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 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.     

Exploratory Factor Analysis With Small Samples and Missing Data

Exploratory factor analysis (EFA) is an extremely popular method for determining the underlying factor structure for a set of variables. Due to its exploratory nature, EFA is notorious for being conducted with small sample sizes, and recent reviews of psychological research have reported that between 40% and 60% of applied studies have 200 or fewer observations. Recent methodological studies have addressed small size requirements for EFA models; however, these models have only considered complete data, which are the exception rather than the rule in psychology. Furthermore, the extant literature on missing data techniques with small samples is scant, and nearly all existing studies focus on topics that are not of primary interest to EFA models. Therefore, this article presents a simulation to assess the performance of various missing data techniques for EFA models with both small samples and missing data. Results show that deletion methods do not extract the proper number of factors and estimate the factor loadings with severe bias, even when data are missing completely at random. Predictive mean matching is the best method overall when considering extracting the correct number of factors and estimating factor loadings without bias, although 2-stage estimation was a close second.     

Variance estimation in factor analysis: An application of the bootstrap

Sampling variability of the estimates of factor loadings is neglected in modern factor analysis. Such investigations are generally normal theory based and asymptotic in nature. The bootstrap, a computer-based methodology, is described and then applied to demonstrate how the sampling variability of the estimates of factor loadings can be estimated for a given set of data. The issue of the number of factors to be retained in a factor model is also addressed. The bootstrap is shown to be an effective data-analytic tool for computing various statistics of interest which are otherwise intractable.     

An item-factor analysis of the Eysenck Personality Questionnaire (E.P.Q.): Will the real personality factors stand up

The study utilised responses to the E.P.Q. from 3 samples of subjects. An initial sample of 239 Independent Study Students, a second sample of 278 student and pupil nurses and an aggregate sample of 617 students of both types. The samples were subjected to principal components analyses followed by varimax and promax rotations. Scree Test, number of significant loadings, replicability across samples and sexes, factor reliability and factor replicability were used as criterion measures to determine the number of real factors present. Nearly all the N, E and L-items were recovered in the analyses of each of the samples. In the two smaller samples, insufficient items were retrieved to indicate the presence of a clear psychoticism factor, but in the aggregate sample one factor contained 19 of the 25 P-items. Scree Test, numbers of significant loadings, replicability over samples and factor reliability criteria indicated that either a 4- or 5-factor solution was a valid possibility. Application of the criterion of factor replicability, using factor comparability coefficients based on factor scores, revealed an exceptionally clear 4-factor structure which was also replicable across sexes. The effects of dissimulation were investigated by dichotomising the male and female samples by L-scores. In both cases the mean scores for P and N for the low L-score group were higher than the corresponding mean scores for the high L-score group, which supports a previous finding that dissimulation tends to artificially lower the P and N-scores. Examination of the effect of dissimulation on the factor structure showed that E and N came out equally clearly in the high and low L-score groups for both males and females. P came out more clearly in the low L-score group for males and in the high L-score group for females, the latter finding being contrary to that of other authors. The investigation confirmed the presence of 4 “real”, replicable factors of P, E, N and L at the first order, showed that P and N are sensitive to dissimulation and illustrated the effectiveness of using factor replicability, measured by factor comparability coefficients based on factor scores, to determine the number of “real” factors in an analysis.     

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 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.     

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.     

Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.     

Some relationships between factors and components

The asymptotic correlations between the estimates of factor and component loadings are obtained for the exploratory factor analysis model with the assumption of a multivariate normal distribution for manifest variables. The asymptotic correlations are derived for the cases of unstandardized and standardized manifest variables with orthogonal and oblique rotations. Based on the above results, the asymptotic standard errors for estimated correlations between factors and components are derived. Further, the asymptotic standard error of the mean squared canonical correlation for factors and components, which is an overall index for the closeness of factors and components, is derived. The results of a Monte Carlo simulation are presented to show the usefulness of the asymptotic results in the data with a finite sample size.The author is indebted to anonymous referees for their comments, corrections and suggestions which have led to the improvement of this article.     

