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.     

General theory and methods for matric factoring

Methods are developed for factoring an arbitrary rectangular matrixS of rankr into the formFP, whereF hasr columns andP hasr rows. For the statistical problem of factor analysis,S may be the score matrix of a population of individuals on a battery of tests. ThenF is a matrix of factor loadings,P is a matrix of factor scores, andr is the number of factor variates. (As in current procedures, there remains a subsequent problem of rotation of axes and interpretation of factors, which is not discussed here.) Methods are also developed for factoring an arbitrary Gramian matrixG of rankr into the formFF, whereF hasr columns andF denotesF transposed. For the statistical problem of factor analysis,G may be the matrix of intercorrelations,R, of a battery of tests, with unity, communalities, or other parameters in the principal diagonal.R is proportional toSS, and it is shown thatS can be factored by factoringR. This may usually be the most economical procedure in practice; it should not be overlooked, however, thatS can be factored directly. The general methods build up anF (andP) in as many stages as desired; as many factors as may be deemed computationally practical can be extracted at a time. Perhaps it will usually be found convenient to extract not more than three factors at a time. Current procedures, like the centroid and principal axes, are special cases of a general method presented here for extracting one factor at a time.     

Multiple group methods for common-factor analysis: Their basis,computation, and interpretation

In a previous paper (1) were developed three basic theorems which were shown to provide numerical routines, as well as algebraic proof, for existing common-factor methods. New multiple routines were also indicated. The first theorem showed how to extract as many common factors as one wished from the correlation matrix in one operation. The second theorem showed how to do the same from the score matrix. The third proved that factoring the correlation matrix was equivalent to factoring the score matrix. A particular application of these theorems is the multiple group factoring method, which the writer first used in practice on some Army attitude scores during World War II. The present paper explains the basic theorems in more detail with special reference to group factoring. Computations are outlined as consisting of five simple matric operations. The meaning of commonfactor analysis is given in terms of the basic theorems, as well as the relationship to inverted factor theory.     

A note on sufficient conditions that a common factor will be determinate in an infinite domain of variables

It is proved for the common factor model withr common factors that under certain condition s which maintain the distinctiveness of each common factor a given common factor will be determinate if there exists an unlimited number of variables in the model each having an absolute correlation with the factor greater than some arbitrarily small positive quantity.The author is indebted to R. P. McDonald for suggesting the proof of Guttman's determinantal equation for the squared multiple correlation in predicting a factor from the observed variables used in the parenthetical note.     

Factor pattern of test items and tests as a function of the correlation coefficient: Content,difficulty, and constant error factors

A dilemma was created for factor analysts by Ferguson (Psychometrika, 1941,6, 323–329) when he demonstrated that test items or sub-tests of varying difficulty will yield a correlation matrix of rank greater than 1, even though the material from which the items or sub-tests are drawn is homogeneous, although homogeneity of such material had been defined operationally by factor analysts as having a correlation matrix of rank 1. This dilemma has been resolved as a case of ambiguity, which lay in (1) failure to specify whether homogeneity was to apply to content, difficulty, or both, and (2) failure to state explicitly the kind of correlation to be used in obtaining the matrix. It is demonstrated that (1) if the material but (2) if content is homogeneous but difficulty is not, the homogeneity of the content can be demonstrated only by using the tetrachoric correlation coefficient in deriving the matrix; and that the use of the phi-coefficient (Pearsonianr) will disclose only the nonhomogeneity of the difficulty and lead to a series ofconstant error factors as contrasted withcontent factors. Since varying difficulty of items (and possibly of sub-tests) is desirable as well as practically unavoidable, it is recommended that all factor analysis problems be carried out with tetrachoric correlations. While no one would want to obtain the constant error factors by factor analysis (difficulty being more easily obtained by counting passes), their importance for test construction is pointed out.     

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.     

Rotation to achieve factorial invariance

Under certain conditions it is reasonable to assume that the same factor pattern matrix will describe the regression of observed on factor scores in different populations. However, ordinary factoring procedures will not reveal in general the existence of such a factor pattern matrix. Two procedures for rotating any number of factor pattern matrices based on different populations to conform to a single best fitting factor pattern matrix are developed in this paper. It is assumed that the same number of factors have been determined for each population. Both procedures will yield oblique results in the various populations. The procedures are illustrated with data taken from the 1939 Holzinger-Swineford monograph. Four groups of individuals are utilized.     

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.     

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).     

Communality of a variable: Formulation by cluster analysis

The communality of a variable represents the degree of its generality acrossn – 1 behaviors. Domain-sampling principles provide a fundamental conception and definition of the communality. This definition may be alternatively stated in eight different ways. Three definitions lead to precise formulas that determine thetrue value of the communality: (i) from thek necessary and sufficient dimensions derived by iterated factoring, (ii) from then – 1 remaining variable-domains, and (iii) fromk' multiple clusters of then variables. Seven definitions provide approximation formulas: (i) one from thek dimensions as initially factored, (ii) one from then – 1 remaining variables, and (iii) five from a single cluster. Rank of the matrix is not a desiratum in some definitions. Using an example designed by Guilford to illustrate multiple-factor analysis, applications of the formulas based on the three precise definitions recover the true communalities, and five approximation formulas each gives values closer than thead hoc estimates usually employed in factor analysis.The writer wishes to express his indebtedness to C. F. Wrigley and H. Kaiser for their many helpful constructive criticisms.     

