Evaluating the impact of partial factorial invariance on selection in two populations

Studies of factorial invariance examine whether a common factor model holds across multiple populations with identical parameter values. Partial factorial invariance exists when some, but not all, parameters are invariant. The literature on factorial invariance is unclear about what should be done if partial invariance is found. One approach to this problem evaluates the impact of partial invariance on accuracy of selection on the basis of a composite of the measures whose factor structure is being studied. Assuming a single-factor model holds, accuracy of selection using the composite is evaluated under varying degrees of partial invariance. A variety of examples are presented with discussion of extensions and limitations.     

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 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.     

A unifying model for the structure of intellectual abilities

Models of the structure of cognitive abilities suggested by Spearman, Thurstone, Guilford, Vernon and Cattell-Horn are reviewed. It is noted that some of the models include a general intellectual factor (g) while others do not. It is also noted that some models are nonhierarchical, while in others more narrow abilities are subsumed under broader abilities in a hierarchical pattern. An empirical study in which a test battery of 16 tests was administered to some 1000 subjects in the 6th grade is reported. Using the LISREL technique to test different models, good support is obtained for oblique primary factors in the Thurstone tradition as well as for the second-order factors fluid intelligence, crystallized intelligence, and general visualization hypothesized by Cattell and Horn. It is also found, however, that the second-order factor of fluid intelligence i is identical with a third-order g-factor. On the basis of these results a three-level model (the HILI-model) is suggested, with the g-factor at the top, two broad factors reflecting the ability to deal with verbal and figural information, respectively, at the second-order level, and the primary factors in the Thurstone and Guilford tradition at the lowest level. It is argued that most previously suggested models are special cases of the HILI-model.     

A note on the identification of restricted factor loading matrices

It is shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions. A necessary (under a regularity assumption) and sufficient condition for local uniqueness is given and a counterexample is provided to a theorem by J. Algina concerning necessary and sufficient conditions for global uniqueness.     

Measurement Invariance,Predictive Invariance,and the Duality Paradox

The statistical literature on bias in psychological testing distinguishes at least two forms of bias: measurement bias and predictive bias. Measurement bias concerns group differences in the relationship between a test and the latent variable to be measured. Predictive bias concerns group differences in the relationship between a test and an external criterion. How are these two forms of bias related? For example. if a test is unbiased in the predictive sense, does this fact support the hypothesis that the test is unbiased in the measurement sense? A theorem is given that describes the conditions under which measurement invariance (lack of bias) is consistent with predictive invariance for the linear case. Paradoxically, these two forms of invariance are shown to be inconsistent under realistic conditions. This duality or inconsistency is illustrated in simulated data. The implications of the duality for group differences research are illustrated in real data involving gender and ethnic differences on the SAT. The phenomenon of duality may force a reinterpretation of common empirical findings of test criterion regression slope invariance. and of invariance in test validities. Other implications are discussed.     

Weak measurement invariance with respect to unmeasured variables: An implication of strict factorial invariance

Measurement bias refers to systematic differences across subpopulations in the relation between observed test scores and the latent variant underlying the test scores. Comparisons of subpopulations with the same score on the latent variable can be expected to have the same observed test score. Measurement invariance is therefore one of the key issues in psychological testing. It has been established that strict factorial invariance (SFI) with respect to a selection variable V almost certainly implies weak measurement invariance with respect to V: given SFI, means and variances of observed scores do not depend on V. It is shown that this result can be extended. SFI in groups derived by selection on V has implications not only for V but also for potentially biasing variables W, if W and the selection variable V and/or if W and the factor underlying the observed test scores are statistically dependent. Given SFI with respect to V and prior knowledge concerning these dependencies, it is not necessary to measure and model variables W in order to exclude them as potentially biasing variables if the investigation focuses on groups selected on V.     

Scale freeness in factor analysis

The notion of scale freeness does not seem to have been well understood in the factor analytic literature. It has been believed that if the loss function that is minimized to obtain estimates of the parameters in the factor model is scale invariant, then the estimates are scale free. It is shown that scale invariance of the loss function is neither a necessary nor a sufficient condition for scale freeness. A theorem that ensures scale freeness in the orthogonal factor model is given in this paper.The authors are grateful for the suggestions of the referees.     

An empirical verification of the Wherry-Gaylord iterative factor analysis procedure

A comparison of the Wherry-Gaylord iterative factor analysis procedure and the Thurstone multiple-group analysis of sub-tests shows that the two methods result in the same factors. The Wherry-Gaylord method has the advantage of giving factor loadings for items. The number of iterations needed can be reduced by doing a factor analysis of sub-tests, re-grouping sub-tests according to factors, and using each group as a starting point for iterations.This research was carried out under Contract No. WSW-2503, between the Department of the Army and Ohio State University. This paper is based on the final report PRS No. 827 under that contract. The opinions expressed herein regarding matters relating to the Department of the Army are those of the authors and are not necessarily official.     

Factorial Structure and Measurement Invariance of the Innovative Behavior Questionnaire

The article presents an analysis of the factorial structure and measurement invariance of the Innovative Behavior Questionnaire, developed by Scott and Bruce. Although the instrument is widely used to capture individuals' innovative behavior, very little evidence concerning its psychometric properties is available. A time‐lagged study among 382 employees was conducted to check the factorial structure of the questionnaire, using confirmatory factor analysis, as well as its measurement invariance across gender and time. One‐factor structure (with correlated error terms of first three items) and strict invariance across time and across gender of the Innovative Behavior Questionnaire were demonstrated. As such, the measure can be used as a reliable tool for capturing individuals' innovative behavior by self‐report.     

