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.     

A general factor in improvement with practice

Results of 10 trials on 6 tests for 50 subjects were analyzed, first, by applying the centroid method to actual improvement or practice scores and, second, by applying a formula developed by Woodrow for determining factor loadings for practice scores from the differences between factor loadings of initial and final scores. Contrary to expectation, the two methods yielded discrepant results, for the explanation of which a hypothesis is advanced. The operation of a general factor was not demonstrated. Tentative interpretations of the factors extracted by the centroid method are offered.     

A confirmatory analysis of the factor stability hypothesis

The procedures recently developed by Jöreskog for studying similarities and differences in factor structures between different groups were applied to data from a study designed by Thurstone to investigate the sampling stability of a hypothesized isolated configuration. The hypothesis of an isolated configuration was rejected but not by much. The results represent a restricted and unique simple structure solution with the factor loadings and uniqueness values invariant over the two samples.     

A method of factor analysis by means of which all coordinates of the factor matrix are given simultaneously

In general, the methods of factor analysis developed during the past five years are based on the reduction of the correlational matrix by successive steps. The first factor loadings are determined and eliminated from the correlational matrix, giving a residual matrix. This process is continued for successive factor loadings until the elements of the last obtained residual matrix may be regarded as due to chance. The method outlined in this paper assumes the maximum number of factorsm in the correlational matrix. Them factor vectors are solved for simultaneously. Once them factor vectors are found, any vectors having only negligible factor loadings may be discarded.     

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.     

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.     

A factor analysis of certain neurotic symptoms

In an attempt to investigate the concept of neurotic tendency thirty-nine of the forty-two most discriminative items in the Thurstone Neurotic Inventory were administered as a questionnaire to a group of five hundred male college students. An analysis of the table of intercorrelations by Thurstone's centroid method showed that eight factors were sufficient to account for the observed intercorrelations with negligible residuals. The eight centroid factors were then transformed into a simple structure. It is concluded that a single trait of neurotic tendency cannot be postulated and tentative hypotheses are formed as to the nature of the primary traits revealed by the analysis.Readministering the same test to the same students a week later showed a high consistency of response both on the test as a whole, and on the individual items.     

A factor analysis of the original Stanford-Binet scale

From the original Stanford-Binet scale, those items passed by between 10 and 90 per cent of a group of ten-year-old children were analyzed by the centroid method. Upon rotation, there appeared a common factor, for which two explanatory hypotheses are offered, the more tenable being that it is an effect of maturation. Primary factors tentatively identified are Number, Space, Imagery, Verbal Relations and Induction. A sixth factor apparently involves a reasoning ability and a seventh can not be interpreted.The writer is indebted to Dr. L. L. Thurstone for his interest and assistance throughout this study and to Dr. Andrew W. Brown, who made possible the collection of data at the Institute for Juvenile Research, Chicago, Illinois.     

Application of the quartimax method of rotation to Thurstone's primary mental abilities study

This study compares a quartimax rotation of the centroid factor loadings for Thurstone's Primary Mental Abilities Test Battery with factorings of the same correlation matrix by Thurstone (simple structure), Zimmerman (revised simple structure), Holzinger and Harman (bi-factor analysis), and Eysenck (group factor analysis). The quartimax results agree very closely with the solutions of Holzinger and Harman and of Eysenck, and reasonably well with the two simple structure analyses. The principal difference is the general factor provided by the quartimax solution. Reproduction of the factorial structure is sufficiently good to justify its use at least as the first stage of rotation. More extensive trial of the method will be needed with more varied data before it will be possible to decide whether quartimax factors meet psychological requirements sufficiently well without further rotation.We wish to thank Professor L. G. Henyey and the University of California Computer Center for making the IBM 701 electronic computer available for this study, and the National Science Foundation for its support of the work of the Computer Center. Professor H. F. Kaiser of the University of Illinois has made helpful criticisms of the paper, and Mr. Louis S. Davis of the University of California has assisted with preparation of the tables. The research was supported in part by the United States Air Force under Contract No. AF 33 (038)-25726 monitored by the Air Force Personnel and Training Research Center. Permission is granted for reproduction, translation, publication, use and disposal in whole and in part by or for the United States Government.A 701 program for calculation of the quartimax and varimax loadings, prepared by Professor H. F. Kaiser, is available in the library of computer programs held by the Computer Center at the University of California (Program No. 464). Mr. J. O. Neuhaus and Mr. K. W. Dickman have prepared a quartimax program for Illiac at the University of Illinois. This Illiac program will be usable on three other computers recently built or under construction: Mistic (Michigan State University), Silliac (University of Sydney), and the machine being constructed by Iowa State College.     

