Canonical/redundancy factoring analysis

The interrelationships between two sets of measurements made on the same subjects can be studied by canonical correlation. Originally developed by Hotelling [1936], the canonical correlation is the maximum correlation betweenlinear functions (canonical factors) of the two sets of variables. An alternative statistic to investigate the interrelationships between two sets of variables is the redundancy measure, developed by Stewart and Love [1968]. Van Den Wollenberg [1977] has developed a method of extracting factors which maximize redundancy, as opposed to canonical correlation.A component method is presented which maximizes user specified convex combinations of canonical correlation and the two nonsymmetric redundancy measures presented by Stewart and Love. Monte Carlo work comparing canonical correlation analysis, redundancy analysis, and various canonical/redundancy factoring analyses on the Van Den Wollenberg data is presented. An empirical example is also provided.Wayne S. DeSarbo is a Member of Technical Staff at Bell Laboratories in the Mathematics and Statistics Research Group at Murray Hill, N.J. I wish to express my appreciation to J. Kettenring, J. Kruskal, C. Mallows, and R. Gnanadesikan for their valuable technical assistance and/or for comments on an earlier draft of this paper. I also wish to thank the editor and reviewers of this paper for their insightful remarks.     

Child behavior disorders by age and sex based on item factoring of the revised Bristol Guides

Standardized teacher observations of 2,527 schoolchildren, selected at random for the revised Bristol Social Adjustment Guides were partitioned into four subsamples consisting of 797 5- to 10-year-old boys, 758 5- to 10-year-old-girls; 508 11- to 15-year-old boys, and 464 11- to 15-year-old girls, respectively. The children were observed by over 900 teachers and rated on 104 indicators of maladaptive behavior. Item scores for each age/ sex sample were subjected to first- and second-order factor analysis, with varimax rotation yielding identical second-order models of behavior disorder across age and sex samples and somewhat different first-order models for each sample. Comparison of derived dimensions with dimensions emergent in other behavior problem research indicated considerable consistency. Moreover, the similarity of the factorially derived dimensions confirmed the cross-age and -sex generality of the syndromes known as unforthcomingness, hostility, and depression, and provided reasonable support for the utility of the syndrome of inconsequence, although it was apparent that inconsequence stands as more a composite of underlying factor dimensions reflecting hyperactive and attention-seeking behaviors. While the withdrawal syndrome found factorial support, its integrity was clearly specific to child age and sex.     

Properties and applications of gramian factoring

The Gramian factorizationG of a GramianR is square and symmetric and has no negative characteristic roots. It is shown to be that square factorization that is, in the least-squares sense, most isomorphic toR, most like a scalarK, and most highly traced, and to be the necessary and sufficient relation between the oblique vectors of an oblique transformation and the orthogonal vectors of the least-squares orthogonal counterpart. A slightly modified Gramian factorization is shown to be the factorization that is most isomorphic to a specified diagonalD, and to be the main part of an iterative procedure for obtaining simplimax, a square factor matrix with simple structure maximized in the sense of having the largest sum of squared diagonal loadings. Several published applications of Gramian factoring are cited.     

Having reasons and the factoring account

It’s natural to say that when it’s rational for me to φ, I have reasons to φ. That is, there are reasons for φ-ing, and moreover, I have some of them. Mark Schroeder calls this view The Factoring Account of the having reasons relation. He thinks The Factoring Account is false. In this paper, I defend The Factoring Account. Not only do I provide intuitive support for the view, but I also defend it against Schroeder’s criticisms. Moreover, I show that it helps us understand the requirements of substantive rationality, or what we are rationally required to do when responding to reasons.     

General theory and methods for matric factoring

Methods are developed for factoring an arbitrary rectangular matrixS of rankr into the formFP, whereF hasr columns andP hasr rows. For the statistical problem of factor analysis,S may be the score matrix of a population of individuals on a battery of tests. ThenF is a matrix of factor loadings,P is a matrix of factor scores, andr is the number of factor variates. (As in current procedures, there remains a subsequent problem of rotation of axes and interpretation of factors, which is not discussed here.) Methods are also developed for factoring an arbitrary Gramian matrixG of rankr into the formFF, whereF hasr columns andF denotesF transposed. For the statistical problem of factor analysis,G may be the matrix of intercorrelations,R, of a battery of tests, with unity, communalities, or other parameters in the principal diagonal.R is proportional toSS, and it is shown thatS can be factored by factoringR. This may usually be the most economical procedure in practice; it should not be overlooked, however, thatS can be factored directly. The general methods build up anF (andP) in as many stages as desired; as many factors as may be deemed computationally practical can be extracted at a time. Perhaps it will usually be found convenient to extract not more than three factors at a time. Current procedures, like the centroid and principal axes, are special cases of a general method presented here for extracting one factor at a time.     

