of the core. Simple structure of the core can be defined with respect to all three modes simultaneously, possibly with different emphases on the different modes. The present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously. Computationally, this approach relies on repeated (two-way) orthomax applied to supermatrices containing the frontal, lateral or horizontal slabs, respectively. The procedure is illustrated by means of a number of exemplary analyses. As a by-product, application of the Three-mode Orthomax procedures to two-way arrays is shown to reveal interesting relations with and interpretations of existing two-way simple structure rotation techniques.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge and two anonymous reviewers for useful comments on an earlier version of this paper.     

Inducing a blockmodel structure of two-mode binary data using seriation procedures

We show that seriation of the rows and columns of a two-mode, binary matrix can be an effective method for producing a reordering of the matrix that reveals a blockmodel structure of the data. The objective criterion of the seriation process is based on Robinson patterning of matrix elements. The key advantages of the proposed method are: (a) it can be used in conjunction with existing two-mode blockmodeling algorithms by facilitating selection of the number of classes for the rows and columns of the matrix and the appropriate types of ideal blocks; (b) the model uses a well-grounded index based on Robinson structure, (c) guaranteed optimal solutions can be obtained for problems of practical size, and (d) the seriation method is frequently capable of producing a solution that has a substantive interpretation with respect to the orderings of the row objects and column items.     

Perceiving gaze from head and eye rotations: An integrative challenge for children and adults

Gaze is an emergent visual feature. A person's gaze direction is perceived not just based on the rotation of their eyes, but also their head. At least among adults, this integrative process appears to be flexible such that one feature can be weighted more heavily than the other depending on the circumstances. Yet it is unclear how this weighting might vary across individuals or across development. When children engage emergent gaze, do they prioritize cues from the head and eyes similarly to adults? Is the perception of gaze among individuals with autism spectrum disorder (ASD) emergent, or is it reliant on a single feature? Sixty adults (M = 29.86 years‐of‐age), thirty‐seven typically developing children and adolescents (M = 9.3 years‐of‐age; range = 7–15), and eighteen children with ASD (M = 9.72 years‐of‐age; range = 7–15) viewed faces with leftward, rightward, or direct head rotations in conjunction with leftward or rightward pupil rotations, and then indicated whether the face was looking leftward or rightward. All individuals, across development and ASD status, used head rotation to infer gaze direction, albeit with some individual differences. However, the use of pupil rotation was heavily dependent on age. Finally, children with ASD used pupil rotation significantly less than typically developing (TD) children when inferring gaze direction, even after accounting for age. Our approach provides a novel framework for understanding individual and group differences in gaze as it is actually perceived—as an emergent feature. Furthermore, this study begins to address an important gap in ASD literature, taking the first look at emergent gaze perception in this population.     

Factor analysis of Wechsler Adult Intelligence Scale-Revised in developmentally disabled persons

The results of previous studies on the factorial structure of Wechsler Intelligence Scales are somewhat inconsistent across normal and pathological samples. To study specific clinical groups, such as developmentally disabled persons, it is useful to examine the factor structure in appropriate samples. A factor analysis was carried out using the principal component method and the Varimax orthogonal rotation on the Wechsler Adult Intelligence Scale (WAIS-R) in a sample of 203 developmentally disabled persons, with a mean age of 25 years 4 months. Developmental disability ranged from mild to moderate. Partially contrasting with previous studies on normal samples, results found a two-factor solution. Wechsler's traditional Verbal and Performance scales seems to be more appropriate for this sample than the alternative three-factor solution.     

Recording horizontal rotations of head and eyes in spontaneous shifts,of gaze

In this paper, a method is described for simultaneous recording of horizontal rotation of the eyes in the head and rotation of the head about its natural vertical axis. Angular positions of eyesand head are obtained as proportionate dc voltages that are summed to obtain a record of the horizontal direction of gaze sensitive to l-deg saccades and accurate to within ±2 deg for at least the central 90 deg of the visual field. The apparatus is so light that it is hardly felt by the S, who, while seated, is free to move in any way he chooses and to perform any task. In addition, the angular position of a target rotating around the head may also be recorded for observation of visual-pursuit behavior.     

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 Comparative Investigation of Rotation Criteria Within Exploratory Factor Analysis

Exploratory factor analysis (EFA) is a commonly used statistical technique for examining the relationships between variables (e.g., items) and the factors (e.g., latent traits) they depict. There are several decisions that must be made when using EFA, with one of the more important being choice of the rotation criterion. This selection can be arduous given the numerous rotation criteria available and the lack of research/literature that compares their function and utility. Historically, researchers have chosen rotation criteria based on whether or not factors are correlated and have failed to consider other important aspects of their data. This study reviews several rotation criteria, demonstrates how they may perform with different factor pattern structures, and highlights for researchers subtle but important differences between each rotation criterion. The choice of rotation criterion is critical to ensure researchers make informed decisions as to when different rotation criteria may or may not be appropriate. The results suggest that depending on the rotation criterion selected and the complexity of the factor pattern matrix, the interpretation of the interfactor correlations and factor pattern loadings can vary substantially. Implications and future directions are discussed.     

Updating of locations during whole-body rotations in patients with hemispatial neglect

Posterior parietal cortex lesions have been associated with both hemispatial neglect and spatialupdating deficits. Currently, the relation between these processes remains poorly understood. We tested the ability of parietal patients with neglect to update remembered target locations during passive whole-body rotations. The rotations and manual pointing responses were executed with and without vision. During the rotation, the remembered location stayed on the same side of the body midline or crossed the midline. Parietal patients generally underestimated rotations, as compared with control groups, but updated targets equally well on either side of the body midline, regardless of the amount of updating required. Once parietal patients have localized a target, they can use self-motion information to update its location, even if it passes into the region they typically neglect. This lack of contralesional updating effects contrasts with impairments in eye position updating found in previous work with parietal patients.     

The factorial structure of the 16PF and EPQ in Egyptian samples: A preliminary study

Three hundred and sixteen Egyptian undergraduates (164 males and 152 females) completed the 16PF questionnaire (Form A) and the EPQ. The Varimax orthogonal rotation procedure applied to the 16PF revealed two factors, neuroticism (N) and extraversion (E). The rotated Varimax factors of the combined 16PF and EPQ inventories showed three factors, i.e. N and E, in addition to psychoticism (P). Coefficients of factor similarity (CFS) are high between the corresponding extracted factors of N, E and P in the male and female groups. The present results therefore support the second-order, broader factors. They are highly replicable and invariant in the two Egyptian samples studied.     

Disparity-processing in Wilde's dot-row phenomenon

Summary In this investigation on Wilde's phenomenon vertical lines were used instead of dots. In three experiments it was found that for the occurrence of Wilde's phenomenon, it is a necessary condition that the distances between all lines including the monocular end-lines are equal. When this is not the case the Panum effect is observed. When the distance between the end-line and its adjacent line deviates slightly from the distance between the central lines, one observes apparent rotation and a Panum effect. The observed depth of the end-line is in accordance with its disparity. The observed rotation is less than can be expected from the position in depth of the end-line. The disparity of the end-line is fully processed. Part of its disparity is used for apparent rotation of the pattern, and part of it is used for a displacement of the whole pattern.