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.     

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.     

Estimation of covariance structure models with parameters subject to functional restraints

This paper demonstrates the feasibility of using the penalty function method to estimate parameters that are subject to a set of functional constraints in covariance structure analysis. Both types of inequality and equality constraints are studied. The approaches of maximum likelihood and generalized least squares estimation are considered. A modified Scoring algorithm and a modified Gauss-Newton algorithm are implemented to produce the appropriate constrained estimates. The methodology is illustrated by its applications to Heywood cases in confirmatory factor analysis, quasi-Weiner simplex model, and multitrait-multimethod matrix analysis.The author is indebted to several anonymous reviewers for creative suggestions for improvement of this paper. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

The nonconvergence of factals: A nonmetric common factor analysis

FACTALS is a nonmetric common factor analysis model for multivariate data whose variables may be nominal, ordinal or interval. In FACTALS an Alternating Least Squares algorithm is utilized which is claimed to be monotonically convergent.In this paper it is shown that this algorithm is based upon an erroneous assumption, namely that the least squares loss function (which is in this case a nonscale free loss function) can be transformed into a scalefree loss function. A consequence of this is that monotonical convergence of the algorithm can not be guaranteed.     

Rotation to simple loadings using component loss functions: The orthogonal case

Component loss functions (CLFs) are used to generalize the quartimax criterion for orthogonal rotation in factor analysis. These replace the fourth powers of the factor loadings by an arbitrary function of the second powers. Criteria of this form were introduced by a number of authors, primarily Katz and Rohlf (1974) and Rozeboom (1991), but there has been essentially no follow-up to this work. A method so simple, natural, and general deserves to be investigated more completely. A number of theoretical results are derived including the fact that any method using a concave CLF will recover perfect simple structure whenever it exists, and there are methods that will recover Thurstone simple structure whenever it exists. Specific CLFs are identified and it is shown how to compare these using standardized plots. Numerical examples are used to illustrate and compare CLF and other methods. Sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them.The author is very indebted to a reviewer for pointing him to the generalized hyperplane count literature and to all the reviewers for valuable comments and suggestions.     

Some Results on Mean Square Error for Factor Score Prediction

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γ _p = Φ ^1/2 Λ _p ′Ψ _p ⁻¹ Λ _p Φ ^1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γ _p or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors. The author is obliged to Maarten Speekenbrink and Peter van Rijn for their assistance with plotting the figures. In addition, I am obliged to the referees for their stimulating remarks.     

Psychometric Qualities and Measurement Invariance of the Modified Self‐Rated Creativity Scale

The 13‐item self‐rated creativity scale (SRCS) initially developed for supervisory rating of employees’ creativity was modified by some researchers and used as a self‐report of creativity. However, it is not clear if the modified SRCS is psychometrically sound. The present study addressed this gap in three studies (N = 1,033). The exploratory factor analysis (Study 1) revealed a two‐factor solution after removing Item 9 due to low factor loading. Confirmatory factor analysis was then used in Study 2 to examine and compare the conceptual one‐factor model with 13 items (Model 1), one‐factor model with 12 items (Model 2), two‐factor model with 12 items (Model 3), and the 12‐item bifactor model with one general factor and two specific factors (Model 4). The results indicated that Model 4 is more superior to all the competing models. Study 3 further confirmed that the bifactor model, showed support to the reliability and convergent validity, and found partial metric invariance across Chinese and Malay undergraduates. Taken together, the modified (12‐item) SRCS is a psychometrically sound tool for self‐rated creativity in the Malaysian context.     

Development and validation of the Indonesian Well‐being Scale

The present study describes the development and validation of a new instrument to measure the well‐being of Indonesian people. Items were generated by taking into account the recent cross‐cultural developments in the literature. Participants (N = 1,028) from a number of provinces in Indonesia completed an online or a paper questionnaire containing a battery of measures, including the new well‐being instrument. The total sample was randomly split into two equal groups. An exploratory factor analysis (n = 516) was conducted on one half to explore the factor structure of the new scale, which resulted in a 24‐item scale with a four‐factor solution. The four factors were (a) Basic Needs, (b) Social Relation, (c) Acceptance, and (d) Spirituality. A confirmatory factor analysis was conducted on the second half (n = 516), which confirmed the four‐factor structure with 20 items emerging as the best and most parsimonious fit of the data. The psychometric properties of the final scale were tested on the whole sample. The new scale displayed sound internal consistency, test‐retest reliability, and divergent, convergent, and discriminant validity. The scale has the potential to be used in future research which examines well‐being in the Indonesia context. Future directions and limitations of the study are discussed.     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.     

