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.     

Concise formulas for the standard errors of component loading estimates

Concise formulas for the asymptotic standard errors of component loading estimates were derived. The formulas cover the cases of principal component analysis for unstandardized and standardized variables with orthogonal and oblique rotations. The formulas can be used under any distributions for observed variables as long as the asymptotic covariance matrix for sample covariances/correlations is available. The estimated standard errors in numerical examples were shown to be equivalent to those by the methods using information matrices.The author is indebted to anonymous reviewers for the corrections and suggestions on this study, which have led to improvements of earlier versions of this article.     

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.     

Oblique Rotaton in Canonical Correlation Analysis Reformulated as Maximizing the Generalized Coefficient of Determination

To facilitate the interpretation of canonical correlation analysis (CCA) solutions, procedures have been proposed in which CCA solutions are orthogonally rotated to a simple structure. In this paper, we consider oblique rotation for CCA to provide solutions that are much easier to interpret, though only orthogonal rotation is allowed in the existing formulations of CCA. Our task is thus to reformulate CCA so that its solutions have the freedom of oblique rotation. Such a task can be achieved using Yanai’s (Jpn. J. Behaviormetrics 1:46–54, 1974; J. Jpn. Stat. Soc. 11:43–53, 1981) generalized coefficient of determination for the objective function to be maximized in CCA. The resulting solutions are proved to include the existing orthogonal ones as special cases and to be rotated obliquely without affecting the objective function value, where ten Berge’s (Psychometrika 48:519–523, 1983) theorems on suborthonormal matrices are used. A real data example demonstrates that the proposed oblique rotation can provide simple, easily interpreted CCA solutions.     

Oblique rotation in correspondence analysis: A step forward in the search for the simplest interpretation

Correspondence analysis (CA) is a popular method that can be used to analyse relationships between categorical variables. It is closely related to several popular multivariate analysis methods such as canonical correlation analysis and principal component analysis. Like principal component analysis, CA solutions can be rotated orthogonally as well as obliquely into a simple structure without affecting the total amount of explained inertia. However, some specific aspects of CA prevent standard rotation procedures from being applied in a straightforward fashion. In particular, the role played by weights assigned to points and dimensions and the duality of CA solutions are unique to CA. For orthogonal simple structure rotation, procedures recently have been proposed. In this paper, we construct oblique rotation methods for CA that take into account these specific difficulties. We illustrate the benefits of our oblique rotation procedure by means of two illustrative examples.     

Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.     

The Infinitesimal Jackknife with Exploratory Factor Analysis

The infinitesimal jackknife, a nonparametric method for estimating standard errors, has been used to obtain standard error estimates in covariance structure analysis. In this article, we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices. Both maximum likelihood estimation and ordinary least squares estimation are considered.     

Asymmetric response time functions during left-/right-facing discriminations of rotated objects: The short and the long of it

When deciding if a rotated object would face to the left or to the right, if imagined at the upright, mental rotation is typically assumed to be carried out through the shortest angular distance to the upright prior to determining the direction of facing. However, the response time functions for left- and right-facing objects are oppositely asymmetric, which is not consistent with the standard explanation. Using Searle and Hamm’s individual differences adaption of Kung and Hamm’s Mixture Model, the current study compares the predicted response time functions derived when assuming that objects are rotated through the shortest route to the upright with the predicted response time functions derived when assuming that objects are rotated in the direction they face. The latter model provides a better fit to the majority of the individual data. This allows us to conclude that, when deciding if rotated objects would face to the left or to the right if imagined at the upright, mental rotation is carried out in the direction that the objects face and not necessarily in the shortest direction to the upright. By comparing results for mobile and immobile object sets we can also conclude that semantic information regarding the mobility of an object does not appear to influence the speed of mental rotation, but it does appear to influence pre-rotation processes and the likelihood of employing a mental rotation strategy.     

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.     

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.     

Exploratory second-order analyses for components and factors

Abstract: Exploratory methods using second‐order components and second‐order common factors were proposed. The second‐order components were obtained from the resolution of the correlation matrix of obliquely rotated first‐order principal components. The standard errors of the estimates of the second‐order component loadings were derived from an augmented information matrix with restrictions for the loadings and associated parameters. The second‐order factor analysis proposed was similar to the classical method in that the factor correlations among the first‐order factors were further resolved by the exploratory method of factor analysis. However, in this paper the second‐order factor loadings were estimated by the generalized least squares using the asymptotic variance‐covariance matrix for the first‐order factor correlations. The asymptotic standard errors for the estimates of the second‐order factor loadings were also derived. A numerical example was presented with simulated results.     

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.     

