Conditions for factor (in)determinacy in factor analysis

The subject of factor indeterminacy has a vast history in factor analysis (Guttman, 1955; Lederman, 1938; Wilson, 1928). It has lead to strong differences in opinion (Steiger, 1979). The current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables. Five conditions are given which rigorously define indeterminacy. It is shown that (un)observable factors are (in)determinate. Specifically, the indeterminacy proof by Guttman is extended to Heywood cases. The results are illustrated by two examples and implications for indeterminacy are discussed.     

A comparison of factor scores under conditions of factor obliquity

Six different methods of computing factor scores were investigated in a simulation study. Population scores created from oblique factor patterns selected from the psychological literature served as the bases for the simulations, and the stability of the different methods was assessed through cross-validation in a subject-sampling model. Results from 5 evaluative criteria indicated that a simplified, unit-weighting procedure based on the factor score coefficients was generally superior to several unit-weighting procedures based on the pattern or structure coefficients. This simplified method of computing factor scores also compared favorably with an exact-weighting scheme based on the full factor score coefficient matrix. Results are discussed with regard to their potential impact on current practice, and several recommendations are offered.     

Longitudinal factor analysis

A model is presented for factor analysing scores on a set of psychological tests administered as both pre- and postmeasures in a study of change. The model assumes that the same factors underlie the tests on each occasion, but that factor scores as well as factor loadings may change between occasions. Factors are defined to be orthogonal between as well as within occasions. A two-stage least squares procedure for fitting the model is described, and generally provides a unique rotation solution for the factors on each occasion.We thank M. W. Browne, R. P. McDonald, R. Pruzek, and the Managing Editor for their constructive comments.     

The perceptual factor

A further study of the perceptual factor, previously isolated in a factor analysis of a battery of fifty-six tests, is made in a manner designed also to determine whether the same seven primaries would be found in a different population of subjects and with another battery of tests. The tests are described, and the results of the analysis are given in detail. Much attention is given to the matter of the orthogonality of primary factors and to their psychological meaningfulness.     

Alpha factor analysis

A distinction is made between statistical inference and psychometric inference in factor analysis. After reviewing Rao's canonical factor analysis (CFA), a fundamental statistical method of factoring, a new method of factor analysis based upon the psychometric concept of generalizability is described. This new procedure (alpha factor analysis, AFA) determines factors which have maximum generalizability in the Kuder-Richardson, or alpha, sense. The two methods, CFA and AFA, each have the important property of giving the same factors regardless of the units of measurement of the observable variables. In determining factors, the principal distinction between the two methods is that CFA operates in the metric of the unique parts of the observable variables while AFA operates in the metric of the common (communality) parts.On the other hand, the two methods are substantially different as to how they establish the number of factors. CFA answers this crucial question with a statistical test of significance while AFA retains only those alpha factors with positive generalizability. This difference is discussed at some length. A brief outline of a computer program for AFA is described and an example of the application of AFA is given.The first version of this paper was prepared while the senior author was a U. S. Public Health Service Fellow at the Center for Advanced Study in the Behavioral Sciences and while the junior author was Director of Research of the Palo Alto Public Schools.     

Model‐related factor score predictors for confirmatory factor analysis

The present paper introduces model‐related (MR) factor score predictors, which reflect specific aspects of confirmatory factor models. The development is mainly based on Schönemann and Steiger's regression score components, but it can also be applied to the factor score coefficients. It is shown that the rotation of factor score predictors has no impact on the covariance matrix reproduced from the corresponding regression component patterns. Thus, regression score components or factor score coefficients can be rotated in order to obtain the required properties. This idea is the basis for MR factor score predictors, which are computed by means of a partial Procrustes rotation towards a target pattern representing the interesting properties of a confirmatory factor model. Two examples demonstrate the construction of MR factor score predictors reflecting specific constraints of a factor model.     

MMPI factor analytic findings for adolescents: item- and scale-level factor structures

Although numerous studies have examined the factor structures of the Minnesota Multiphasic Personality Inventory (MMPI) in investigations of adults, very little attention has been directed to this issue in adolescent samples. In this study, we investigated the item-level and scale-level factor structures using a sample of 1,762 adolescents receiving psychiatric services at the time of their MMPI assessment. Results identified 17 factors related to item responses and 5 factors based on scale-level data. Findings are discussed in relation to results obtained in adult populations and the potential utility of factor analytically derived content scales for adolescent MMPI interpretation.     

Note on factor analysis

Certain assumptions and procedures basic to factor analysis are examined from the point of view of the mathematician. It is demonstrated that the Hotelling method does not yield meaningful traits, and an example from the theory of gas mixtures with convertible components is cited as evidence. The justification of current methods for determining the adequacy of the reproduction of a correlation matrix by a factorial matrix is questioned, and ax ² criterion, practical only for a small matrix, is proposed. By means of a hypothetical example from geometry, it is shown that results of a Hotelling analysis are necessarily relative to the population at hand. The factorial effects of the adjunction of a total test to a group of tests are considered. Some of the general considerations and questions raised are pertinent to types of analysis other than the Hotelling.The calculations which lie in the background of thisNote, but of which only relatively few are in evidence, were made possible by a grant from the Carnegie Corporation of New York to whom grateful acknowledgment is hereby made.     

