A weighted procrustes criterion

The Procrustes criterion is a common measure for the distance between two matricesX andY, and can be interpreted as the sum of squares of the Euclidean distances between their respective column vectors. Often a weighted Procrustes criterion, using, for example, a weighted sum of the squared distances between the column vectors, is called for. This paper describes and analyzes the performance of an algorithm for rotating a matrixX such that the column-weighted Procrustes distance toY is minimized. The problem of rotatingX intoY such that an aggregate measure of Tucker's coefficient of congruence is maximized is also discussed.We wish to thank Richard A. Harshman and C. F. Jeff Wu for valuable discussions in the early stages of this work. We would also like to thank Jos ten Berge, John Gower, and the Editor, Associate Editor, and referees whose comments and suggestions greatly improved this paper.     

Distance stability analysis in multidimensional scaling using the jackknife method

Stability or sensitivity analysis is an important topic in data analysis that has received little attention in the application of multidimensional scaling (MDS), for which the only available approaches are given in terms of a coordinate‐based analytical jackknife methodology. Although in MDS the prime interest is in assessing the stability of the points in the configuration, this methodology may be influenced by imprecisions resulting from the inherently necessary Procrustes method. This paper proposes an analytical distance‐based jackknife procedure to study stability and cross‐validation in MDS in terms of the jackknife distances, which is not influenced by the Procrustes method. For each object, the corresponding jackknife estimated points are considered as naturally clustered points, and stability and cross‐validation are analysed in terms of the MDS distances arising from the jackknife procedure, on the basis of a weighted cluster‐MDS algorithm. A jackknife‐relevant configuration is also proposed for cross‐validation in terms of coordinates, in a cluster‐MDS framework.     

A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.     

On weighted procrustes and hyperplane fitting in factor analytic rotation

A weighted collinearity criterion for Procrustean rotation is developed, and it is shown that special cases with respect to the choice of weights and a vector norm are forms of hyperplane fitting, classical oblique Procrustes, etc.; a family of Procrustean transformation procedures is thereby generated. Numerical illustrations utilizing the Holzinger-Swineford data are presented.     

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.     

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.     

Minimization of a class of matrix trace functions by means of refined majorization

A procedure is described for minimizing a class of matrix trace functions. The procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization. It contains a recently proposed algorithm by Koschat and Swayne for weighted Procrustes rotation as a special case. A number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization-based procedure. The research of H. A. L. Kiers has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences.     

Procrustes matching by congruence coefficients

Matching by Procrustes methods involves the transformation of one matrix to match with another. A special least squares criterion, the congruence coefficient, has advantages as a criterion for some factor analytic interpretations. A Procrustes method maximizing the congruence coefficient is given. This solution is identical to Mosier's [1939] approximate solution, but is an exact solution for maximum congruence.     

探索性因素分析、目标旋转与因素结构的一致性

在因素结构水平上评估跨群体的一致性是在心理学研究中常常遇见的一个问题,对此问题的解答,可以选择探索性因素分析→目标旋转→一致性评估这一途径。本文首先介绍正交目标旋转的简单原理,然后介绍其在心理研究中的应用以及相关软件和程序。目标旋转之后,结构一致性的量化可以采用一致性系数等指标,这些指标可采用一定的实证分布、近似处理或经验标准进行统计检验。之后采用一项实证数据,演示探索性因素分析、目标旋转以及结构一致性的评估方法。     

Relations Between Conscious And Unconscious Manifestations Of Motivation In Children

Interest strength in 36 attitudes was measured on 89 public school students by means of four indirect behavioral tests. Standard scores on Word Association and Information were combined to give the Integrated (conscious) scores, and those on Selective Memory and Autism were combined to give the Unintegrated (unconscious) scores. Eighteen hypothesized drive factors and four device factors were extracted. The centroid was partitioned, and the Unintegrated portion was rotated by Procrustes for pure drive factors. This portion was projected upon the Integrated realm by reassembling the centroid matrix, and rotating the whole to the Procrustes position obtained for the part. Overall results appeared to support the "structural" rather than the "topographic" approaches t o psychoanalytic theory.     

A general solution for a class of weakly constrained linear regression problems

This paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights. Global optimality is obtained from inequalities rather than from partial derivatives of a Lagrangian function. Applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances, in Procrustes analysis, and in ridge regression analysis. The similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed.The author is obliged to Henk Kiers and Willem Heiser for helpful comments.     

Semi-sparse PCA

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

     

THE PROCRUSTES CLASS OF FACTOR-ANALYTIC TRANSFORMATIONS

The Procrustes method of deriving a reference vector structure from a principal-axes factor matrix is shown to be a class of solutions all of which obtain a transformation matrix which “rotates” the principal-axes matrix as close as possible to some desired matrix H. Various factor transformation methods, such as Promax, eigenvector rotation, as well as the classical Procrustes, differ chiefly in their rationale and method of generating H.     

