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.     

Piagetian tasks measure intelligence and intelligence tests assess cognitive development: A reanalysis

A reanalysis of the intercorrelations of Wechsler subtests, achievements tests, and Piagetian tasks originally published by Stephens et al. (1972) has corrected errors in the original design and has used models which allow both differences and similarities to appear. Unit weighted composites of the two sets of tests were formed after first partialling chronological age out of the complete table of intercorrelations. The correlation between the composites is .876 which corrects, using split-half estimates of reliabilities, to .918. Next the matrix of partial correlations was factored and the factors rotated to a hierarchical structure. Piagetian tasks contribute almost equally to the definition of the general factor in intelligence along with the Wechsler subtests and the achievement tests. The general factor, furthermore, makes the single largest contribution to the variance of most of the measures of both sets. The smaller group factors consist of an Academic Achievement factor and three Piagetian factors. Of the latter the largest and most clearly identified is the factor called Operational Thought in the earlier analysis, but is here described simply as Conservation. It is concluded that the evidence for communality in function far outweighs the differences between intelligence tests and Piagetian tasks.     

Rotational Uniqueness Conditions Under Oblique Factor Correlation Metric

In an addendum to his seminal 1969 article Jöreskog stated two sets of conditions for rotational identification of the oblique factor solution under utilization of fixed zero elements in the factor loadings matrix (Jöreskog in Advances in factor analysis and structural equation models, pp. 40–43, 1979). These condition sets, formulated under factor correlation and factor covariance metrics, respectively, were claimed to be equivalent and to lead to global rotational uniqueness of the factor solution. It is shown here that the conditions for the oblique factor correlation structure need to be amended for global rotational uniqueness, and, hence, that the condition sets are not equivalent in terms of unicity of the solution.     

Multiple rectilinear prediction and the resolution into components: II

Given a battery ofn tests that has already been resolved intor orthogonal common factors andn unique factors, procedures are outlined for computing the following types of linear multiple regressions directly from the factor loadings: (i) the regression of any one test on then?1 remaining tests; (ii) all then different regressions of ordern?1 for then tests, computed simultaneously; (iii) the regression of any common factor on then tests; (iv) the regressions of all the common factors on then tests computed simultaneously; (v) the regression of any unique factor on then tests; (vi) the regressions of all the unique factors on then tests, computed simultaneously. Multiple and partial correlations are then determined by ordinary formulas from the regression coefficients. A worksheet with explicit instructions is provided, with a completely worked out example. Computing these regressions directly from the factor loadings is a labor-saving device, the efficiency of which increases as the number of tests increases. The amount of labor depends essentially on the number of common factors. This is in contrast to computations based on the original test intercorrelations, where the amount of labor increases more than proportionately as the number of tests increases. The procedures evaluate formulas developed in a previous paper (2). They are based essentially on a shortened way of computing the inverse of the test intercorrelation matrix by use of the factor loadings.     

Factors of style and personality

In an attempt to find relationships between psychological and linguistic variables, style samples of short stories, 300 words each, were analyzed according to formal criteria and the results were correlated with scores on personality tests. The number of significant correlations supported the hypothesis that style is related to personality. A factor analysis, using the principal component solution and Varimax rotation (Harman, 1967), of the correlation matrix resulted in six identifiable factors of style, three factors of psychological tests, and a large number of small factors, each represented only by two to five experimental variables with significant loadings. A significant loading for interpretation purposes was defined in agreement with Guilford (1956) as 0.30 or greater, positive or negative. A close examination of style factors led to the tentative differentiation of a basic language factor as resulting from grammatical constraint, and several factors of subjective style of individuals. Some of these later factors had enough loadings on personality variables to permit cautious psychological interpretation.     

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.     

A new iterative method for correcting erroneous communality estimates in factor analysis

A new method for correcting erroneous communality estimates is applicable to any completed orthogonal factor solution. It seeks, by direct correction of factor loadings, to make the residuals conform to the chance error criteria of zero mean and zero skewness for each row separately. Two numerical examples, with one and two factors, respectively, are presented. The method can be used as a short cut for Dwyer's extension in adding variables to a matrix. It can also be used as a short cut in cross-validation factor studies. Successful use on problems with many variables and numerous factors is claimed. Factors can be made oblique,after correction, if desired.     