Confirming the Factor Structure of Attention Deficit Hyperactivity Disorder Symptoms in Adult, Nonclinical Samples

Three separate models have been proposed to describe the factor structure of attention deficit hyperactivity disorder (ADHD) in the past 20 years. The Diagnostic and Statistical Manual of Mental Disorders-III (DSM-III, 1980) proposed 3 separate factors of inattention, hyperactivity, and impulsivity. The DSM-III-R (1987) proposed a single factor. The DSM-IV (1994) described the disorder as having 2 factors: hyperactivity/impulsivity and inattention. The following 2 studies employed confirmatory factor analysis to compare each of these models and 3 alternative models. University students completed questionnaires that assessed each of the 18 symptoms listed in the DSM-IV for ADHD. The 3-factor model fit the data significantly better than each of the other models in both studies. These findings suggest that a 3-factor model of inattention, hyperactivity, and impulsivity describes adult ADHD symptoms better than current alternatives.     

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.     

Dimensionality of stress experiences: Factorial structure of the Perceived Stress Questionnaire (PSQ) in a population‐based Swedish sample

We investigated the factorial structure of the Perceived Stress Questionnaire (PSQ‐recent; Levenstein, Prantera, Varvo et al., 1993) in a large (N = 1516; 35–95 years) population‐based Swedish sample (Nilsson, Adolfsson, Bäckman et al., 2004; Nilsson, Bäckman, Erngrund et al., 1997). Exploratory principal components analysis (PCA) was conducted on a first, randomly drawn subsample (n = 506). Next, the model based on the PCA was tested in a second sample (n = 505). Finally, a third sample (n = 505) was used to cross‐validate the model. Five components were extracted in the PCA (eigenvalue > 1) and labeled “Demands,” “Worries/Tension,” “Lack of joy,” “Conflict,” and “Fatigue,” respectively. Twenty‐one out of the 30 original PSQ items were retained in a confirmatory factor analysis (CFA) model that included the five (first‐order) factors and, additionally, a general (second‐order) stress factor, not considered in prior models. The model showed reasonable goodness of fit [χ²(184) = 511.2, p < 0.001; CFI = 0.904; RMSEA = 0.059; and SRMR = 0.063]. Multigroup confirmatory factor analyses supported the validity of the established model. The results are discussed in relation to prior investigations of the factorial structure of the PSQ.     

The Yale-Brown Obsessive-Compulsive Scale: Confirmatory Factor Analytic Findings

The psychometric properties of the Yale-Brown Obsessive-Compulsive Scale (Y-BOCS) have recently been heavily examined. Specifically, a number of researchers have been interested in determining the factor structure of this scale to find whether it best forms a one, two, or three factor model. The present study continued this examination by considering different scaling models using confirmatory factor analysis with a sample of individuals diagnosed with Obsessive-Compulsive Disorder (OCD). One hundred and forty-six individuals diagnosed with OCD participated and were administered the Y-BOCS and scales measuring depression (Hamilton Depression Rating Scale) and anxiety (Hamilton Anxiety Rating Scale). It was found that the Y-BOCS forms two different two-factor models. One model consisted of an obsessions and compulsions factor; the other composed of disturbance and symptom severity factors. It was likewise found that depression and anxiety were related to both factors in one model (disturbance and symptom severity). The finding that depression was related to obsessions and anxiety to compulsions was found, as in a previous factor analysis of the Y-BOCS. These findings suggest that OCD may be best characterized as a multidimensional syndrome that may not be adequately examined by a single unitary factor as described in the Y-BOCS.     