On component analyses

Principal components analysis can be redefined in terms of the regression of observed variables upon component variables. Two criteria for the adequacy of a component representation in this context are developed and are shown to lead to different component solutions. Both criteria are generalized to allow weighting, the choice of weights determining the scale invariance properties of the resulting solution. A theorem is presented giving necessary and sufficient conditions for equivalent component solutions under different choices of weighting. Applications of the theorem are discussed that involve the components analysis of linearly derived variables and of external variables.Preparation of this article was supported in part by National Institute of Aging Grant NIA-AG03164-03 to William Meredith.     

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.     

The relation between results obtainable with raw and corrected correlation coefficients in multiple factor analysis

This paper presents three theorems concerning the relation between results with obtained and corrected correlation coefficients in the Thurstone method of multiple factor analysis. (1) The rank of the correlational matrix, and thus the number of factors involved, is unaffected by correcting the obtained coefficients for attenuation. (2) The communality of a variable when the obtained coefficients have been corrected for attenuation is equal to the communality with obtained coefficients divided by the reliability coefficient of the variable. (3) The relationship is shown between the factorial matrix of a correlational matrix of raw correlation coefficients and the factorial matrix of a correlational matrix of corrected correlation coefficients, and a method of obtaining either of these factorial matrices from the other is indicated.     

Multialternative decision field theory: a dynamic connectionist model of decision making

The authors interpret decision field theory (J. R. Busemeyer & J. T. Townsend, 1993) as a connectionist network and extend it to accommodate multialternative preferential choice situations. This article shows that the classic weighted additive utility model (see R. L. Keeney & H. Raiffa, 1976) and the classic Thurstone preferential choice model (see L. L. Thurstone, 1959) are special cases of this new multialternative decision field theory (MDFT), which also can emulate the search process of the popular elimination by aspects (EBA) model (see A. Tversky, 1969). The new theory is unique in its ability to explain several central empirical results found in the multialternative preference literature with a common set of principles. These empirical results include the similarity effect, the attraction effect, and the compromise effect, and the complex interactions among these three effects. The dynamic nature of the model also implies strong testable predictions concerning the moderating effect of time pressure on these three effects.     

Factor analysis; some effects of chance error

Ignorance concerning the standard error of individual factor loadings and their differences has been a major obstacle to the proper interpretation of factorial results. The effects of three types of experimental error (selection of variables, selection of subjects and selection of scores) are here reported. In order to handle the errors of rotation systematically, it has been necessary to introduce a new semi-analytical criterion for the attainment of simple structure. Variability in results which may theoretically be eliminated is discussed under the heading of non-essential error.     

L. L. Thurstone: A new assessment

Most psychologists have heard of Thurstone, and know him as a pioneer of mathematical psychology and factor analysis. But his appearances in history books are rare and peripheral, as a man of technical prowess rather than intellectual achievement, and at best he is seen as a link in the tradition culminating in signal detection and mathematical learning theory. Nineteen eighty-seven is the centenary of his birth, so it is a timely moment to correct this picture. Thurstone was influenced by G. H. Mead, and his work is best placed within the pragmatist movement, not as moving towards the mechanistic psychology that now dominates psychology. Viewed in this way, his true status and the reasons for his relative neglect become apparent. This is illustrated by a detailed consideration of his mathematical learning models.     

Centroid orthogonal transformations

An initial transformation for facilitating the analysis of a factor problem is discussed. Such a transformation is used in the rotational procedure developed by L. L. Thurstone in an accompanying article.     

Invariance in Measurement and Prediction Revisited

Borsboom (Psychometrika, 71:425–440, 2006) noted that recent work on measurement invariance (MI) and predictive invariance (PI) has had little impact on the practice of measurement in psychology. To understand this contention, the definitions of MI and PI are reviewed, followed by results on the consistency between the two forms of invariance in the general case. The special parametric cases of factor analysis (strict factorial invariance) and linear regression analyses (strong regression invariance) are then described, along with findings on the inconsistency between the two forms of invariance in this context. Two numerical examples of inconsistency are reviewed in detail. The impact of violations of MI on accuracy of selection is illustrated. Finally, reasons for the slow dissemination of work on invariance are discussed, and the prospects for altering this situation are weighed. This paper is based on the Presidential Address given at the International Meeting of the Psychometric Society in Tokyo, Japan, on July 11, 2007. This research was supported by National Institute of Mental Health grants 1P30 MH 068685-01A1 and RO1 MH64707-01.     

THELMA GWINN THURSTONE

This paper examines the life and career of Thelma Gwinn Thurstone—her career strategies and her contributions to psychological testing, intelligence theory, and instruction—based on her publications and a series of personal interviews with her and her colleagues. Thurstone's contributions, with her husband, L. L. Thurstone, included the development of the American Council on Education's Psychological Examinations and the Primary Mental Abilities test batteries. Her own work included the development of instructional materials using the common factor theory of intelligence. Thurstone, wife of a preeminent scholar and mother of three sons, pursued career strategies that facilitated her continued professional activity for six decades. Further research on measurement history and also on women's career strategies are suggested.     

What Invariance Is and How to Test for It

Causal assessment is the problem of establishing whether a relation between (variable) X and (variable) Y is causal. This problem, to be sure, is widespread across the sciences. According to accredited positions in the philosophy of causality and in social science methodology, invariance under intervention provides the most reliable test to decide whether X causes Y. This account of invariance (under intervention) has been criticised, among other reasons, because it makes manipulations on the putative causal factor fundamental for the causal methodology; consequently, the argument goes, the account is ill-suited to those contexts where manipulations are not performed, for instance, the social sciences. The article aims to extend the account of invariance (under intervention), in a way that manipulations on the putative causal factors are not methodologically fundamental, and yet invariance remains key for causal assessment both in experimental and non-experimental contexts.