Orthogonal procrustes rotation maximizing congruence

Procedures for assessing the invariance of factors found in data sets using different subjects and the same variables are often using the least squares criterion, which appears to be too restrictive for comparing factors.Tucker's coefficient of congruence, on the other hand, is more closely related to the human interpretation of factorial invariance than the least squares criterion. A method maximizing simultaneously the sum of coefficients of congruence between two matrices of factor loadings, using orthogonal rotation of one matrix is presented. As shown in examples, the sum of coefficients of congruence obtained using the presented rotation procedure is slightly higher than the sum of coefficients of congruence using Orthogonal Procrustes Rotation based on the least squares criterion.The author is obliged to Lewis R. Goldberg for critically reviewing the first draft of this paper.     

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.     

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.     

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.     

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。     

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.     

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 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.     

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.     

Bootstrap confidence intervals for principal covariates regression

Principal covariate regression (PCOVR) is a method for regressing a set of criterion variables with respect to a set of predictor variables when the latter are many in number and/or collinear. This is done by extracting a limited number of components that simultaneously synthesize the predictor variables and predict the criterion ones. So far, no procedure has been offered for estimating statistical uncertainties of the obtained PCOVR parameter estimates. The present paper shows how this goal can be achieved, conditionally on the model specification, by means of the bootstrap approach. Four strategies for estimating bootstrap confidence intervals are derived and their statistical behaviour in terms of coverage is assessed by means of a simulation experiment. Such strategies are distinguished by the use of the varimax and quartimin procedures and by the use of Procrustes rotations of bootstrap solutions towards the sample solution. In general, the four strategies showed appropriate statistical behaviour, with coverage tending to the desired level for increasing sample sizes. The main exception involved strategies based on the quartimin procedure in cases characterized by complex underlying structures of the components. The appropriateness of the statistical behaviour was higher when the proper number of components were extracted.     

A Profile-Based Framework for Factorial Similarity and the Congruence Coefficient

We present a novel profile-based framework for understanding factorial similarity in the context of exploratory factor analysis in general, and for understanding the congruence coefficient (a commonly used index of factor similarity) specifically. First, we introduce the profile-based framework articulating factorial similarity in terms of 3 intuitive components: general saturation similarity, differential saturation similarity, and configural similarity. We then articulate the congruence coefficient in terms of these components, along with 2 additional profile-based components, and we explain how these components resolve ambiguities that can be—and are—found when using the congruence coefficient. Finally, we present secondary analyses revealing that profile-based components of factorial are indeed linked to experts' actual evaluations of factorial similarity. Overall, the profile-based approach we present offers new insights into the ways in which researchers can examine factor similarity and holds the potential to enhance researchers' ability to understand the congruence coefficient.     

Global optimality of the successive Maxbet algorithm

The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.The authors are obliged to Henk Kiers for critical comments on a previous draft.     

A family of association coefficients for metric scales

Four types of metric scales are distinguished: the absolute scale, the ratio scale, the difference scale and the interval scale. A general coefficient of association for two variables of the same metric scale type is developed. Some properties of this general coefficient are discussed. It is shown that the matrix containing these coefficients between any number of variables is Gramian. The general coefficient reduces to specific coefficients of association for each of the four metric scales. Two of these coefficients are well known, the product-moment correlation and Tucker's congruence coefficient. Applications of the new coefficients are discussed.     

Negative congruence effects in letter and pseudo-letter recognition: the role of similarity and response conflict

Letters and pseudo-letters were presented in three experiments using a sequential same-different task. While first items were always presented in isolation, the second item was either presented in isolation or surrounded by geometrical non-target shapes that could be congruent or incongruent to the target. In two experiments, a physical sameness criterion was used. In Experiment 1, in one condition, different pairs were always distinct in shape, in another they were similar in shape. Negative congruence effects were obtained for different pairs that are similar. In Experiment 2, this effect is replicated within participants. In this experiment, similar and dissimilar stimuli were mixed. The results were explained in terms of the difficulty of responding different to stimuli that are similar in shape: when the second item is surrounded by a congruent shape, the similarity is emphasized, making this response even more difficult. In Experiment 3, the same stimuli were presented using a categorical sameness criterion. This reduces the role of physical similarity and thus eliminates the response conflict. As a result, negative congruence effects were no longer observed. Taken together, the three experiments identify another source of negative congruence effects besides the ones recently reported in the literature.     

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.     

Stability of nonlinear principal components analysis: an empirical study using the balanced bootstrap

Principal components analysis (PCA) is used to explore the structure of data sets containing linearly related numeric variables. Alternatively, nonlinear PCA can handle possibly nonlinearly related numeric as well as nonnumeric variables. For linear PCA, the stability of its solution can be established under the assumption of multivariate normality. For nonlinear PCA, however, standard options for establishing stability are not provided. The authors use the nonparametric bootstrap procedure to assess the stability of nonlinear PCA results, applied to empirical data. They use confidence intervals for the variable transformations and confidence ellipses for the eigenvalues, the component loadings, and the person scores. They discuss the balanced version of the bootstrap, bias estimation, and Procrustes rotation. To provide a benchmark, the same bootstrap procedure is applied to linear PCA on the same data. On the basis of the results, the authors advise using at least 1,000 bootstrap samples, using Procrustes rotation on the bootstrap results, examining the bootstrap distributions along with the confidence regions, and merging categories with small marginal frequencies to reduce the variance of the bootstrap results.     

Thinking Ahead: Assuming Linear Versus Nonlinear Personality-Criterion Relationships in Personnel Selection

Recent studies suggest that the form of some personality–performance relationships may be curvilinear, meaning that traditional top-down selection is inefficient in capitalizing on underlying personality–performance relationships. This study examines how mean performance is affected by how well the selection method is aligned with the nature of personality–criterion relationships. A simulation manipulated the linearity or nonlinear inflection point of predictor–criterion relationships, and several selection approaches were implemented that varied in level of congruence with these relationships. Results indicate that incongruence can produce notable decrements in mean performance under some conditions. Some evidence also suggests that decrements can be greater when linearity is assumed but relationships are nonlinear (vs. when nonlinearity is assumed but relationships are linear), selection ratios are smaller, and a single predictor is used.     

Dissociating congruence effects in letters versus shapes: Kanji and kana

In order to consolidate the dissociation in feature integration found in earlier studies between letters of the Roman alphabet and corresponding pseudo-letters, the present study used kanji and kana and corresponding pseudo-letters, presented visually, either in isolation or surrounded by congruent or incongruent shape, as targets in a choice-response task with three different response criteria. When the criterion was shape, congruence effects were obtained for both real and pseudo-letters. With the second and third response criteria this result was found for pseudo-letters, but not for letters. These criteria either involved distinguishing between letters and visually similar pseudo-letters or distinguishing between visually similar letters. The dissociation of congruence effects between letters and pseudo-letters was therefore shown to depend on visual similarity between targets, independent of their category. This effect was found to be robust for kanji but not for kana, which is related to distinctions between these two writing systems.