Exploring item and higher order factor structure with the Schmid-Leiman solution: Syntax codes for SPSS and SAS

To ease the interpretation of higher order factor analysis, the direct relationships between variables and higher order factors may be calculated by the Schmid-Leiman solution (SLS; Schmid & Leiman, 1957). This simple transformation of higher order factor analysis orthogonalizes first-order and higher order factors and thereby allows the interpretation of the relative impact of factor levels on variables. The Schmid-Leiman solution may also be used to facilitate theorizing and scale development. The rationale for the procedure is presented, supplemented by syntax codes for SPSS and SAS, since the transformation is not part of most statistical programs. Syntax codes may also be downloaded from www.psychonomic.org/archive/.     

Hierarchical structure of dysfunctional beliefs in obsessive-compulsive disorder

The Obsessive Beliefs Questionnaire was developed as a comprehensive measure of dysfunctional beliefs, which cognitive models consider to be etiologically related to obsessive-compulsive disorder. Obsessive Beliefs Questionnaire subscales tend to be highly correlated, which raises the question of whether obsessive-compulsive-related beliefs are hierarchically structured, consisting of lower-order factors loading on 1 or more higher-order factors. To investigate the nature and relative importance of these factors, a hierarchical factor analysis was conducted (n = 202 obsessive-compulsive disorder patients), using a Schmid-Leiman transformation. Results indicated a higher-order (general factor) and 3 lower-order factors: (i) responsibility and overestimation of threat, (ii) perfectionism and intolerance of uncertainty and (iii) importance and control of thoughts. The high-order factor accounted for more variance in Obsessive Beliefs Questionnaire scores (22%) than did the lower-order factors (6-7%), thereby underscoring the importance of the higher-order factor. Despite the importance of the higher-order factor, the lower-order factors significantly predicted unique variance in measures of obsessive-compulsive symptoms, including severity ratings of compulsions. These finding suggest that cognitive models of obsessive-compulsive disorder should take into consideration the hierarchic structure of obsessive-compulsive-related beliefs.     

双因子模型：多维构念测量的新视角

双因子模型是一种既有全局因子又有局部因子的模型,近年来有了许多应用。本文讨论了双因子模型和高阶因子模型在数学模型、参数之间的关系,概念上和应用上的差异;概述了双因子模型在信度研究、平衡量表、探索性因子分析和项目反应理论中的应用。作为例子,在Rosenberg自尊量表结构的研究中,通过双因子模型分析了自尊特质效应与项目表述方法效应。     

Evaluating the factor structure of the Psychological Performance Inventory

This study assesses the construct validity of a measure of mental toughness, Loehr's Psychological Performance Inventory. Performers (N = 408, 303 men, 105 women, M age = 24.0 yr., SD = 6.7) drawn from eight sports (artistic rollerskating, basketball, canoeing, golf, rugby league, rugby union, soccer, swimming), and competing at either international, national, county and provincial, or club and regional standards. They completed the 42-item Psychological Performance Inventory during training camps. Principal components analysis provided minimal support for the factor structure. Instead, the exploratory analysis yielded a 4-factor 14-item model (PPI-A). A single factor underlying mental toughness (G(MT)) was identified with higher-order exploratory factor analysis using the Schmid-Leiman procedure. Psychometric analysis of the model, using confirmatory analysis techniques, fitted the data well. Collectively satisfying absolute and incremental fit index benchmarks, the inventory possesses satisfactory psychometric properties, with adequate reliability and convergent and discriminant validity. The results lend preliminary support to the factorial validity and reliability of the model; however, further investigation of its stability is required before recommending practitioners use changes in scores as an index for evaluating effects of training in psychological skills.     

Factor Structure and Psychometric Properties of the Modified Childhood Anxiety Sensitivity Index in a Sample of Croatian Children and Adolescents

Anxiety sensitivity (AS) refers to a person’s tendency to fear anxiety-related symptoms due to the belief that they have harmful consequences. The Childhood Anxiety Sensitivity Index (CASI) is a well accepted operationalization of the AS construct in children and adolescents. This study evaluated the factor structure, gender stability and psychometric properties of the CASI, modified to a 5-point scale, in a sample of Croatian children and adolescents (N = 1,679). Exploratory and confirmatory analysis and a Schmid-Leiman solution confirmed the multidimensional and hierarchical structure of the CASI, which consisted of three lower-order factors and a single higher-order factor. Further, the modification of the CASI to a 5-point scale resulted in higher reliability, while maintaining acceptable levels of factor stability and validity.     