Estimating Standard Errors in Exploratory Factor Analysis

The article describes 6 issues influencing standard errors in exploratory factor analysis and reviews 7 methods of computing standard errors for rotated factor loadings and factor correlations. These 7 methods are the augmented information method, the nonparametric bootstrap method, the infinitesimal jackknife method, the method using the asymptotic distributions of unrotated factor loadings, the sandwich method, the parametric bootstrap method, and the jackknife method. Standard error estimates are illustrated using a personality study with 537 men and an intelligence study with 145 children.     

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.     

The use of proprioception and tactile information in haptic search

To investigate how tactile and proprioceptive information are used in haptic object discrimination we conducted a haptic search task in which participants had to search for either a cylinder, a bar or a rotated cube within a grid of aligned cubes. Tactile information from one finger is enough to detect a cylinder amongst the cubes. For detecting a bar or a rotated cube amongst cubes touch alone is not enough. For the rotated cube this is evident because its shape is identical to that of the non-targets, so proprioception must provide information about the orientation of the fingers and hand when touching it. For the bar one either needs proprioceptive information about the distance and direction of a single finger’s movements along the surfaces, or proprioceptive information from several fingers when they touch it simultaneously. When using only one finger, search times for the bar were much longer than those for the other two targets. When the whole hand or both hands were used the search times were similar for all shapes. Most errors were made when searching for the rotated cube, probably due to systematic posture-related biases in judging orientation on the basis of proprioception. The results suggest that tactile and proprioceptive information are readily combined for shape discrimination.     

Rotation to Simple Loadings Using Component Loss Functions: The Oblique Case

Component loss functions (CLFs) similar to those used in orthogonal rotation are introduced to define criteria for oblique rotation in factor analysis. It is shown how the shape of the CLF affects the performance of the criterion it defines. For example, it is shown that monotone concave CLFs give criteria that are minimized by loadings with perfect simple structure when such loadings exist. Moreover, if the CLFs are strictly concave, minimizing must produce perfect simple structure whenever it exists. Examples show that methods defined by concave CLFs perform well much more generally. While it appears important to use a concave CLF, the specific CLF used is less important. For example, the very simple linear CLF gives a rotation method that can easily outperform the most popular oblique rotation methods promax and quartimin and is competitive with the more complex simplimax and geomin methods. The author would like to thank the editor and three reviewers for helpful suggestions and for identifying numerous errors.     

Cross‐modal facilitation of mental rotation: Effects of modality and complexity

Performance in a visual mental rotation (MR) task has been reported to predict the ability to recognize retrograde‐transformed melodies. The current study investigated the effects of melodic structure on the MR of sequentially presented visual patterns. Each trial consisted of a five‐segment sequentially presented visual pattern (standard) followed by a five‐tone melody that was either identical in structure to the standard or its retrograde. A visual target pattern was either the rotated version of the standard or unrelated to it. The task was to indicate whether the target pattern was a rotated version of the standard or not. Periodic patterns were not rotated but melodies facilitated the rotation of non‐periodic patterns. For these, rotation latency was determined by a quantitative index of complexity (number of runs). This study provides the first experimental confirmation for cross‐modal facilitation of MR.     

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.     

Discrimination learning of rotated faces: a Markov analysis of coding and association processes]

30 subjects participated in a discrimination experiment learning face-letter associations under four rotation conditions (45 degrees, 90 degrees, 135 degrees, 180 degrees). Under each condition two thirds of the faces were presented twice, upright and rotated away from the vertical; the remaining faces were presented once, upright or rotated. Learning is described by a joint Markov model: For faces that are presented twice it assumes a separate association and encoding process (two-stage-model), for faces that are presented once it assumes an association process (all-or-none-model). The Markov model fits the data for all four rotation conditions. The angle of rotation does not affect learning for faces that are presented once. For faces that are presented twice it influences both the association and the encoding process. For the angles employed, the effect of rotation can be approximated linearly. The results suggest that the encoding of a rotated face differs increasingly from an upright face as a function of these angles of rotation. This confirms analogous conclusions from mental rotation experiments.     

Spatial representation by blind and sighted children

Spatial representation by 72 blind and blindfolded sighted children between the ages of 6 and 11 was tested in two experiments by mental rotation of a raised line under conditions of clockwise varied directions.Experiment 1 showed that the two groups were well matched on tactual recognition and scored equally badly on matching displays to their own mentally rotated position.Experiment 2 found the sighted superior in recall tests. There was a highly significant interaction between sighted status and degree of rotation. Degree of rotation affected only the blind. Their scores were significantly lower for rotating to oblique and to the far orthogonal directions than to near orthogonal test positions. On near orthogonals the blind did not differ from the sighted.Age was a main effect, but it did not interact with any other variable. Older blind children whose visual experience dated from before the age of 6 were superior to congenitally blind subjects, but not differentially more so on oblique directions.The results were discussed in relation to hypotheses about the nature of spatial representation and strategies by children whose prior experience derived from vision or from touch and movement.