The Ethnic Adornment Survey (EAS): Measuring Individuals’ Use of Clothing to Express Ethnic Identity

The present paper introduces the Ethnic Adornment Survey as a measure of the relative frequency with which individuals use clothing to express their ethnic identity. Results of exploratory factor analyses in a pilot study (n = 174) and confirmatory factor analyses in a main study (n = 217) revealed that the Ethnic Adornment Survey measures two positively correlated, yet distinct, ethnic adornment factors (i.e., socially desirable and undesirable aspects). Moreover, results of regression analyses indicated that (1) self-esteem was positively predicted by socially desirable aspects of ethnic adornment (p’s < .05 in both studies); whereas (2) self-esteem was negatively predicted by socially undesirable aspects of ethnic adornment (p’s < .05 in the pilot study and < .10 in the main study). Implications for research on ethnic identity are discussed.     

Multigroup confirmatory factor analysis and structural invariance with age of the Behavior Rating Inventory of Executive Function (BRIEF)—French version

The parent and teacher forms of the French version of the Behavioral Rating Inventory of Executive Function (BRIEF) were used to evaluate executive function in everyday life in a large sample of healthy children (N = 951) aged between 5 and 18.

Several psychometric methods were applied, with a view to providing clinicians with tools for score interpretation. The parent and teacher forms of the BRIEF were acceptably reliable. Demographic variables (such as age and gender) were found to influence the BRIEF scores. Confirmatory factor analysis was then used to test five competing models of the BRIEF's latent structure. Two of these models (a three-factor model and a two-factor model, both based on a nine-scale structure) had a good fit. However, structural invariance with age was only obtained with the two-factor model.

The French version of the BRIEF provides a useful measure of everyday executive function and can be recommended for use in clinical research and practice.     

Narcissistic Personality Inventory: Structure of the adapted Dutch version

Barelds, D. P. H. & Dijkstra, P. (2009). Narcissistic Personality Inventory: Structure of the adapted Dutch version. Scandinavian Journal of Psychology, 51, 132–138. The present study examined the structure of a Dutch adaptation of the 40‐item Narcissistic Personality Inventory ( Raskin & Terry, 1988 ) in a community sample (n = 460) and a student sample (n = 515). Altering the response format of the NPI to a Likert‐scale had no apparent effect on the responses. Confirmatory factor analyses supported neither the four‐factor structure reported by Emmons (1984) , nor the seven‐factor structure reported by Raskin and Terry (1988) . Instead, exploratory factor analyses supported either a single‐factor solution (general narcissism), or a two‐factor solution (Authority/Power and Self‐Admiration). The validity of the NPI was supported by its relations with sex, age, personality, self‐esteem, shame, guilt and social desirability.     

Homogeneity analysis withk sets of variables: An alternating least squares method with optimal scaling features

Homogeneity analysis, or multiple correspondence analysis, is usually applied tok separate variables. In this paper we apply it to sets of variables by using sums within sets. The resulting technique is called OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: nominal, ordinal, and numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and nonlinear multivariate analysis techniques are shown to be special cases of OVERALS. An application to data from an epidemiological survey is presented.This research was partly supported by SWOV (Institute for Road Safety Research) in Leidschendam, The Netherlands.     

Invariance of the “NEO-PI-R” factor structure across exploratory and confirmatory factor analyses

Confirmatory factor analyses (CFA) carried out in the last decades to test the Five-factor simple structure have generally obtained negative results. McCrae, Zonderman, Costa, Bond, and Paunonen (1996) suggested that CFA limitations were the main cause of such negative results. They also claim that Procrustes Orthogonal rotation method is an adequate procedure to test the replicability of the Five-factor structure. The aim of the present paper is twofold: (A) comparing several Exploratory factor procedures (including the Procrustes rotation) to test what is the most appropriate one to analyse the replicability of the Five-factor structure, and (B) replicating the CFA results of the McCrae et al.’s study with a larger number of subjects. The normative American (N = 1000), and Spanish (2000) standardization samples of the NEO-PI-R, together with an independent university sample (N = 948) were analysed. Results were replicated in the three samples, and suggest that: (1) structural properties of the Five-factor model, as measured by the NEO-PI-R, are invariant irrespective of the Exploratory factor procedure used, and (2) when CFA limitations are surpassed, the Five-factor structure is supported.