Stability of factor loadings

Each of 17 tests was embedded in six different eight-test matrices. Stability of general factor loadings were determined over the six contexts. Correlations of factor loadings ranged from 0.52 to 0.94 with an average value of 0.83.     

A factor simplicity index

We propose an index for assessing the degree of factor simplicity in the context of principal components and exploratory factor analysis. The new index, which is called Loading Simplicity, is based on the idea that the communality of each variable should be related to few components, or factors, so that the loadings in each variable are either zero or as far from zero as possible. This index does not depend on the scale of the factors, and its maximum and minimum are only related to the degree of simplicity in the loading matrix. The aim of the index is to enable the degree of simplicity in loading matrices to be compared.The author would like to thank the review team for their insights and recommendations. This work was supported by a grant SEC2001-3821-C05-C02 from the Spanish Ministry of Science and Technology.     

The central intellective factor

Psychometrika - The proof of the existence of “g” is more than a methodological problem and concerns the very core of psychological theory. The principles of noegenesis should be...     

Regression among factor scores

Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.     

Orthogonal inter-battery factor analysis

It is the purpose of this paper to present a method of analysis for obtaining (i) inter-battery factors and (ii) battery specific factors for two sets of tests when the complete correlation matrix including communalities is given. In particular, the procedure amounts to constructing an orthogonal transformation such that its application to an orthogonal factor solution of the combined sets of tests results in a factor matrix of a certain desired form. The factors isolated are orthogonal but may be subjected to any suitable final rotation, provided the above classification of factors into (i) and (ii) is preserved. The general coordinate-free solution of the problem is obtained with the help of methods pertaining to the theory of linear spaces. The actual numerical analysis determined by the coordinate-free solution turns out to be a generalization of the formalism of canonical correlation analysis for two sets of variables. A numerical example is provided.This investigation has been supported by the U.S. Office of Naval Research under Contract Nonr-2752(00).     

Revisiting the factor structure of the WAIS-R: insights through nested factor modeling

Past attempts to model via confirmatory factor analysis (CFA) the Wechsler Adult Intelligence Scale-Revised (WAIS-R) intersubtest covariation have used an oblique factor or a higher order modeling approach. The attempts have failed to yield adequate model fit, based on current CFA recommendations. Using the WAIS-R standardization data, it is demonstrated that the WAIS-R can be better conceptualized as measuring a first-order general factor and three orthogonal group-level factors. The results are discussed in relation to Verbal and Performance Intelligence scoring and failed attempts to find relationships between VIQ/PIQ difference scores and external criteria. Because Arithmetic and Digit Span did not share any variance with the other VIQ subtests, independent of General Intelligence, clinicians should reconsider interpreting a VIQ score that includes information from Arithmetic and Digit Span in nonclinical populations. Researchers are encouraged to model intelligence factors as nested factor models, considering their superior model fit, and the increased clarity in the interpretations of relationships between IQ indices and criteria.     

Modeling differentiation of cognitive abilities within the higher-order factor model using moderated factor analysis

Receptive foreign language proficiency is usually measured with reading and listening comprehension tasks. A novel approach to assess such proficiencies – viewing comprehension – is based on the presentation of short instructional videos followed by one or more comprehension questions concerning the preceding video stimulus. In order to evaluate a newly developed viewing comprehension test 485 German high school students completed reading, listening, and viewing comprehension tests, all measuring the receptive proficiency in English as a foreign language. Fluid and crystallized intelligence were measured as predictors of performance. Relative to traditional comprehension tasks, the viewing comprehension task has similar psychometric qualities. The three comprehension tests are very highly but not perfectly correlated with each other. Relations with fluid and crystallized intelligence show systematic differences between the three comprehension tasks. The high overlap between foreign language comprehension measures and between crystallized intelligence and language comprehension ability can be taken as support for a uni-dimensional interpretation. Implications for the assessment of language proficiency are discussed.     

Recovery of weak factor loadings in confirmatory factor analysis under conditions of model misspecification

This article presents the results of two Monte Carlo simulation studies of the recovery of weak factor loadings, in the context of confirmatory factor analysis, for models that do not exactly hold in the population. This issue has not been examined in previous research. Model error was introduced using a procedure that allows for specifying a covariance structure with a specified discrepancy in the population. The effects of sample size, estimation method (maximum likelihood vs. unweighted least squares), and factor correlation were also considered. The first simulation study examined recovery for models correctly specified with the known number of factors, and the second investigated recovery for models incorrectly specified by underfactoring. The results showed that recovery was not affected by model discrepancy for the correctly specified models but was affected for the incorrectly specified models. Recovery improved in both studies when factors were correlated, and unweighted least squares performed better than maximum likelihood in recovering the weak factor loadings.