Quantifying the Strength of General Factors in Psychopathology: A Comparison of CFA with Maximum Likelihood Estimation,BSEM, and ESEM/EFA Bifactor Approaches

Whether or not importance should be placed on an all-encompassing general factor of psychopathology (or p factor) in classifying, researching, diagnosing, and treating psychiatric disorders depends (among other issues) on the extent to which comorbidity is symptom-general rather than staying largely within the confines of narrower transdiagnostic factors such as internalizing and externalizing. In this study, we compared three methods of estimating p factor strength. We compared omega hierarchical and explained common variance calculated from confirmatory factor analysis (CFA) bifactor models with maximum likelihood (ML) estimation, from exploratory structural equation modeling/exploratory factor analysis models with a bifactor rotation, and from Bayesian structural equation modeling (BSEM) bifactor models. Our simulation results suggested that BSEM with small variance priors on secondary loadings might be the preferred option. However, CFA with ML also performed well provided secondary loadings were modeled. We provide two empirical examples of applying the three methodologies using a normative sample of youth (z-proso, n = 1,286) and a university counseling sample (n = 359).     

贝叶斯结构方程模型及其研究现状

在心理学研究中结构方程模型(Structural Equation Modeling, SEM)被广泛用于检验潜变量间的因果效应, 其估计方法有频率学方法(如, 极大似然估计)和贝叶斯方法两类。近年来由于贝叶斯统计的流行及其在结构方程建模中易于处理小样本、缺失数据及复杂模型等方面的优势, 贝叶斯结构方程模型发展迅速, 但其在国内心理学领域的应用不足。主要介绍了贝叶斯结构方程模型的方法基础和优良特性, 及几类常用的贝叶斯结构方程模型及其应用现状, 旨在为应用研究者介绍新的研究工具。     

Factor analytic models: viewing the structure of an assessment instrument from three perspectives

The overarching purpose of this article is to present a nonmathematical introduction to the application of confirmatory factor analysis (CFA) within the framework of structural equation modeling as it applies to psychological assessment instruments. In the interest of clarity and ease of understanding, I model exploratory factor analysis (EFA) structure in addition to first- and second-order CFA structures. All factor analytic structures are based on the same measuring instrument, the Beck Depression Inventory-II (BDI-II; Beck, Steer, & Brown, 1996). Following a "walk" through the general process of CFA modeling, I identify several common misconceptions and improper application practices with respect to both EFA and CFA and tender caveats with a view to preventing further proliferation of these pervasive practices.     

Validating the Beck Depression Inventory-II for Hong Kong Community Adolescents

The primary purpose of this study was to test for the validity of a Chinese version of the Beck Depression Inventory-II (C-BDI-II) for use with Hong Kong community (i.e., nonclinical) adolescents. Based on a randomized triadic split of the data (N = 1460), we conducted exploratory factor analysis on Group1 (n = 486) and confirmatory factor analysis (CFA) within the framework of structural equation modeling on Groups 2 (n = 487) and 3 (n = 487); the second CFA served as a cross-validation of the determined factor structure. Factor analytic results, based on a 4-factor structure that comprised 1 2nd-order general factor of Depression and 3 1st-order factors representing Negative Attitude, Performance Difficulty, and Somatic Elements, replicated those reported previously for Canadian (Byrne & Baron, 1993), Swedish (Byrne, Baron, Larsson, & Melin, 1995), and Bulgarian (Byrne, Baron, & Balev, 1998) nonclinical adolescents. Based on this cross-validated factor structure, findings related to internal consistency reliability, stability over a 6-month time lag, and relations with relevant external criteria provided strong support for the valid use of the C-BDI-II in measuring depressive symptoms for Hong Kong community adolescents.     

