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.     

Measures of invariance and comparability in factor analysis for fixed variables

New procedures are presented for measuring invariance and matching factors for fixed variables and for fixed or different subjects. Two of these, the coefficient of invariance for factor loadings and the coefficient of factor similarity, utilize factor scores computed from the different sets of factor loadings and one of the original standard score matrices. Another, the coefficient of subject invariance, is obtained by using one of the sets of factor loadings in conjunction with the different standard score matrices. These coefficients are correlations between factor scores of the appropriate matrices. When the best match of factors is desired, rather than degree of resemblance, the method of assignment is proposed.     

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.     

Use of internal consistency coefficients for estimating reliability of experimental task scores

Reliabilities of scores for experimental tasks are likely to differ from one study to another to the extent that the task stimuli change, the number of trials varies, the type of individuals taking the task changes, the administration conditions are altered, or the focal task variable differs. Given that reliabilities vary as a function of the design of these tasks and the characteristics of the individuals taking them, making inferences about the reliability of scores in an ongoing study based on reliability estimates from prior studies is precarious. Thus, it would be advantageous to estimate reliability based on data from the ongoing study. We argue that internal consistency estimates of reliability are underutilized for experimental task data and in many applications could provide this information using a single administration of a task. We discuss different methods for computing internal consistency estimates with a generalized coefficient alpha and the conditions under which these estimates are accurate. We illustrate use of these coefficients using data for three different tasks.     

Heteroscedastic one‐factor models and marginal maximum likelihood estimation

In the present paper, a general class of heteroscedastic one‐factor models is considered. In these models, the residual variances of the observed scores are explicitly modelled as parametric functions of the one‐dimensional factor score. A marginal maximum likelihood procedure for parameter estimation is proposed under both the assumption of multivariate normality of the observed scores conditional on the single common factor score and the assumption of normality of the common factor score. A likelihood ratio test is derived, which can be used to test the usual homoscedastic one‐factor model against one of the proposed heteroscedastic models. Simulation studies are carried out to investigate the robustness and the power of this likelihood ratio test. Results show that the asymptotic properties of the test statistic hold under both small test length conditions and small sample size conditions. Results also show under what conditions the power to detect different heteroscedasticity parameter values is either small, medium, or large. Finally, for illustrative purposes, the marginal maximum likelihood estimation procedure and the likelihood ratio test are applied to real data.     

Concurrent validity of the WISC-R factor scores

This study investigated the efficacy of predicting academic achievement as measured by the Wide Range Achievement Test using the factor scores of the Wechsler Intelligence Scale for Children-Revised as potential predictors. Both instruments were administered to 200 school children referred for psychological evaluations. Factor scores for the WISC-R were computed according to equations provided by Gutkin (1978). These factor scores then were regressed in a stepwise manner on the Reading, Spelling, and Arithmetic standard scores from the Wide Range Achievement Test. Stepwise regressions also were performed using the traditional Verbal and Performance IQs as the potential predictors. A comparison was made between the results obtained by use of the factors or the more common scores. The results indicated that the Freedom From Distractibility factor score significantly aided in the prediction of Reading and Spelling achievement and was of primary importance in predicting Arithmetic achievement. When the stepwise results using the different scores were contrasted, the factor scores accounted for appreaciably more variance in Arithmetic than did the traditional Verbal and Performance IQs. Derived regression equations are reported along with a discussion of the interpretability of the Freedom From Distractibility factor.     

Evidence on age and ethnic status bias in factor scores and the comparison score for the AAMD adaptive behavior scale-school edition

This report analyzes the contribution of gender, ethnic status, age, and school classification to the five factor scores and the comparison score of the Adaptive Behavior Scale-School Edition (ABS-SE). These factor scores were derived from extensive analysis of the performance of subjects of different ages and different levels of mental retardation. The comparison score evolved from discriminant analysis of the factor scores and is computed as a weighted sum of the three factor scores from Part One of the ABS-SE. The results of the ANOVAs conducted to test the main and interaction effects showed significant mean differences between normal, mildly retarded, and moderately to severely retarded subjects over the age range from 3 through 16 years on the ABS-SE factor and comparison scores. In general, there was no significant contribution of either gender or ethnic status to scores from ages 7 through 16, but there was a significant difference attributable to ethnic status with a meaningful amount of explained variance in the community and self-sufficiency and comparison scores of subjects aged 3–6. Although these differences were significant, children 3–6 years old classified as white did not necessarily perform better on all factor scores than those of minority ethnic groups. Discussion of the results in the context of contemporary criteria for test bias and of competing explanations for ethnic group differences in performance of young children on the ABS-SE follows the presentation of the findings. The results provide additional evidence for the validity of the ABS-SE factor and comparison scores and show that in general the factor and comparison scores are not affected by gender or ethnic status.     

