A non-graphical method for transforming an arbitrary factor matrix into a simple structure factor matrix

The most commonly used method of factoring a matrix of intercorrelations is the centroid method developed by L. L. Thurstone. It is, however, necessary to transform the centroid matrix of factor loadings into a simple structure matrix in order to facilitate the interpretation of the factor loadings. Current methods for effecting this transformation are chiefly graphical and require considerable experience and personal judgment. This paper presents a new method for transforming an arbitrary factor matrix into a simple structure matrix by methods almost completely objective. The theory underlying the method is developed and approximation procedures are derived. The method is applied to a matrix of factor loadings previously analyzed by Thurstone.     

A simple procedure for approximate factor analysis

A variation of the centroid method is described and illustrated. By the application of new rules for reflecting signs, it may be possible to reduce to insignificance the factor loadings of tests showing insignificant correlation (original or residual) with clusters of tests having relatively high intercorrelations. As a result, a factor common to any one of these clusters may be revealed by the centroid method itself with little or no need for rotation of axes or further calculations.     

The measurement of speed in motor performance

When the centroid method of factor analysis was applied to two sets of data on athletic performances, three significant factors emerged: strength, velocity, and dead weight. Scores on this speed factor were predicted by the multiple regression technique, the factor loadings on the speed factor being used as the criterion correlations, and these predicted scores were correlated with each of the other variables. When the original tables, augmented by the new speed variable, were refactored, the computed speed factor fell on the speed axis as a primary trait. It is thus shown that it is possible to isolate and measure a factor which appears in variables under consideration only as a compound.Much of the expense of the statistical computations connected with this study was financed by a grant from the Carnegie Foundation for the Advancement of Teaching.     

A revised orthogonal rotational solution for Thurstone's original primary mental abilities test battery

By extension of the rotational process, meaningful orthogonally related positions were found for all of the thirteen centroid factors which Thurstone extracted from his original PMA intercorrelations. Most of the original primary ability factors were more sharply delineated and corresponded more closely to the Army Air Force factors that bear similar names (demonstrating greater invariance from analysis to analysis). While such different results obtained by two investigators applying the same methods on the same data may initiate some concern, the results strengthen rather than weaken the idea that more psychological meaningfulness and greater invariance will result if centroid axes are rotated, using the concepts of a simple structure and positive manifold.Thurstone considered loadings between ±.20 as negligible and considered only loadings of at least .40 in naming factors.     

On the sampling errors of factor loadings

The results of three empirical studies on the sampling fluctuation of centroid factor loadings are reported. The first study is based on data which happened to be available on 8 variables for 700 cases and which were factored to three factors for subsamples. The second study is based on fictitious data for 2500 cases which provided separate analyses on 25 samples for each of three situations: 5 variables, one factor; 5 variables, two factors; and 6 variables, three factors. The third study, based on real data for 9 variables and 7000 cases, involves separate factorization for 35 samples of 200 cases. The three studies agree in showing that the sampling behavior of first centroid factor loadings is much like that of correlation coefficients, whereas the sampling fluctuations for loadings beyond the first are disturbingly large.     

The relation between factor loadings and biserial correlations in item analysis

The first centroid factor loadings obtained from various interitem relations are compared with item discrimination indices commonly used in item analysis. Depending upon what type of matrix is factored, the factor loadings are shown to be related to point biserial and biserial correlations.     

Measures of invariance and comparability in factor analysis for fixed variables

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.     

A factorial study of number ability

In order to investigate certain hypotheses concerning the nature of number ability, and, secondarily, the nature of perceptual speed, a battery of thirty-four tests was given to 223 Chicago high school seniors and the data were factored by the centroid method. Seven primary factors were identifiable upon rotation. Several deductions are made relative to the interpretation of the factors and relative to the consistency of the data with the hypotheses which were to be tested. I wish to express my great appreciation of the aid of Professor L. L. Thurstone whose generosity made this study possible. Grateful acknowledgment is made of the aid of Mr. Ledyard Tucker in the use of the I.B.M. machines for obtaining the intercorrelations and the centroid factor loadings, and to both him and Mr. Harold Bechtoldt for aid in the testing of subjects.     