Structural equation modeling: reviewing the basics and moving forward

This tutorial begins with an overview of structural equation modeling (SEM) that includes the purpose and goals of the statistical analysis as well as terminology unique to this technique. I will focus on confirmatory factor analysis (CFA), a special type of SEM. After a general introduction, CFA is differentiated from exploratory factor analysis (EFA), and the advantages of CFA techniques are discussed. Following a brief overview, the process of modeling will be discussed and illustrated with an example using data from a HIV risk behavior evaluation of homeless adults (Stein & Nyamathi, 2000). Techniques for analysis of nonnormally distributed data as well as strategies for model modification are shown. The empirical example examines the structure of drug and alcohol use problem scales. Although these scales are not specific personality constructs, the concepts illustrated in this article directly correspond to those found when analyzing personality scales and inventories. Computer program syntax and output for the empirical example from a popular SEM program (EQS 6.1; Bentler, 2001) are included.     

A multi-sample investigation of the higher-order latent dimensionality of the Sport-Multidimensional Perfectionism Scale-2

ObjectiveExamine the higher-order latent dimensionality of the Sport-Multidimensional Perfectionism Scale-2 (Sport-MPS-2: Gotwals & Dunn, 2009).DesignCorrelational.MethodA total of 1605 athletes (562 female) from eight independent samples completed the Sport-MPS-2. Athletes in one sample (n = 239) also completed a portion of the Multidimensional Inventory of Perfectionism in Sport (MIPS: Stoeber, Otto, & Stoll, 2006). The correlation matrices among the Sport-MPS-2 subscales for five samples were analyzed with exploratory factor analyses. The covariance matrices for the subscales in the three remaining samples (including the sample that completed the MIPS) were analyzed with confirmatory factor analyses and exploratory structural equation modeling (ESEM: Asparouhov & Muthén, 2009).ResultsTwo highly interpretable factors—labelled Perfectionistic Strivings and Perfectionistic Concerns—were obtained for each data set.ConclusionTheorists note the importance of using multiple indicators to measure perfectionistic strivings and perfectionistic concerns in sport. The current factor-analytic and ESEM results indicate that the six subscales comprising the Sport-MPS-2 may help to achieve this objective.     

Structural Equation Modeling With AMOS,EQS, and LISREL: Comparative Approaches to Testing for the Factorial Validity of a Measuring Instrument

Using a confirmatory factor analytic (CFA) model as a paradigmatic basis for all comparisons, this article reviews and contrasts important features related to 3 of the most widely-used structural equation modeling (SEM) computer programs: AMOS 4.0 (Arbuckle, 1999), EQS 6 (Bentler, 2000), and LISREL 8 (Joreskog & Sorbom, 1996b). Comparisons focus on (a) key aspects of the programs that bear on the specification and testing of CFA models-preliminary analysis of data, and model specification, estimation, assessment, and misspecification; and (b) other important issues that include treatment of incomplete, nonnormally-distributed, or categorically-scaled data. It is expected that this comparative review will provide readers with at least a flavor of the approach taken by each program with respect to both the application of SEM within the framework of a CFA model, and the critically important issues, previously noted, related to data under study.     

Rejoinder to MacCallum, Edwards, and Cai (2012) and Rindskopf (2012): Mastering a new method

This rejoinder discusses the general comments on how to use Bayesian structural equation modeling (BSEM) wisely and how to get more people better trained in using Bayesian methods. Responses to specific comments cover how to handle sign switching, nonconvergence and nonidentification, and prior choices in latent variable models. Two new applications are included. The first one revisits the Kaplan (2009) science model by considering priors on primary parameters. The second one applies BSEM to the bifactor model that was hypothesized in the original Holzinger and Swineford (1939) study. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Analysis of the factor structure of the Sociocultural Attitudes Towards Appearance Questionnaire (SATAQ-3) in Spanish secondary-school students through exploratory structural equation modeling

The aims of the present study were: (1) to assess the factor structure of the SATAQ-3 in Spanish secondary-school students by means of exploratory factor analysis (EFA), confirmatory factor analysis (CFA) and exploratory structural equation modeling (ESEM) models; and (2) to study its invariance by sex and school grade. ESEM is a technique that has been proposed for the analysis of internal structure that overcomes some of the limitations of EFA and CFA. Participants were 1559 boys and girls in grades seventh to tenth. The results support the four-factor solution of the original version, and reveal that the best fit was obtained with ESEM, excluding Item 20 and with correlated uniqueness between reverse-keyed items. Our version shows invariance by sex and grade. The differences between scores of different groups are in the expected direction, and support the validity of the questionnaire. We recommend a version excluding Item 20 and without reverse-keyed items.     