The factor structure and dimensional scoring of the generalized anxiety disorder questionnaire for DSM-IV

Despite favorable psychometric properties, the Generalized Anxiety Disorder Questionnaire for the Diagnostic and Statistical Manual of Mental Disorders (4th ed.) (GAD-Q-IV) does not have a known factor structure, which calls into question use of its original weighted scoring system (usually referred to as the dimensional score). Analyses appropriate to categorical item responses in a large sample of undergraduates were used to establish the scale's factor structure. Analyses indicated that a one-factor structure resulted in good fit. A scoring method based on this one-factor structure was compared with a variety of alternative scoring procedures, and it was found that a method based on factor scores did relatively well but that the previously suggested dimensional score failed to perform better than a simple sum of items. Results support the general unity of the measure but raise doubts regarding its scoring and response options.     

Doing it all Bass-Ackwards: The development of hierarchical factor structures from the top down

A simple method is presented for examining the hierarchical structure of a set of variables, based on factor scores from rotated solutions involving one to many factors. The correlations among orthogonal factor scores from adjoining levels can be viewed as path coefficients in a hierarchical structure. The method is easily implemented using any of a wide variety of standard computer programs, and it has proved to be extremely useful in a number of diverse applications, some of which are here described.     

Using reliability generalization methods to explore measurement error: an illustration using the MMPI-2 PSY-5 scales

Reliability generalization (RG) is a meta-analytic technique that allows for the systematic examination of variation in score reliability for different samples of test takers; this procedure is based on the recognition that reliability is not a stable property of a test but is sample dependent. As a demonstration of an RG analysis, I obtained 63 reliability coefficients for each of the MMPI-2 (Butcher et al., 2001) Personality Psychopathology 5 (Harkness, McNulty, & Ben-Porath, 1995) scales. The overall variability of alpha coefficients supports the argument that reliability is sample dependent and underscores the need for researchers to calculate reliability estimates based on their research samples rather than simply citing published alpha coefficients as evidence of score reliability. I observed statistically significant mean reliability differences for scores across the 5 scales, with the highest level of reliability observed for scores on the measure of Negative Emotionality and the lowest levels of reliability observed for scores on the measures of Aggression and Disconstraint. There was no evidence that the sex-composition of a sample was systematically related to score reliability, and there were no statistically significant differences in reliability between scores obtained with the English version of the test and those obtained with translated forms. However, reliability was consistently lower for scores on some scales when the data were obtained in nonclinical settings as opposed to clinical ones. Sample size was not significantly correlated with reliability estimates. RG methods have the potential for deepening the level of understanding about the role of reliability in the evaluation and use of personality tests.     

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.     

Relations of factor score estimates to their use

Considerations of factor score estimates have concentrated on internal characteristics. This report considers external characteristics of four methods for determining factor score estimates; that is, relations of these estimates to measures on attributes not entered into the factor analysis. These external characteristics are important for many uses of factor score estimates. Findings are that different ones of the methods are appropriate for different uses.Supported in part by the Personnel and Training Branch of the Office of Naval Research under contract number 00014-67-A-0305-0003.     

On robusiness of the normal-theory based asymptotic distributions of three reliability coefficient estimates

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

Descriptive axioms for common factor theory,image theory and component theory

Through an extension of work by Guttman, common factor theory, image theory, and component theory are derived from distinct minimum subsets of assumptions chosen out of a set of five possible assumptions. It is thence shown that the problem of indeterminacy of factor scores in the common factor model is precisely reflected in the problem of the non-orthogonality of anti-images. Indeed, image scores are determinate for the same reason that the usual estimates of factor scores are determinate, and image scores cannot be used as though they were factor scores for the same reason that factor score estimates cannot be used as though they were factor scores.     

The comparability of intelligence test results: Group- and individual-level comparisons of seven intelligence tests

A significant body of research has demonstrated that IQs obtained from different intelligence tests substantially correlate at the group level. Yet, there is minimal research investigating whether different intelligence tests yield comparable results for individuals. Examining this issue is paramount given that high-stakes decisions are based on individual test results. Consequently, we investigated whether seven current and widely used intelligence tests yielded comparable results for individuals between the ages of 4–20 years. Results mostly indicated substantial correlations between tests, although several significant mean differences at the group level were identified. Results associated with individual-level comparability indicated that the interpretation of exact IQ scores cannot be empirically supported, as the 95% confidence intervals could not be reliably replicated with different intelligence tests. Similar patterns also appeared for the individual-level comparability of nonverbal and verbal intelligence factor scores. Furthermore, the nominal level of intelligence systematically predicted IQ differences between tests, with above- and below-average IQ scores associated with larger differences as compared to average IQ scores. Analyses based on continuous data confirmed that differences appeared to increase toward the above-average IQ score range. These findings are critical as these are the ranges in which diagnostic questions most often arise in practice. Implications for test interpretation and test construction are discussed.     