A cluster‐based factor rotation

A new oblique factor rotation method is proposed, the aim of which is to identify a simple and well‐clustered structure in a factor loading matrix. A criterion consisting of the complexity of a factor loading matrix and a between‐cluster dissimilarity is optimized using the gradient projection algorithm and the k‐means algorithm. It is shown that if there is an oblique rotation of an initial loading matrix that has a perfect simple structure, then the proposed method with Kaiser's normalization will produce the perfect simple structure. Although many rotation methods can also recover a perfect simple structure, they perform poorly when a perfect simple structure is not possible. In this case, the new method tends to perform better because it clusters the loadings without requiring the clusters to be perfect. Artificial and real data analyses demonstrate that the proposed method can give a simple structure, which the other methods cannot produce, and provides a more interpretable result than those of widely known rotation techniques.     

Testing a simple structure hypothesis in factor analysis

It is assumed that the investigator has set up a simple structure hypothesis in the sense that he has specified the zero loadings of the factor matrix. The maximum-likelihood method is used to estimate the factor matrix and the factor correlation matrix directly without the use of rotation methods, and the likelihood-ratio technique is used to test the simple structure hypothesis. Numerical examples are presented.The work was supported by a grant (NSF-GB 1985) from the National Science Foundation to Educational Testing Service. Reproduction in whole or in part for any purpose of the United States Government is permitted.The work was carried out when the author was Visiting Research Statistician at Educational Testing Service. The author wishes to thank Dr. Frederic M. Lord for many helpful suggestions throughout the course of this study.     

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 mathematical theory of factorial invariance under selection

It is first demonstrated that Aitken's selection formulas are equivalent to a linear transformation in the factor space. On this basis the Thomson-Ledermann theorem concerning the invariance of the number of common factors under selection, and a theorem concerning the invariance of factor loadings under selection are derived. A mathematical proof of the results of Thurstone, which are concerned with the invariance of simple structure under selection, is given. The paper provides a conclusive answer to the question, considered by Thurstone and Thomson, whether a multivariate selection is always reducible to successive univariate selections.     

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.     

Rotation to simple loadings using component loss functions: The orthogonal case

Component loss functions (CLFs) are used to generalize the quartimax criterion for orthogonal rotation in factor analysis. These replace the fourth powers of the factor loadings by an arbitrary function of the second powers. Criteria of this form were introduced by a number of authors, primarily Katz and Rohlf (1974) and Rozeboom (1991), but there has been essentially no follow-up to this work. A method so simple, natural, and general deserves to be investigated more completely. A number of theoretical results are derived including the fact that any method using a concave CLF will recover perfect simple structure whenever it exists, and there are methods that will recover Thurstone simple structure whenever it exists. Specific CLFs are identified and it is shown how to compare these using standardized plots. Numerical examples are used to illustrate and compare CLF and other methods. Sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them.The author is very indebted to a reviewer for pointing him to the generalized hyperplane count literature and to all the reviewers for valuable comments and suggestions.     

An empirical verification of the Wherry-Gaylord iterative factor analysis procedure

A comparison of the Wherry-Gaylord iterative factor analysis procedure and the Thurstone multiple-group analysis of sub-tests shows that the two methods result in the same factors. The Wherry-Gaylord method has the advantage of giving factor loadings for items. The number of iterations needed can be reduced by doing a factor analysis of sub-tests, re-grouping sub-tests according to factors, and using each group as a starting point for iterations.This research was carried out under Contract No. WSW-2503, between the Department of the Army and Ohio State University. This paper is based on the final report PRS No. 827 under that contract. The opinions expressed herein regarding matters relating to the Department of the Army are those of the authors and are not necessarily official.     

A simple method of factor analysis

A simple method for extracting correlated factors simultaneously is described. The method is based on the idea that the centroid pattern coefficients for the sections of unit rank of the complete matrix may be interpreted as structure values for the entire matrix. Only the routine centroid average process is required.     

A factor analysis of women's measurements taken for garment and pattern construction

In order to facilitate garment and pattern construction, a research which involved taking a minimum of fifty-five measurements on several thousand women was conducted by the Bureau of Home Economics. Partial correlations with age held constant were computed for a representative group of 4,128 of the women. The correlations among twenty-nine of these variables served as the basis of the present study. By a combination of the multiple-group and the centroid method of factoring, five factors were extracted. After twenty-nine rotations, simple structure was evident, and the factors were interpreted as bone length, size of joints, circumference below the waist, circumference of extremities, and circumference above the waist. The intercorrelations of the primaries were computed, and two second-order factors were extracted. One of these seems to be primarily related to the growth of fatty tissue and the other to the development of the bones.The author wishes to express her appreciation to Professor L. L. Thurstone and to the members of the Psychometric Laboratory Staff for advice relative to this study.     

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.     

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.