A new algorithm for the least-squares solution in factor analysis

A new algorithm to obtain the least-squares or MINRES solution in common factor analysis is presented. It is based on the up-and-down Marquardt algorithm developed by the present authors for a general nonlinear least-squares problem. Experiments with some numerical models and some empirical data sets showed that the algorithm worked nicely and that SMC (Squared Multiple Correlation) performed best among four sets of initial values for common variances but that the solution might sometimes be very sensitive to fluctuations in the sample covariance matrix.Numerical computation was made on a NEAC S-1000 computer in the Computer Center, Osaka University.     

Relations amongm sets of measures

The problem of determining linear functions for two sets of variables so as to maximize the correlation between the two functions has been solved by Hotelling. This article presents a more efficient computational solution for the case of two sets of variables and a generalized solution for any number of sets. Applications are discussed and a numerical example is included to demonstrate the solution for more than two sets.     

Tandem criteria for analytic rotation in factor analysis

Two related orthogonal analytic rotation criteria for factor analysis are proposed. Criterion I is based upon the principle that variables which appear on the same factor should be correlated. Criterion II is based upon the principle that variables which are uncorrelated should not appear on the same factor. The recommended procedure is to rotate first by criterion I, eliminate the minor factors, and then rerotate the remaining major factors by criterion II. An example is presented in which this procedure produced a rotational solution very close to expectations whereas a varimax solution exhibited certain distortions. A computer program is provided.     

An inter-battery method of factor analysis

The inter-battery method of factor analysis was devised to provide information relevant to the stability of factors over different selections of tests. Two batteries of tests, postulated to depend on the same common factors, but not parallel tests, are given to one sample of individuals. Factors are determined from the correlation of the tests in one battery with the tests in the other battery. These factors are only those that are common to the two batteries. No communality estimates are required. A statistical test is provided for judging the minimum number of factors involved. Rotation of axes is carried out independently for the two batteries. A final step provides the correlation between factors determined by scores on the tests in the two batteries. The correlations between corresponding factors are taken as factor reliability coefficients.This research was jointly supported by Princeton University and the Office of Naval Research under contract N6onr-270-20 and the National Science Foundation under grant NSF G-642; Harold Gulliksen, principal investigator. The preparation of this paper and the accompanying material has been aided by the Educational Testing Service. The author is grateful to Professors Harold Gulliksen and Samuel S. Wilks for their many most helpful comments and suggestions.     

Linear relations amongk sets of variables

A family of solutions for linear relations amongk sets of variables is proposed. It is shown how these solutions apply fork=2, and how they can be generalized from there tok3.The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.     

Evidence for a third factor on the Positive and Negative Affect Schedule in a college student sample

The Positive and Negative Affect Schedule (PANAS) was originally developed to measure two orthogonal dimensions of affect. The present study examined the factor structure of the PANAS in a sample of 302 undergraduates. maximum Likelihood factor analysis was used to compare two- and three-factor solutions to self-rated affect. The two-factor solution resulted in confirmation of the two factors of Positive and Negative Affect hypothesized to underlie the schedule. When, however, a three-factor solution was specified, the Positive Affect factor was retained, while the Negative Affect factor split into two lower-order factors generally consistent with the Upset and Afraid factors described by Mehrabian in 1997. These findings highlight the need for research to consider the possible influence of a third affective dimension, such as Dominance-Submissiveness on self-rated affective experience.     

Further examination of the Self-efficacy Scale

The purpose of this study was to examine further the factorial validity of the Self-efficacy Scale via component and subsequent correlational analyses. 651 undergraduates enrolled in an introductory psychology course voluntarily completed the Self-efficacy Scale and the Bem Sex-role Inventory. A principal component analysis with an orthogonal rotation produced a two-factor solution which was remarkably similar to the factor structures reported previously. The two factors (General Self-efficacy and Social Self-efficacy) accounted for about 34% of the total variance. Further, the correlation coefficients indicated that General Self-efficacy was more strongly related to masculine traits than to feminine traits, as defined by the Bem Sex-role Inventory.     