A comparison of four methods of constructing factor scores

Four least-squares methods for constructing factor scores have been described in the literature. The formal properties of these scores are developed, and they are compared in terms of four generally desirable properties of constructed factor scores. In particular, it is shown that two of the methods yield scores that are conditionally unbiassed, and univocal in the sense of Guilford and Michael, though not orthogonal, while one of the other methods yields orthogonal scores.It is shown that constructed factor scores cannot be simultaneously univocal and orthogonal, unless we choose the special basis in factor space given by Canonical Factor Analysis.The general problem of choosing between the methods is discussed, on the basis of the theoretical relations obtained.EDITOR'S NOTE: The reader will quickly discover that this article develops several of the generalizations given in the second part of the preceding article, On Factors and Factor Scores. Independent development of the same generalizations is, of course, not a new phenomenon. Because the Presidential Address automatically is accepted for publication and given space in the December issue, it was decided that the only fair thing to do was to print this article in the same issue.     

A confirmatory analysis of the factor structure and cross-age invariance of the Wechsler Adult Intelligence Scale-Third Edition

In the Wechsler Adult Intelligence Scale-Third Edition (WAIS-III; D. Wechsler, 1997), the manual reports several confirmatory factor analyses in support of the instrument's latent factor structure. In practice, examiners frequently compare an examinee's score from a current administration of the WAIS-III with the results from a previous test administration. Implicit in test-retest score comparisons is evidence that scores retain similar interpretive meaning across time. Establishing an instrument's factorial invariance provides the foundation for this practice. This study investigated the factorial invariance of the WAIS-III across the instrument's 13 age groups. The overall results from this study generally support both configural and factorial invariance of the WAIS-III when the 11 primary tests are administered.     

Approaches to univocal factor scores

Two current methods of deriving common-factor scores from tests are briefly examined and rejected. One of these estimates a score from a multiple-regression equation with as many terms as there are tests in the battery. The other limits the equation to a few tests heavily saturated with the desired factor, with or without tests used to suppress the undesired factors. In the proposed methods, the single best test for each common factor is the starting point. Such a test ordinarily has a very few undesired factors to be suppressed, frequently only one. The suppression test should be univocal, or nearly so. Fortunately, there are relatively univocal tests for factors that commonly require suppression. Equations are offered by which the desired-factor test and a single suppression test can be weighted in order to achieve one or more objectives. Among the objectives are (1) maximizing the desired factor variance, (2) minimizing the undesired factor variance, (3) a compromise, in which the undesired variance is materially reduced without loss in desired variance, and (4) a change to any selected ratio of desired to undesired variance. A more generalized solution is also suggested. The methods can be extended in part to the suppression of more than one factor. Equations are derived for the suppression of two factors.     

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.     

The dependability of the general factor of intelligence: Why small, single-factor models do not adequately represent g

Floyd, Shands, Rafael, Bergeron and McGrew (2009) used generalizability theory to test the reliability of general-factor loadings and to compare three different sources of error in them: the test battery size, the test battery composition, the factor-extraction technique, and their interactions. They found that their general-factor loadings were moderately to strongly dependable. We replicated the methods of Floyd et al. (2009) in a different sample of tests, from the Minnesota Study of Twins Reared Apart (MISTRA). Our first hypothesis was that, given the greater diversity of the tests in MISTRA, the general-factor loadings would be less dependable than in Floyd et al. (2009). Our second hypothesis, contrary to the positions of Floyd et al. (2009) and Jensen and Weng (1994), was that the general factors from the small, randomly-formed test batteries would differ substantively from the general factor from a well-specified hierarchical model of all available tests. Subtests from MISTRA were randomly selected to form independent and overlapping batteries of 2, 4 and 8 tests in size, and the general-factor loadings of eight probe tests were obtained in each battery by principal components analysis, principal factor analysis and maximum likelihood estimation. Results initially indicated that the general-factor loadings were unexpectedly more dependable than in Floyd et al. (2009); however, further analysis revealed that this was due to the greater diversity of our probe tests. After adjustment for this difference in diversity, and consideration of the representativeness of our probe tests versus those of Floyd et al. (2009), our first hypothesis of lower dependability was confirmed in the overlapping batteries, but not the independent ones. To test the second hypothesis, we correlated g factor scores from the random test batteries with g factor scores from the VPR model; we also calculated special coefficients of congruence on the same relation. Consistent with our second hypothesis, the general factors from small non-hierarchical models were found to not be reliable enough for the purposes of theoretical research. We discuss appropriate standards for the construction and factor analysis of intelligence test batteries.