A necessary and sufficient formula for matric factoring

For the purpose of extracting factors from matrices, it is proved that a certain formula is both necessary and sufficient. In factor analysis, the formula may be applied either to the correlation matrix, or directly to the score matrix (assuming the communality problem is solved). As many factors as desired can be extracted in one operation. Having such a compact formulation is useful for teaching as well as computing purposes, since it includes all techniques of factor extraction as special cases.On leave from the Israel Institute for Applied Social Research. This research was facilitated in part by a grant from the Lucius N. Littauer Foundation to the American Committee for Social Research in Israel, Inc.     

A method for factoring large numbers of items

The computation of intercorrelation matrices involving large numbers of variables and the subsequent factoring of these matrices present a formidable task. A method for estimating factor loadings without computing the intercorrelation matrix is developed. The estimation procedure is derived from a theoretical model which is shown to be a special case of the multiple-group centroid method of factoring. Empirical checks have indicated that the model, even though it makes some stringent assumptions, can be applied to a variety of variables found in psychological factoring problems. It has been found to be particularly useful in factoring test items.     

A multiple group method of factoring the correlation matrix

There are a number of methods of factoring the correlation matrix which require the calculation of a table of residual correlations after each factor has been extracted. This is perhaps the most laborious part of factoring. The method to be described here avoids the computation of residuals after each factor has been computed. Since the method turns on the selection of a set of constellations or clusters of test vectors, it will be calleda multiple group method of factoring. The method can be used for extracting one factor at a time if that is desired but it will be considered here for the more interesting case in which a number of constellations are selected from the correlation matrix at the start. The result of this method of factoring is a factor matrixF which satisfies the fundamental relationFF'=R.This study is one of a series of investigations in the development of multiple factor analysis and application to the study of primary mental abilities. We wish to acknowledge the financial assistance from the Social Science Research Committee of The University of Chicago which has made possible the work of the Psychometric Laboratory.     

Leadership styles and Deming's behavior factors

Although total quality management scholars, including W. Edwards Deming, consider leadership to be important, there has been little work which considers linkages between specific leadership styles and TQM behaviors/policies. This paper examines the relationship between a full range of leadership styles (Bass & Avolio, 1994) including laissez-faire, management-by-exceptionpassive, management-by-exception-active, transactional and transformational leadership, and five specific TQM behavior factors derived from Deming's (1986) Fourteen Points, i.e., change agency, teamwork, trust-building, short-term goal eradication and continuous improvement.The authors gratefully acknowledge the helpful suggestions of Boas Shamir, Don Spangler, Bruce Avolio, Francis Yammarino, and Susan Kang to earlier versions of this paper.     

Reinforcement frequency and contingency as factors in fixed-ratio behavior

Two variables often confounded in fixed-ratio schedules are reinforcement frequency and response requirement. These variables were isolated by a technique that yoked the distributions of reinforcements in time for one group of pigeons to those of pigeons responding on various fixed-ratio schedules. The contingencies for the yoked birds were then manipulated by adding various tandem fixed-ratio requirements to their schedules. Post-reinforcement pause was approximately equal for the yoked and ratio pigeons, and was relatively insensitive to changes in the tandem requirement. Terminal response rate increased with increases in the tandem requirement, even though reinforcement rate was invariant. This increase was attributed to the progressive interference of the tandem requirement with the differential reinforcement of long interresponse times.     

Friend influence on prosocial behavior: The role of motivational factors and friendship characteristics

This study examined motivation (prosocial goals), individual characteristics (sex, ethnicity, and grade), and friendship characteristics (affective quality, interaction frequency, and friendship stability) in relation to middle adolescents' prosocial behavior over time. Ninth- and 10th-grade students (N = 208) attending a suburban, mid-Atlantic public high school and having at least 1 reciprocated friendship were followed for 1 year. Hierarchical regression analyses revealed that a friend's behavior is related to an individual's prosocial goal pursuit, which in turn, is related to an individual's prosocial behavior. Further, the affective quality of a friendship and the frequency with which friends interact moderate relations of a friend's prosocial behavior to an individual's prosocial goal pursuit.