Factor analysis of the army air forces sheppard field battery of experimental aptitude tests

A factor analysis was made of 39 experimental printed aptitude tests and seven reference tests selected from the Army Air Forces Aircrew Classification Battery. Thirteen factors were extracted and two independent orthogonal rotational solutions were completed. Twelve factors were interpreted. Of these, seven were clearly identifiable with previously known factors: numerical, perceptual-speed, spatial-relations, visualization, visual-memory, paired-associates-memory, and length-estimation factors. A planning factor was not as clearly identifiable. A reasoning factor was probably a composite of two or more factors that failed to separate. A new factor possibly has to do with orientation with respect to the points of the compass. Two factors were doublets, each apparently specific to one kind of test. Better conceptions were gained of the spatial-relations and visualization factors and of the kinds of tests that measure them best. Efforts to improve measures of unique factors were not uniformly successful. The attempt to duplicate a psychomotor test rather directly by analogy in printed form failed almost completely.For support of a large part of the investigation the senior author is indebted to the Social Sciences Research Council for a grant-in-aid. Dr. Fruchter supervised the computational work in the extraction of factors and in one of the rotational solutions. Much of the work was done while he was employed by the Air Training Command Human Resources Research Center. While the opinions and conclusions are those of the authors, they wish to express their appreciation to that organization and especially to Dr. John T. Dailey for making the facilities possible and to Mr. William B. Lecznar for technical and computational assistance.     

The curvex: Simple order structure revealed in ratings of complexity, interestingness, and pleasantness

An alternative interpretation is offered of some factor analytic studies of ratings of aesthetic stimuli. Earlier interpretation has been in terms of two orthogonal factors, Evaluation and Activity. The re-interpretation is based on the circular fan-like structure of the factor plot, reflected in a high degree of order in the correlation matrix (the curvex structure), the semantically meaningful progression of the bi-polar adjective scales around the circle, and the fact that the stimulus objects in some cases are located along an ordered continuum in the factor space. Coombs and Kao's application of unfolding concepts to factor analysis of preferential choice data and Guttman's concept of order factor analysis is applied. It is proposed that the ratings reflect one underlying stimulus dimension, objective complexity, and that the rating scales represent ideal points along this dimension. The scales Pleasing, Interesting and Complex are hypothesized to reach their ideal point at increasingly higher levels of objective complexity. Ten of the 13 factor analyses studied were in agreement with the hypothesis.     

Criterion Validity is Maintained When Items are Evaluatively Neutralized: Evidence from a Full‐scale Five‐factor Model Inventory

The original version and an evaluatively neutralized version (with items rephrased to reduce popularity) of a personality inventory were compared. The results revealed (i) similar criterion validity across three different sets of self‐rated behaviours, (ii) stronger relations to the rated social desirability of criteria for the original version and (iii) less correlation between factors for the neutralized version. We take the results to indicate that evaluative neutralization is a viable technique for reducing social desirability in self‐ratings. Implications for test construction are discussed. Copyright © 2014 European Association of Personality Psychology     

The Assessment of Dimensionality for Use in Item Response Theory

The application of item response theory (IRT) models requires the identification of the data's dimensionality. A popular method for determining the number of latent dimensions is the factor analysis of a correlation matrix. Unlike factor analysis, which is based on a linear model, IRT assumes a nonlinear relationship between item performance and ability. Because multidimensional scaling (MDS) assumes a monotonic relationship this method may be useful for the assessment of a data set's dimensionality for use with IRT models. This study compared MDS, exploratory and confirmatory factor analysis (EFA and CFA, respectively) in the assessment of the dimensionality of data sets which had been generated to be either one- or two-dimensional. In addition, the data sets differed in the degree of interdimensional correlation and in the number of items defining a dimension. Results showed that MDS and CFA were able to correctly identify the number of latent dimensions for all data sets. In general, EFA was able to correctly identify the data's dimensionality, except for data whose interdimensional correlation was high.     

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.     

An adolescent symptom checklist for juvenile deliquents

A rating instrument was developed to assess the personality functioning of problem adolescents who have a brief history of official delinquency. Subjects were 186 residents at a short-term diagnostic detention facility. The instrument that emerged from a factor analysis was tested for reliability, validity, and cross-situational generality. Two major orthogonal factors emerged, Apathy-Withdrawal (I) and Anger-Defiance (II). Each factor was found to be congruent with the corresponding dimension on behavior rating instruments that had previously yielded two orthogonal factors and that had been shown to be relevant to a large number of psychological and educational issues. Significant correlations between the factors and a series of global ratings measuring behavior and measures indicative of the frequency and seriousness of official delinquency provided further evidence of construct and discriminant validity as well as evidence of cross-setting consistency of personality functioning.