Determining a simple structure when loadings for certain tests are known

A rigorous and an approximate solution are found for the problem: Given a primary trait matrix forn tests andr ₁ traits, and a matrix for the samen tests andr ₂ reference axes, to discover the transformation which will transform the second matrix into the first, or primary trait matrix. Formulas for determining the limits of the effect of using the approximate solution are presented. The method is applied to a set of twenty hypothetical tests, defined by their loadings on four orthogonal primary traits. After factoring the inter-correlations of these variables by Thurstone's centroid method, approximating the diagonals, the original hypothetical matrix is reproduced with a root mean square discrepancy of .014 by assuming as known the primary trait loadings of only the first eight tests. The method is applied to the results of factoring two batteries of 14 tests, having 8 tests in common, to give the factor loadings of the two batteries on the same reference axes. The method provides a means of comparing directly and quantitatively the results of two different factor studies, provided they have tests in common, and of testing the stability of simple structure under changes in the battery. The relations of the method here developed to certain problems in multiple correlation are shown.     

The optimality of the centroid method

The aim of this note is to show that the centroid method has two optimality properties. It yields loadings with the highest sum of absolute values, even in absence of the constraint that the squared component weights be equal. In addition, it yields scores with maximum variance, subject to the constraint that none of the squared component weights be larger than 1.This research is financed by NSERC of Canada. The author is grateful to Michel Tenenhaus for pointing the similarity of the procedures in the centroid method and Q-mode PCA in L₁. The author also thanks the editor and associate editor for providing shorter proofs of the theorems, along with the referees for their helpful comments.     

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 simple orthogonal multiple factor approximation procedure

This paper describes a simple orthogonal multiple factor approximation procedure that involves no inversion of the sign of negative residuals, the estimation of only as many communalities as there are factors, and none or only a few minor rotations of the axes in an attempt to obtain a meaningful solution. It also suggests a technique for the estimation of those communalities that must be estimated. The factor loadings obtained by means of this procedure, which we shall designate as the pre-selection procedure, are affected by the order in which the factors are obtained, showing a reduction in variance accounted for by each successive factor, as is characteristic of the centroid, bi-factor, and principal factor solutions. The entire procedure takes considerably less time than that involved in the orthodox centroid method alone.     

A comparative study of three methods of rotation

Three methods of rotation (the graphical, the Thurstone analytical, and the direct-rotational) were applied to the matrix of centroid loadings for 35 variables, to determine which method is the most efficient from theoretical and practical standpoints. The direct-rotational method provided the most information for determining the rank of the configuration and was most economical with respect to time required to reach a rotational solution. The analytical method required the least number of judgmental decisions and was the most objective. The graphical method was the most laborious but had a slight advantage with regard to the number of near-zero loadings in the rotational solution.Systems Development Corp.     

Influence curves for factor loadings

Influence curves for the initial and rotated loadings are derived for the maximum likelihood factor analysis (MLFA) model. Cook's distances based on the empirical influence curves of factor loadings are proposed for the identification of influential observations. The distances are shown to be invariant under scale transformation and factor rotation. We find that an observation with a very large Cook's distance based on the sample influence curve may not necessarily exert an excessive influence on the factor loadings pattern but may change the ordering of the factors. The issue of the switching of factors is also studied by means of the empirical influence curve and factor scores.     

TECHNIQUE FOR MINIMIZING THE INSTRUMENTAL FACTOR

A technique is described for using orthogonal factor scores to compare the item-scores obtained from two or more media so as to minimize the instrumental factor problem. It is compared with the hypothesis matrix technique using the item-scores of a behavior rating scale and a structured psychiatric interview scale. The direct factor score method proved to be more effective in several aspects, mainly because it yielded larger loadings and, therefore, resulted in the extraction of more and better delineated factors.     

The Moral Judgments of Chinese Students

Abstract

The Defining Issues Test (DIT) was translated into Chinese and administered to 84 male and 84 female Chinese subjects in Hong Kong. No significant sex differences in all of the stage scores derived from DIT ratings and rankings were found. In addition, the Pearson correlation matrix of Stage 2 through 6 scores for the females but not for the males approximated a simplex-like pattern. The unrotated factor loadings of the Principal Components Analysis for the female subjects provided clear support to the claim of an ordered sequence of Kohlberg's stages measured by the DIT. The factor loadings on the higher stages (4, 5, and 6), however, were less discriminating and consistent in the male factor pattern.     