A generalized linear factor model approach to the hierarchical framework for responses and response times

We show how the hierarchical model for responses and response times as developed by van der Linden (2007), Fox, Klein Entink, and van der Linden (2007), Klein Entink, Fox, and van der Linden (2009), and Glas and van der Linden (2010) can be simplified to a generalized linear factor model with only the mild restriction that there is no hierarchical model at the item side. This result is valuable as it enables all well‐developed modelling tools and extensions that come with these methods. We show that the restriction we impose on the hierarchical model does not influence parameter recovery under realistic circumstances. In addition, we present two illustrative real data analyses to demonstrate the practical benefits of our approach.     

Bayesian inference for graphical factor analysis models

We generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off-diagonal elements. The resulting model is named graphical factor analysis model. Allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables, which can be used both in the confirmatory and exploratory context. We first present a sufficient condition for global identifiability of this class of models with a generic number of factors, thereby extending the results in Stanghellini (1997) and Vicard (2000). We then consider the issue of model comparison and show that fast local computations are possible for this purpose, if the conditional independence graphs on the residuals are restricted to be decomposable and a Bayesian approach is adopted. To achieve this aim, we propose a new reversible jump MCMC method to approximate the posterior probabilities of the considered models. We then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years. We acknowledge support from M.U.R.S.T. of Italy and from the European Science Foundation H.S.S.S. Network. We are grateful to the referees and the Editor for many useful suggestions and comments which led to a substantial improvement of the paper. We also thank Nanny Wermuth for stimulating discussions and Kenneth A. Bollen for kindly providing us with the data-set.     

Heterogeneous factor analysis models: A bayesian approach

Multilevel factor analysis models are widely used in the social sciences to account for heterogeneity in mean structures. In this paper we extend previous work on multilevel models to account for general forms of heterogeneity in confirmatory factor analysis models. We specify various models of mean and covariance heterogeneity in confirmatory factor analysis and develop Markov Chain Monte Carlo (MCMC) procedures to perform Bayesian inference, model checking, and model comparison.We test our methodology using synthetic data and data from a consumption emotion study. The results from synthetic data show that our Bayesian model perform well in recovering the true parameters and selecting the appropriate model. More importantly, the results clearly illustrate the consequences of ignoring heterogeneity. Specifically, we find that ignoring heterogeneity can lead to sign reversals of the factor covariances, inflation of factor variances and underappreciation of uncertainty in parameter estimates. The results from the emotion study show that subjects vary both in means and covariances. Thus traditional psychometric methods cannot fully capture the heterogeneity in our data.     

Higher-order exploratory factor analysis of the Reynolds Intellectual Assessment Scales with a referred sample

The factor structure of the Reynolds Intellectual Assessment Scales (RIAS; [Reynolds, C.R., & Kamphaus, R.W. (2003). Reynolds Intellectual Assessment Scales. Lutz, FL: Psychological Assessment Resources, Inc.]) was investigated with a large (N = 1163) independent sample of referred students (ages 6-18). More rigorous factor extraction criteria (viz., Horn's parallel analysis (HPA); [Horn, J.L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.], and Minimum Average Partial (MAP) analysis; [Velicer, W.F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327.]), in addition to those used in RIAS development, were investigated. Exploratory factor analyses using both orthogonal and oblique rotations and higher-order exploratory factor analyses using the Schmid and Leiman [Schmid, J., and Leiman, J.M. (1957). The development of hierarchical factor solutions. Psychometrika, 22, 53-61.] procedure were conducted. All factor extraction criteria indicated extraction of only one factor. Oblique rotations resulted in different results than orthogonal rotations, and higher-order factor analysis indicated the largest amount of variance was accounted for by the general intelligence factor. The proposed three-factor solution was not supported. Implications for the use of the RIAS with similarly referred students are discussed.     

Orthogonal higher order structure and confirmatory factor analysis of the French Wechsler Adult Intelligence Scale (WAIS-III)

According to the most widely accepted Cattell-Horn-Carroll (CHC) model of intelligence measurement, each subtest score of the Wechsler Intelligence Scale for Adults (3rd ed.; WAIS-III) should reflect both 1st- and 2nd-order factors (i.e., 4 or 5 broad abilities and 1 general factor). To disentangle the contribution of each factor, we applied a Schmid-Leiman orthogonalization transformation (SLT) to the standardization data published in the French technical manual for the WAIS-III. Results showed that the general factor accounted for 63% of the common variance and that the specific contributions of the 1st-order factors were weak (4.7%-15.9%). We also addressed this issue by using confirmatory factor analysis. Results indicated that the bifactor model (with 1st-order group and general factors) better fit the data than did the traditional higher order structure. Models based on the CHC framework were also tested. Results indicated that a higher order CHC model showed a better fit than did the classical 4-factor model; however, the WAIS bifactor structure was the most adequate. We recommend that users do not discount the Full Scale IQ when interpreting the index scores of the WAIS-III because the general factor accounts for the bulk of the common variance in the French WAIS-III. The 4 index scores cannot be considered to reflect only broad ability because they include a strong contribution of the general factor.     

Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices

The Schmid–Leiman (S–L; Psychometrika 22: 53–61, 1957) transformation is a popular method for conducting exploratory bifactor analysis that has been used in hundreds of studies of individual differences variables. To perform a two-level S–L transformation, it is generally believed that two separate factor analyses are required: a first-level analysis in which k obliquely rotated factors are extracted from an observed-variable correlation matrix, and a second-level analysis in which a general factor is extracted from the correlations of the first-level factors. In this article, I demonstrate that the S–L loadings matrix is necessarily rank deficient. I then show how this feature of the S–L transformation can be used to obtain a direct S–L solution from an unrotated first-level factor structure. Next, I reanalyze two examples from Mansolf and Reise (Multivar Behav Res 51: 698–717, 2016) to illustrate the utility of ‘best-fitting’ S–L rotations when gauging the ability of hierarchical factor models to recover known bifactor structures. Finally, I show how to compute direct bifactor solutions for non-hierarchical bifactor structures. An online supplement includes R code to reproduce all of the analyses that are reported in the article.     

The direct product model for the mtmm matrix parameterized as a second order factor analysis model

The composite direct product model for the multitrait-multimethod matrix is reparameterized as a second-order factor analysis model. This facilitates the use of widely available computer programs such as LISREL and LISCOMP for fitting the model.Bruce Bloxom. Paul Horst and Karl Jöreskog contributed helpful comments to an earlier version of this paper. Their suggestions are gratefully acknowledged.     

Exploratory Bifactor Analysis: The Schmid-Leiman Orthogonalization and Jennrich-Bentler Analytic Rotations

Analytic bifactor rotations have been recently developed and made generally available, but they are not well understood. The Jennrich-Bentler analytic bifactor rotations (bi-quartimin and bi-geomin) are an alternative to, and arguably an improvement upon, the less technically sophisticated Schmid-Leiman orthogonalization. We review the technical details that underlie the Schmid-Leiman and Jennrich-Bentler bifactor rotations, using simulated data structures to illustrate important features and limitations. For the Schmid-Leiman, we review the problem of inaccurate parameter estimates caused by the linear dependencies, sometimes called “proportionality constraints,” that are required to expand a p correlated factors solution into a (p + 1) (bi)factor space. We also review the complexities involved when the data depart from perfect cluster structure (e.g., item cross-loading on group factors). For the Jennrich-Bentler rotations, we describe problems in parameter estimation caused by departures from perfect cluster structure. In addition, we illustrate the related problems of (a) solutions that are not invariant under different starting values (i.e., local minima problems) and (b) group factors collapsing onto the general factor. Recommendations are made for substantive researchers including examining all local minima and applying multiple exploratory techniques in an effort to identify an accurate model.     

Evaluating the invariance of the factor structure of the EPQ-R-N among adolescents

We examined the factor structure of the Neuroticism scale of the Eysenck Personality Questionnaire (EPQ-R-N; S. B. G. Eysenck, Eysenck & Barrett, 1985) and its factor invariance across sex and racial/ethnic groups in a sample of 1,979 adolescents. Using confirmatory factor analyses, we compared a hierarchical model to previous models of the EPQ-R-N and to single-factor and 3-factor structures. The hierarchical factor structure in which a general factor coexists with 3 group factors (depression, social concerns, and worry) was superior to alternative models. The general factor accounted for more than 60% of the variance in EPQ-R-N total scores and was invariant across sex and ethnicity. The 3 group factors varied across ethnicity and sex. We discuss the implications of these findings for conceptualization and assessment of neuroticism using the EPQ-R-N.     

Multivariate genetic analysis of specific cognitive abilities in the Colorado adoption project at age 7

A multivariate hierarchical model of specific cognitive abilities was fitted to data from 7-year-old adopted and nonadopted sibling pairs in the Colorado Adoption Project in order to assess differential genetic influence on specific mental abilities. Model fitting results and Schmid-Leiman (Schmid & Leiman, 1957) transformations reveal significant heritable variation for verbal, spatial, and memory factors independent of general cognitive ability for the eight ability tests examined. In contrast, environmental influences are primarily measure-specific. The results suggest genetic effects in middle childhood that differentially influence mental ability scores.     