Characteristics factor structures of the Japanese version of the State-Trait Anxiety Inventory: coexistence of positive-negative and state-trait factor structures

Previous factor studies of the State-Trait Anxiety Inventory (STAI; Spielberger, Gorsuch, & Lushene, 1970) have reported certain typical factors that are state-trait (S-T) 2-factor solutions and positively-negatively (P-N) worded item 2-factor solutions in addition to 4-factor solutions (positively and negatively worded state factors, positively and negatively worded trait factors). We explored the possibility that these factor structures are included in a factor space. Responses to the Japanese version of the STAI in a sample of 848 male workers were factor analyzed. The first-order factors obtained from principal-component analysis were almost equal to the previous 4 factors, except for a minor factor, and their second-order factors were the P-N factors. However, the S-T factors were also obtained from the same first-order factors by the oblique Procrustes rotation. Moreover, coexistence of these two 2-factor structures was determined in the same factor space by the orthogonal Procrustes rotation.     

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.     

Orthogonal procrustes rotation for two or more matrices

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower's method of generalized Procrustes analysis are suggested.     

Characteristic Factor Structures of the Japanese Version of the State-Trait Anxiety Inventory: Coexistence of Positive-Negative and State-Trait Factor Structures

Previous factor studies of the State-Trait Anxiety Inventory (STAI; Spielberger, Gorsuch, & Lushene, 1970) have reported certain typical factors that are state-trait (S-T) 2-factor solutions and positively-negatively (P-N) worded item 2-factor solutions in addition to 4-factor solutions (positively and negatively worded state factors, positively and negatively worded trait factors). We explored the possibility that these factor structures are included in a factor space. Responses to the Japanese version of the STAI in a sample of 848 male workers were factor analyzed. The first-order factors obtained from principal-component analysis were almost equal to the previous 4 factors, except for a minor factor, and their second-order factors were the P-N factors. However, the S-T factors were also obtained from the same first-order factors by the oblique Procrustes rotation. Moreover, coexistence of these two 2-factor structures was determined in the same factor space by the orthogonal Procrustes rotation.     

The processing of spatial information in short-term memory: Insights from eye tracking the path length effect

Effects of averageness and symmetry on the judgment of facial attractiveness were investigated using a generalized Procrustes method and multiple regression analyses. Participants (n = 114) rated attractiveness of 96 photographs of faces with neutral expressions. Through a generalized Procrustes method, the faces and their mirror-reversed versions were represented as points on a hyperplane. Both averageness and symmetry of each individual were defined as distances on the plane. A multiple regression analysis was performed to examine the effect of symmetry and averageness for each gender. For male faces, both symmetry and averageness affected attractiveness ratings positively , and there was no difference between the effects of averageness and symmetry. On the other hand, for female faces only averageness affected attractiveness, whereas symmetry did not. However, these effects were not large.     

Relationships among accumulated lifetime life events,prosocial skills,and behavior problems in elementary school children

Relationships among life events, prosocial skills, and behavior problems for 91 first through fourth grade children were examined. The data were based on mothers' reports of their children's lifetime life events, current prosocial skills, and current behavior problems. Higher prosocial skills scores significantly predicted lower Externalizing behavior problem scores. Higher weighted life events scores significantly predicted higher Internalizing behavior problem scores. The weighted life event scores×prosocial skills scores interaction did not significantly predict either Externalizing or Internalizing behavior problem scores. The more parsimonious main effects conceptualization suggests that life events and prosocial skills are orthogonal in young, elementary school children.     

Simultaneous nonlinear prediction

A previous article was concerned with simultaneous linear prediction [1]. There one was given a set of predictor tests or items and one predicted a set of predictands (also tests or items, or perhaps criteria.) We proposed asimultaneous prediction which was a certain weighted sum of the predictors. In the present article the constraint that the prediction be a weighted sum is relaxed. We seek a general function of the predictors which will maximize the quantity chosen for measuring prediction efficiency. This quantity is the same as the one used in linear prediction and we justify this approach by showing it is the appropriate one when there is only one predictand. In order to solve the problem we restrict consideration to a vector of predictors having only a finite number of possible values, i.e., it possesses discrete probability distribution weights. This can be applied in the case of dichotomous items for instance. It may also be used in continuous distributions as an approximation, by first dividing the original range of values into a finite number of intervals. Then one attributes to the interval the weight corresponding to the probability mass it underlies in the original distribution.This work was initiated at Stanford University under contract 2-10-065 with U. S. Office of Education and was partly revised at the Université de Montréal.I wish to express my gratitude to Professor Herbert Solomon, Stanford University, for his unfailing assistance at all stages of my work and specially for bringing to my attention the problem of nonlinear prediction in the present context.