Investigating Group Differences on Cognitive Tests Using Spearman's Hypothesis: An Evaluation of Jensen's Method

Jensen has posited a research method to investigate group differences in cognitive tests. This method consists of first extracting a general intelligence factor by means of exploratory factor analysis. Secondly, similarity of factor loadings across groups is evaluated in an attempt to ensure that the same constructs are measured. Finally, the correlation is computed between the loadings of the tests on the general intelligence factor and the mean differences between groups on the tests. This part is referred to as a test of "Spearman's Hypothesis", which essentially states that differences in g account for the main part of differences in observed scores. Based on the correlation, inferences are made with respect to group differences in general intelligence.

The validity of these inferences is investigated and compared to the validity of inferences based on multi-group confirmatory factor analysis. For this comparison, population covariance matrices are constructed which incorporate violations of the central assumption underlying Jensen's method concerning the existence of g and/or violations of Spearman's Hypothesis. It is demonstrated that Jensen's method is quite insensitive to the violations. This lack of specificity is observed consistently for all types of violations introduced in the present study. Multi-group confirmatory factor analysis emerges as clearly superior to Jensen's method.     

Scaling Variances of Latent Variables by Standardizing Loadings: Applications to Working Memory and the Position Effect

The standardization of loadings gives a metric to the corresponding latent variable and thus scales the variance of this latent variable. By assigning an appropriately estimated weight to all the loadings on the same latent variable it can be achieved that the average squared loading is 1 as the result of standardization. As a consequence, there is comparability of the variances of the latent variables of a confirmatory factor model. A precondition of comparability is that the latent variables must have loadings of the same manifest variables and that the variances are estimated with respect to the same covariance matrix. The usefulness of this standardization method is demonstrated by applying it for the evaluation of the sources of performance in a working memory task and for the evaluation of the impact of the position effect on performance in completing a reasoning measure. In these examples the scaled variances of the latent variables provided useful information.     

On the relations among regular, equal unique variances, and image factor analysis models

We investigate under what conditions the matrix of factor loadings from the factor analysis model with equal unique variances will give a good approximation to the matrix of factor loadings from the regular factor analysis model. We show that the two models will give similar matrices of factor loadings if Schneeweiss' condition, that the difference between the largest and the smallest value of unique variances is small relative to the sizes of the column sums of squared factor loadings, holds. Furthermore, we generalize our results and discus the conditions under which the matrix of factor loadings from the regular factor analysis model will be well approximated by the matrix of factor loadings from Jöreskog's image factor analysis model. Especially, we discuss Guttman's condition (i.e., the number of variables increases without limit) for the two models to agree, in relation with the condition we have shown, and conclude that Schneeweiss' condition is a generalization of Guttman's condition. Some implications for practice are discussed.Kentaro Hayashi is a visiting Assistant Professor, Department of Mathematics, Bucknell University, Lewisburg PA 17837, and Peter M. Bentler is Professor, Departments of Psychology and Statistics, University of California, Los Angeles CA 90095-1563. (Emails: Khayashi@bucknell.edu, bentler@ucla.edu) Parts of this paper were discussed in a session on Factor Analysis (J. ten Berge, Chair) at the IFCS-98 International Conference, Rome, July, 1998. This work was supported by National Institute on Drug Abuse grant DA 01070. The authors thank Professors Hans Schneeweiss and Ke-Hai Yuan, and four anonymous referees, for their invaluable comments which led to an improved version of this paper.     

Emotional intelligence,coping and exam‐related stress in Canadian undergraduate students

The associations among emotional intelligence (EI), coping, personality and exam‐related stress in a group of 475 Canadian undergraduate students were examined. Stress was measured at the start of the semester and again in the pre‐exam period. Higher levels of stress were associated with lower scores on EI components, and higher scores on emotion‐focused coping and neuroticism. A scale‐level factor analysis of the EI and coping subscales produced three composite factors, which each had high loadings from at least one EI and one coping subscale. The associations of the Emotion Regulation factor (high loadings of several EI components and emotion‐focused coping) and the Task Focus factor (high loadings of Adaptability EI and task‐focused coping) with personality, stress and subjective wellbeing (SWB) were examined using structural equation modelling. The results showed that these factors mediated the effect of personality on stress and SWB.