Evidence for a general factor model of ADHD in adults

Objective: To examine factor structures of Diagnostic and Statistical Manual of Mental Disorders (4th ed.) symptoms of ADHD in adults. Method: Two sets of models were tested: (a) models with inattention and hyperactivity/impulsivity as separate but correlated latent constructs and (b) hierarchical general factor models with a general factor for all symptoms and separate specific factors for inattention and hyperactivity/impulsivity. Participants were 751 adults with ADHD. Two models with correlated factors and two general factor models of ADHD symptoms were tested. Results: The general factor model provided a better fit of the data than the correlated models. The general factor model with one general and three (inattention, motoric, and verbal hyperactivity/impulsivity) specific factors best accounted for the adults' symptoms. Conclusion: These results suggest a unitary component to ADHD symptoms as well as dimensional specific factors. The replication of a general factor in adults suggests continuity of symptom presentation from childhood into adulthood. Clinical implications are discussed.     

The big none: No evidence for a general factor of personality in chimpanzees, orangutans, or rhesus macaques

We examined whether a general factor of personality (GFP) was present in chimpanzees, orangutans, or rhesus macaques. We used confirmatory factor analysis (CFA) to model correlations among first-order factors as arising from a GFP. We then conducted principal axis factor analyses (PFA) of the first-order factors to extract a single higher-order factor and then to extract two oblique higher-order factors. The CFA model fit was poor for chimpanzees and orangutans, but not rhesus macaques. The single higher-order factors extracted via PFA did not resemble the GFP in all three species. The oblique higher-order factors extracted via PFA were only weakly correlated in all three species. These results do not support the existence of a GFP in nonhuman primates.     

Psychological distress: A hierarchical factor model of the Multiple Affect Adjective Check List (MAACL)

Recent investigations of the structure of psychological distress have indicated that hierarchical models can accommodate both unitary and multifaceted conceptions of distress. The present study tested the hierarchical framework suggested by Zuckerman, Lubin, and Rinck (1983) for the Multiple Affect Adjective Check List (MAACL), a commonly used measure of psychological distress. One- and two-factor models were estimated using maximum-likelihood methods. Results indicated that the two-factor solution, with correlated positive and negative affect factors, provided a significantly better fit to the data than did the omnibus one-factor solution. These results provide further support for hierarchical models of distress.     

Dimensionality of posttraumatic stress symptoms: a confirmatory factor analysis of DSM-IV symptom clusters and other symptom models

Recent exploratory [Taylor, S., Kuch, K., Koch, W. J., Crockett, D. J., & Passey, G. (1998). The structure of posttraumatic stress symptoms. Journal of Abnormal Psychology, 107, 154-160.] and confirmatory [Buckley, T. C., Blanchard, E. B., & Hickling, E. J. (1998). A confirmatory factor analysis of posttraumatic stress symptoms. Behaviour Research and Therapy, 36, 1091-1099; King, D. W., Leskin, G. A., King, L. A., & Weathers, F. W. (1998). Confirmatory factor analysis of the clinician-administered PTSD scale: evidence for the dimensionality of posttraumatic stress disorder. Psychological Assessment, 10, 90-96.] factor analytic investigations suggest that the three symptom clusters of posttraumatic stress disorder (PTSD) as defined in the Diagnostic and Statistical Manual [4th ed.; DSM-IV; American Psychiatric Association (1994). Diagnostic and statistical manual of mental disorders (4th ed.). Washington, DC: Author.] may not provide the best conceptualization of symptom dimensionality. However, the alternative models have not been in agreement, nor have they been compared against each other or models based on the DSM-IV. The purpose of the present investigation was to test a series of dimensional models suggested by these recent factor analytic investigations and the DSM-IV. Using data collected with the PTSD Checklist--Civilian Version [Weathers, F. W., Litz, B. T., Huska, J. A., & Keane, T. M. (1994). PCL-C for DSM-IV. Boston: National Center for PTSD--Behavioral Science Division.] from 349 referrals to a primary care medical clinic, we used confirmatory factor analysis to evaluate a: (1) hierarchical four-factor model, (2) four-factor intercorrelated model, (3) hierarchical three-factor model, (4) three-factor intercorrelated model, and (5) hierarchical two-factor model. The hierarchical four-factor model (comprising four first-order factors corresponding to reexperiencing, avoidance, numbing, and hyperarousal all subsumed by a higher-order general factor) provided the best overall fit to the data; although, all models met some standards specified for good model fit. More research is needed to establish the dimensional nature of PTSD symptoms and to assess whether identified dimensions differ as a function of the trauma experience. Implications for assessment, diagnosis, and treatment are also discussed.