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 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 chimpanzee (Pan troglodytes) analogue of cross-national generalization of personality structure: zoological parks and an African sanctuary

Six personality factors, including five resembling the human Big Five, had previously been identified in a separate group of zoo-housed chimpanzees. Comparability of chimpanzee personality factor structure was examined in two highly contrasting habitats: zoos and a large African sanctuary. Questionnaires for the zoo chimpanzees were in English, while most for the chimpanzees in the sanctuary were in French. Differences between the two settings were sufficiently extensive to make them analogous to cross-national human personality studies. Internal consistencies for five of the six factors did not differ between the two samples. The patterns of correlations between the unit-weighted factors were also similar for the two samples. Data from these two samples were pooled and factor analyzed. The resulting factor structure was then rotated to the factor structure described in the original study of chimpanzee personality. Dominance, Extraversion, Dependability, and Agreeableness had high congruences. Emotionality and Openness did not, but the items that had the highest loadings were consistent with the factors' definitions. Finally, sex and age effects for all factors generalized across habitats.     

Unavailability Effects on Message Processing: A Theoretical Analysis an Empirical Test

This study is concerned with the ability of standard indexes, of socioeconomic status and similar devices to measure major dimensions of social stratification for whites and blacks. Factor analyses of survey data for samples of both races uncovered five matching factors and two others unique to a single sample, all of which seem to represent important components of stratification. Most of the indexes were relevant to three matching factors, and some of these measures consistently loaded them in both samples. However, several of the pertinent indexes only loaded white versions of these factors, and some failed to load any. Few, if any, indexes were relevant to or loaded the remaining factors. Additionally, some indexes were not pertinent to and did not load any factor. In contrast to the indexes, other variables loaded all but one of the factors. The main conclusions are that: (1) the coverage of the indexes is restricted; (2) the devices are not uniformly applicable to blacks; and (3) the complexity of this domain requires separate measures of each factor, achievable by combining standard indexes and new variables.     

Psychometric Properties of the Depressive Experiences Questionnaire in a College Population

The Depressive Experiences Questionnaire (DEQ; Blatt, D'Afflitti, & Quinlan, 1976) was developed to permit the study of the continuities between normal and pathological forms of depression. Since its publication over 10 years ago, the DEQ has been increasingly used in a wide range of investigations of depression in clinical and nonclinical samples and as a measure of two major personality dimensions in a number of studies of differential responsiveness to various life stressors. This study provides an analysis of the DEQ responses in a large sample of undergraduate subjects in an attempt to articulate more clearly the psychometric properties of the DEQ. The DEQ consists of 66 items describing experiences frequently reported by depressed individuals. In the original development of the DEQ, factor analyses using male and female undergraduate samples yielded three orthogonal factors, Dependency, Self-Criticism, and Efficacy (Blatt et al., 1976). Most subsequent studies have used the factor scoring coefficients from the female sample to score men and women's DEQs. The factor loadings in the new male and female samples were very similar to one another and to the original female factor loadings. Thus, the continued use of the scoring parameters from the original female sample is recommended. Factor scores for Dependency and Self-Criticism continue to be uncorrelated in both men and women. Normative data are presented from the new samples, with men scoring higher on Self-Criticism and women scoring higher on Dependency.     

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.     

Selecting polychoric instrumental variables in confirmatory factor analysis: An alternative specification test and effects of instrumental variables

The polychoric instrumental variable (PIV) approach is a recently proposed method to fit a confirmatory factor analysis model with ordinal data. In this paper, we first examine the small-sample properties of the specification tests for testing the validity of instrumental variables (IVs). Second, we investigate the effects of using different numbers of IVs. Our results show that specification tests derived for continuous data are extremely oversized at all sample sizes when applied to ordinal variables. Possible modifications for ordinal data are proposed in the present study. Simulation results show that the modified specification tests with all available IVs are able to detect model misspecification. In terms of estimation accuracy, the PIV approach where the IVs outnumber the endogenous variables by one produces a lower bias but a higher variation than the PIV approach with more IVs for correctly specified factor loadings at small samples.     

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.