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Flora et al.Factor analysis assumptionsWest, S. G., Finch, J. F., and Curran,P. J. (1995). “Structural equationmodels with non-normal variables:problems and remedies,”in Structural Equation Modeling:Concepts, Issues, and Applications,ed. R. H. Hoyle (ThousandOaks, CA: SAGE Publications),56–75.Wirth, R. J., and Edwards, M. C.(2007). Item factor analysis: currentapproaches and future directions.Psychol. Methods 12, 58–79.Yang-Wallentin, F., Jöreskog, K. G., andLuo, H. (2010). Confirmatory factoranalysis of ordinal variables withmisspecified models. Struct. Equ.Modeling 17, 392–423.Yuan, K.-H., and Bentler, P. M. (1998).Structural equation modeling withrobust covariances. Sociol. Methodol.28, 363–396.Yuan, K.-H., Marshall, L. L., and Bentler,P. M. (2002). A unified approachto exploratory factor analysis withmissing data, nonnormal data, andin the presence of outliers. Psychometrika67, 95–122.Yuan, K.-H., and Zhong, X. (2008).Outliers, leverage observations, andinfluential cases in factor analysis:using robust procedures to minimizetheir effect. Sociol. Methodol. 38,329–368.Conflict of Interest Statement: Theauthors declare that the research wasconducted in the absence of any commercialor financial relationships thatcould be construed as a potential conflictof interest.Received: 27 October 2011; paper pendingpublished: 21 November 2011; accepted:13 February 2012; published online: 01March 2012.Citation: Flora DB, LaBrish C andChalmers RP (2012) Old and new ideasfor data screening and assumption testingfor exploratory and confirmatory factoranalysis. Front. Psychology 3:55. doi:10.3389/fpsyg.2012.00055This article was submitted to Frontiersin Quantitative Psychology and Measurement,a specialty of Frontiers in Psychology.Copyright © 2012 Flora, LaBrish andChalmers. This is an open-access articledistributed under the terms of the CreativeCommons Attribution Non CommercialLicense, which permits noncommercialuse, distribution, and reproductionin other forums, provided theoriginal authors and source are credited.www.frontiersin.org March 2012 | Volume 3 | Article 55 | 121
ORIGINAL RESEARCH ARTICLEpublished: 27 March 2013doi: 10.3389/fpsyg.2013.00109On the relevance of assumptions associated with classicalfactor analytic approaches †Daniel Kasper and Ali Ünlü*Chair for Methods in Empirical Educational Research, TUM School of Education and Centre for International Student Assessment, Technische Universität München,Munich, GermanyEdited by:Jason W. Osborne, Old DominionUniversity, USAReviewed by:Andrew Jones, American Board ofSurgery, USAEvgueni Borokhovski, ConcordiaUniversity, Canada*Correspondence:Ali Ünlü, Chair for Methods inEmpirical Educational Research, TUMSchool of Education and Centre forInternational Student Assessment,Technische Universität München,Lothstrasse 17, 80335 Munich,Germany.e-mail: ali.uenlue@tum.de†The research reported in this paper isbased on the dissertation thesis byKasper (2012).A personal trait, for example a person’s cognitive ability, represents a theoretical conceptpostulated to explain behavior. Interesting constructs are latent, that is, they cannot beobserved. Latent variable modeling constitutes a methodology to deal with hypotheticalconstructs. Constructs are modeled as random variables and become components of astatistical model. As random variables, they possess a probability distribution in the populationof reference. In applications, this distribution is typically assumed to be the normaldistribution. The normality assumption may be reasonable in many cases, but there aresituations where it cannot be justified. For example, this is true for criterion-referencedtests or for background characteristics of students in large scale assessment studies. Nevertheless,the normal procedures in combination with the classical factor analytic methodsare frequently pursued, despite the effects of violating this “implicit” assumption are notclear in general. In a simulation study, we investigate whether classical factor analyticapproaches can be instrumental in estimating the factorial structure and properties of thepopulation distribution of a latent personal trait from educational test data, when violationsof classical assumptions as the aforementioned are present.The results indicate thathaving a latent non-normal distribution clearly affects the estimation of the distribution ofthe factor scores and properties thereof. Thus, when the population distribution of a personaltrait is assumed to be non-symmetric, we recommend avoiding those factor analyticapproaches for estimation of a person’s factor score, even though the number of extractedfactors and the estimated loading matrix may not be strongly affected. An application to theProgress in International Reading Literacy Study (PIRLS) is given. Comments on possibleimplications for the Programme for International Student Assessment (PISA) complete thepresentation.Keywords: factor analysis, latent variable model, normality assumption, factorial structure, criterion-referenced test,large scale educational assessment, Programme for International Student Assessment, Progress in InternationalReading Literacy Study1. INTRODUCTIONEducational research is concerned with the study of processes oflearning and teaching. Typically, the investigated processes are notobservable, and to unveil these, manifest human behavior in testsituations is recorded. According to Lienert and Raatz (1998, p. 1)“a test [. . .] is a routine procedure for the investigation of one ormore empirically definable personality traits” (translated by theauthors), and to satisfy a minimum of quality criteria, a test isrequired to be objective, reliable, and valid.In this paper we deal with factor analytic methods for assessingconstruct validity of a test, in the sense of its factorial validity (e.g.,Cronbach and Meehl, 1955; Lienert and Raatz, 1998). Factorialvalidity refers to the factorial structure of the test, that is, to thenumber (and interpretation) of underlying factors, the correlationstructure among the factors, and the correlations of each test itemwith the factors. There are a number of latent variable models thatmay be used to analyze the factorial structure of a test – for generalizedlatent variable modeling covering a plethora of models asspecial cases of a much broader framework, see Bartholomew et al.(2011) and Skrondal and Rabe-Hesketh (2004). This paper focuseson classical factor analytic approaches, and it examines how accuratelydifferent methods of classical factor analysis can estimatethe factorial structure of test data, if assumptions associated withthe classical approaches are not satisfied. The methods of classicalfactor analysis will include principal component analysis (PCA;Pearson, 1901; Hotelling, 1933a,b; Kelley, 1935), exploratory factoranalysis (EFA; Spearman, 1904; Burt, 1909; Thurstone, 1931,1965), and principal axis analysis (PAA; Thurstone, 1931, 1965).More recent works on factor analysis and related methods are Harman(1976), McDonald (1985), Cudeck and MacCallum (2007),and Mulaik (2009). Further references, to more specific topics infactor analysis, are given below, later in the text. 11 For the sake of simplicity and for the purpose and analysis of this paper, we wantto refer to all of these approaches (PCA, EFA, PAA) collectively as classical factoranalysis/analytic methods. Albeit it is known that PCA differs from factor analysisin important aspects, and that PAA rather represents an alternative estimationprocedure for EFA. PCA and EFA are different technically and conceptually. PCAwww.frontiersin.org March 2013 | Volume 4 | Article 109 | 122
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