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.     

Maximizing predictive efficiency for a fixed total testing time

For any fixed total time of testing it is possible, through proper item-and-time allotment, to combine tests into a battery so that the multiple correlation with a pre-assigned criterion will be maximized. By holding constant the ratio of the length in number of items to the time length for each test, a set of general equations has been derived which will yield this maximum value of the multipleR and will enable one to determine, in any given case, the optimal fraction of total testing time that should be devoted to each type of test under consideration. The set of general equations is applied to a two-test-battery problem to obtain the optimal length of each type of test for one hour total testing time. If two other tests had been selected for the two-test sample problem, different subdivisions of the total time would generally occur. The manner in which the results would change when using other tests with different initial reliability, validity, and intercorrelation values is briefly presented. Some general implications of this method of battery development are also discussed.The writer is indebted to Max Woodbury for his assistance and especially to Dr. N. J. F. Van Steenberg and Dr. Anna S. Henriques, who provided valuable guidance and aid in the development of the solution to this problem. This paper is a revision of a thesis submitted in 1939 at the University of Utah in partial fulfillment of the requirements for the master's degree.     

Matrix correlation

A correlational measure for ann byp matrixX and ann byq matrixY assesses their relation without specifying either as a fixed target. This paper discusses a number of useful measures of correlation, with emphasis on measures which are invariant with respect to rotations or changes in singular values of either matrix. The maximization of matrix correlation with respect to transformationsXL andYM is discussed where one or both transformations are constrained to be orthogonal. Special attention is focussed on transformations which causeXL andYM to ben bys, wheres may be any number between 1 and min (p, q). An efficient algorithm is described for maximizing the correlation betweenXL andYM where analytic solutions do not exist. A factor analytic example is presented illustrating the advantages of various coefficients and of varying the number of columns of the transformed matrices.This research was supported by grant APA 0320 from the Natural Sciences and Engineering Research Council of Canada. The authors wish to acknowledge valuable discussion of this problem with Jan de Leeuw, University of Leiden.     

On the symmetric treatment of an asymmetric approach to factor analysis

On the assumption that a partitioning can be found such that three mutually exclusive test vector configurations span the same factor space, a procedure is derived whereby symmetric parts of the correlation matrix are estimated from functions of asymmetric parts treated symmetrically. This yields an explicit matrix formula for communality estimation which generalizes earlier work by Albert. Conventional factoring methods, with all their computational and fitting advantages, can be applied once the symmetric portions of the correlation matrix have been estimated. Extension to four subgroups of test vectors allows for a matrix generalization of the old tetrad difference criterion to the multiple-factor case.     

A note on the discovery of ag factor by means of Thurstone's centroid method of analysis

A fictitious factor matrix including 16 tests and 3 factors, one of which was ag factor, was prescribed. From it two typical factor problems, including errors of sampling, were derived. Students in training, without awareness of the factor patterns, arrived at essentially correct solutions by the use of Thurstone's centroid method with rotation of axes. Errors in the calculated factor matrix were very close in size to the sampling errors in the correlation coefficients. It is concluded that ag factor need not escape detection by Thurstone's procedures if the criteria of complete simple structure are not demanded.     

A Unified Latent Curve,Latent State-Trait Analysis of the Developmental Trajectories and Correlates of Positive Orientation

The subgrouping strategy has been employed in a number of studies (Fredriksen &; Gilbert, 1960; Fredriksen &; Melville, 1964; Ghiselli, 1956; Klein, Rock &; Evans, 1968; Rock, Evans &; Klein, 1969; Malnig, 1969). In each of these studies it was shown that after partitioning a sample into g subgroups and performing a separate regression analysis within each group, one could identify one or more subgroups having a larger multiple correlation than the total sample regression. An important question is whether this type of result will be maintained under cross validation.     

Note on the computation of the inverse of a triangular matrix

A simplified method of computing the inverse of a triangular matrix is presented. It is useful with the multiple-group method of factoring the correlation matrix as well as with other factor-analysis and multiple-correlation problems.     

Examination of the reliability and validity of the personality assessment inventory

The reliability, discriminant validity, and construct validity of the Personality Assessment Inventory (PAI) — a multidimensional self-report measure of abnormal personality traits — were examined within the Australian context. Subjects included 151 normals, 30 alcoholics, and 30 schizophrenic patients. A subsample of 70 nonpsychiatric adults responded to the PAI items twice over a test-retest interval of 28 days. The resulting median retest coefficient was 0.7, indicating less than optimal stability. The median alpha (KR₂₁) coefficient was 0.8, suggesting somewhat narrow measurement scales. A significant multivariate main effect was obtained across groups after the effects of age and gender were removed. Multiple comparisons for each of the PAI scales revealed significant differences between the respective groups, as discussed. A higher-order scale factoring did not strongly support the purported PAI structure. In reanalyses of the correlation matrices included in the Professional Manual, the purported PAI factor structure was unable to be replicated for the standardization clinical sample (N=1246), and a confirmatory factor analysis using the normative (validation) correlational data (N=1000) revealed poor fit indices, raising further concerns about construct validity.     

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.     

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.