Factor Structure of Subjective Well-Being in Iran

Subjective well-being is predominantly conceived as having 3 components: life satisfaction, positive affect, and negative affect. This article reports 2 studies that seek to investigate the factor structure of subjective well-being in Iran. One-, two-, and three-factor models of subjective well-being were evaluated using confirmatory factor analysis (CFA) and exploratory structural equation modeling (ESEM). The results of Study 1 (N = 2,197) and Study 2 (N = 207) show that whereas the 1- and 2-factor models do not fit the data well, the 3-factor model provides an adequate fit. These results indicate that the 3 components of subjective well-being constitute 3 interrelated, yet distinct, factors. The analyses demonstrate how traditional CFA and ESEM can be combined to obtain a clear picture of the measurement model of subjective well-being and generate new insights about individual items and cross-loadings needed to derive more parsimonious measures. Nuances relating to the assessment of subjective well-being in more collectivist and Muslim countries are discussed.     

Revisiting the Empirical Distinction Between Hedonic and Eudaimonic Aspects of Well-Being Using Exploratory Structural Equation Modeling

The distinction between hedonic (i.e., subjective well-being) and eudaimonic (i.e., psycho-social functioning) components of well-being is questioned by some researchers on the grounds that these two aspects of well-being are highly correlated. However, I argue that previous research has relied on confirmatory factor analysis (CFA), which is likely to overestimate interfactor correlations, because cross-loadings are constrained to be zero in CFA. In contrast, the new method of exploratory structural equation modeling (ESEM) does not constrain cross-ladings to zero, which results in more accurate factor intercorrelations. The present study used ESEM to reinvestigate the relationship between hedonic and eudaimonic aspects of well-being in a sample of 3986 American adults. The results showed that the ESEM model fitted the data better than the CFA model. As expected, interfactor correlations obtained with ESEM were substantially smaller than those obtained with CFA, indicating greater factor distinctiveness. These results suggest that hedonic and eudaimonic factors are correlated yet largely independent from each other. The results also suggest that ESEM is a more appropriate method than CFA in the study of multi-dimensional constructs, such as mental well-being.     

Bayesian structural equation modeling: A more flexible representation of substantive theory

This article proposes a new approach to factor analysis and structural equation modeling using Bayesian analysis. The new approach replaces parameter specifications of exact zeros with approximate zeros based on informative, small-variance priors. It is argued that this produces an analysis that better reflects substantive theories. The proposed Bayesian approach is particularly beneficial in applications where parameters are added to a conventional model such that a nonidentified model is obtained if maximum-likelihood estimation is applied. This approach is useful for measurement aspects of latent variable modeling, such as with confirmatory factor analysis, and the measurement part of structural equation modeling. Two application areas are studied, cross-loadings and residual correlations in confirmatory factor analysis. An example using a full structural equation model is also presented, showing an efficient way to find model misspecification. The approach encompasses 3 elements: model testing using posterior predictive checking, model estimation, and model modification. Monte Carlo simulations and real data are analyzed using Mplus. The real-data analyses use data from Holzinger and Swineford's (1939) classic mental abilities study, Big Five personality factor data from a British survey, and science achievement data from the National Educational Longitudinal Study of 1988. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Multidimensional structure and measurement invariance of school engagement

This study examined the multidimensional structure and measurement invariance of a school engagement instrument using confirmatory factor analysis (CFA), exploratory structural equation modeling (ESEM), bifactor CFA (BCFA), and bifactor ESEM (BESEM). Participants consisted of 1731 students in Grades 9 - 11 from the 4-H Study of Positive Youth Development in the United States. The use of the CFA, ESEM, BCFA, and BESEM models was expected to provide more insight into the cross-loading and hierarchical structures of school engagement. We found empirical evidence to support the (a) tripartite factor structure of school engagement, (b) existence of cross-loadings and hierarchical structures, (c) measurement invariance across gender (male vs female) and race (European American vs African American), and (d) expected latent means differences by gender.     

