Some properties of the communality in multiple factor theory

Several theorems concerning properties of the communaltiy of a test in the Thurstone multiple factor theory are established. The following theorems are applicable to a battery ofn tests which are describable in terms ofr common factors, with orthogonal reference vectors.1. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors.2. The communality of a testj is equal to the square of the multiple correlation of testj with ther reference vectors and then—1 remaining tests. Corollary: The square of the multiple correlation of a testj with then—1 remaining tests is equal to or less than the communality of testj. It cannot exceed the communality.3. The square of the multiple correlation of a testj with then—1 remaining tests equals the communality of testj if the group of tests containsr statistically independent ests teach with a communality of unity.4. With correlation coefficients corrected for attenuation, when the number of tests increases indefinitely while the rank of the correlational matrix remains unchanged, the communality of a testj equals the square of the multiple correlation of testj with then—1 remaining tests.5. With raw correlation coefficients, it is shown in a special case that the square of the multiple correlation of a testj with then—1 remaining tests approaches the communality of testj as a limit when the number of tests increases indefinitely while the rank of correlational matrix remains the same. This has not yet been proved for the general case.The author wishes to express his appreciation of the encouragement and assistance given him by Dr. L. L. Thurstone.     

Maximum likelihood solution to factor analysis when some factors are completely specified

In the factor-analytic model, let some of the factors be known (i.e., the factor loadings are given in advance; they maye.g. be obtained from some previous analyses). However, their covariance matrix may, or may not, be known. The remaining factors (if any) are assumed to be uncorrelated among themselves and to the first set. For this model, the maximum likelihood equations are obtained and an iterative method for the solution is proposed.The work was done while the author was at Deutsches Rechenzentrum, Darmstadt, Germany.     

The determination of the factor loadings of a given test from the known factor loadings of other tests

A technique is indicated by which approximations to the factor loadings of a new test may be obtained if factor loadings of a given group of tests and the correlations of the new test with the other tests are known. The technique is applicable to any orthogonal system and is especially adapted to cases in which a _ji a _jk = 0 wheni k. Application is also made to the simultaneous determination of the factor weights of a group of tests in which no additional common factor is present. The technique is useful in adding tests to a completed factorial solution and in using factorial solutions involving errors to give results which are approximately correct.     

A semi-orthogonal dependent factor solution

The paper presents a general framework for the dependent factor method in which judgmental as well as analytic criteria may be employed.The procedure involves a semi-orthogonal transformation of an oblique solution comprising a number of reference factors and a number of experimental factors (composites). It determines analytically the residual factors in the experimental field keeping the reference field constant. The method is shown to be a generalization of the multiple factor procedure [Thurstone, 1947] in so far as it depends on the use of generalized inverse in derivingpartial structure from total pattern. It is also shown to provide an example of the previously empty category (Case III) of the Harris-Kaiser generalization [1964]. A convenient computational procedure is provided. It is based on an extension of Aitken's [1937] method of pivotal condensation of a triple-product matrix to the evaluation of a matrix of the formH — VA ^–1 U' (for a nonsingularA).     

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.     

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 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.     

Estimating the standard errors of rotated factor loadings by jackknifing

The jackknife by groups and modifications of the jackknife by groups are used to estimate standard errors of rotated factor loadings for selected populations in common factor model maximum likelihood factor analysis. Simulations are performed in whicht-statistics based upon these jackknife estimates of the standard errors are computed. The validity of thet-statistics and their associated confidence intervals is assessed. Methods are given through which the computational efficiency of the jackknife may be greatly enhanced in the factor analysis model.Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by NIH Special Research Resources Grant RR-3.The author wishes to thank his doctoral committee co-chairmen, Drs James W. Frane and Robert I. Jennrich, UCLA, for their contributions to this research.     

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.     

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.     

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.     

An exploration of genetic g: Hierarchical factor analysis of cognitive data from the western reserve twin project

Hierarchical factor analyses involving Schmid-Leiman transformations (Schmid & Leiman, 1957) were conducted on specific cognitive abilities data collected in a sample of 148 identical (MZ) and 135 same-sex fraternal (DZ) twin pairs. Two main questions were addressed: First, are genetic influences on specific cognitive abilities simply a reflection of their g loading, or are different sets of genes affecting separate abilities? Second, to the extent that specific cognitive abilities are affected by common genetic variance, how similar is the common genetic factor to a phenotypic factor reflecting g? Model fitting results suggest that genetic influences on specific abilities are a reflection of both general intelligence and genetic influences specific to separate abilities and that loadings on the common genetic factor are more highly correlated with phenotypic g loadings than are common environmental factor loadings.     

Alpha-maximized factor analysis (alphamax): Its relation to alpha and canonical factor analysis

An alpha-O coefficient of internal consistency is defined for an observed score composite. Maximizing alpha-O leads to a system of psychometric (vs. statistical) factor analysis in which successive factors describe dimensions of successively less internal-consistency. Factoring stops when alpha-O is zero or less. In contrast to Kaiser-Caffrey's alpha-C analysis, when the factored matrix is rank 1, alpha-O does not reach unity; it can approach unity only as the number of variables reach infinity. The relative usefulness and domains of generalization of alpha-C and alpha-O are compared. Basically, alpha-C analysis is concerned with the representativeness of factors while alpha-O analysis is concerned with the assessibility of factors. Consequently, either system of factoring can and should be summarized by both the alpha-C and alpha-O coefficients. Not surprisingly, alpha-O analysis is computationally analogous to Rao's canonical factor analysis.     

EM algorithms for ML factor analysis

The details of EM algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models. The algorithm is essentially the same for both cases and involves only simple least squares regression operations; the largest matrix inversion required is for aq ×q symmetric matrix whereq is the matrix of factors. The example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other; such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.     

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.     

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 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.     

On the interpretation of common factors: A criticism and a statement

The concept of simple structure is criticized for lack of objectivity and for failure to produce invariance of (a) factor loadings under change of battery, and (b) individual scores on primary traits under change of tests. It is also criticized on the grounds that simple structure yields at best the factors which were put in and that, by suitable manipulation of tests, any set of factors may be represented in a battery by tests which will yield a simple structure. A procedure for rotation is developed which locates the first rotated axis to pass through a cluster of tests which, by hypothesis, contain a common factor and to project all tests into a hyperplane orthogonal to this factor. The second factor is then located to pass through the projections of a second cluster of tests defining the second factor, and so on, until all hypothecated factors have been located. Any residual interrelations may then be rotated graphically to the most plausible arrangement.     

Factors in the learning behavior of the albino rat

The table of intercorrelations published by Dr. R. L. Thorndike in theGenetic Psychology Monographs, 1935,17, No. 1, has been reanalyzed into ten factors. Rotation of the reference vectors resulted in a configuration which, for all practical purposes, may be said to show a simple structure and (except for one variable on one factor) a positive manifold. Five of the factors can be interpreted with some confidence, one only with considerable caution; three factors are specific to the apparatus employed, and for the one remaining factor, no interpretation is attempted; it appears to be a residual plane.     

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.