Testing the Factor Structure and Measurement Invariance Across Gender of the Big Five Inventory Through Exploratory Structural Equation Modeling

Confirmatory factor analyses (CFAs) typically fail to support the a priori 5-factor structure of Big Five self-report instruments, due in part to the overly restrictive CFA assumptions. We show that exploratory structural equation modeling (ESEM), an integration of CFA and exploratory factor analysis, overcomes these problems in relation to responses to the 44-item Big Five Inventory (BFI) administered to a large Italian community sample. ESEM fitted the data better and resulted in less correlated factors than CFA, although ESEM and CFA factor scores correlated at near unity with observed raw scores. Tests of gender invariance with a 13-model taxonomy of full measurement invariance showed that the factor structure of the BFI is gender-invariant and that women score higher on Neuroticism, Agreeableness, Extraversion, and Conscientiousness. Through ESEM one could address substantively important issues about BFI psychometric properties that could not be appropriately addressed through traditional approaches.     

Confirmatory Factor Analysis and Profile Analysis via Multidimensional Scaling

This paper describes the Confirmatory Factor Analysis (CFA) parameterization of the Profile Analysis via Multidimensional Scaling (PAMS) model to demonstrate validation of profile pattern hypotheses derived from multidimensional scaling (MDS). Profile Analysis via Multidimensional Scaling (PAMS) is an exploratory method for identifying major profiles in a multi-subtest test battery. Major profile patterns are represented as dimensions extracted from a MDS analysis. PAMS represents an individual observed score as a linear combination of dimensions where the dimensions are the most typical profile patterns present in a population. While the PAMS approach was initially developed for exploratory purposes, its results can later be confirmed in a different sample by CFA. Since CFA is often used to verify results from an exploratory factor analysis, the present paper makes the connection between a factor model and the PAMS model, and then illustrates CFA with a simulated example (that was generated by the PAMS model) and at the same time with a real example. The real example demonstrates confirmation of PAMS exploratory results by using a different sample. Fit indexes can be used to indicate whether the CFA reparameterization as a confirmatory approach works for the PAMS exploratory results.     

探索性结构方程建模(ESEM):EFA和CFA的整合

探索性结构方程建模(ESEM)是在测量模型部分使用了类似于EFA模型的SEM.作为一种高级统计方法,ESEM整合了EFA和CFA两种因子分析方法的功能和优点.通过ESEM,研究者既可以灵活地探索因子结构,又可以系统地验证因子模型,为潜变量的关系分析提供更适宜的测量模型.ESEM已经在某些社科领域的研究中得到应用,是一种值得推介的因子分析方法.ESEM的具体应用问题,例如因子旋转方法的选用、测验信度评价等,仍有待探讨.     

Recovering Substantive Factor Loadings in the Presence of Acquiescence Bias: A Comparison of Three Approaches

Researchers are often advised to write balanced scales (containing an equal number of positively and negatively worded items) when measuring psychological attributes. This practice is recommended to control for acquiescence bias (ACQ). However, little advice has been given on what to do with such data if the researcher subsequently wants to evaluate a 1-factor model for the scale. This article compares 3 approaches for dealing with the presence of ACQ bias, which make different assumptions: an ipsatization approach based on the work of Chan and Bentler (CB; 1993), a confirmatory factor analysis (CFA) approach that includes an ACQ factor with equal loadings (Billiet & McClendon, 2000; Mirowsky & Ross, 1991), and an exploratory factor analysis (EFA) approach with a target rotation (Ferrando, Lorenzo-Seva, & Chico, 2003). We also examine the “do nothing” approach which fits the 1-factor model to the data ignoring the presence of ACQ bias. Our main findings are that the CFA method performs best overall and that it is robust to the violation of its assumptions, the EFA and the CB approaches work well when their assumptions are strictly met, and the “do nothing” approach can be surprisingly robust when the ACQ factor is not very strong.