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Flora et al.Factor analysis assumptionstheir variation and covariation; that is, covariation between anytwo observed variables is due to them being influenced by thesame factor. This idea was introduced by Spearman (1904) and,largely due to Thurstone (1947), evolved into the common factormodel, which remains the dominant paradigm for factor analysistoday. Factor analysis is traditionally a method for fitting modelsto the bivariate associations among a set of variables, with EFAmost commonly using Pearson product-moment correlations andCFA most commonly using covariances. Use of product-momentcorrelations or covariances follows from the fact that the commonfactor model specifies a linear relationship between the factors andthe observed variables.Lawley and Maxwell (1963) showed that the common factormodel can be formally expressed as a linear model with observedvariables as dependent variables and factors as explanatory orindependent variables:y j = λ 1 η 1 + λ 2 η 2 + λ k η k + ε jwhere y j is the jth observed variable from a battery of p observedvariables, η k is the kth of m common factors, λ k is the regressioncoefficient, or factor loading, relating each factor to y j , and ε j is theresidual, or unique factor, for y j . (Often there are only one or twofactors, in which case the right hand side of the equation includesonly λ 1 η 1 + ε j or only λ 1 η 1 + λ 2 η 2 + ε j .) It is convenient to workwith the model in matrix form:y = Λη + ε, (1)where y isavectorofthep observed variables, Λ is a p × m matrixof factor loadings, η isavectorofm common factors, and ε isavectorofp unique factors 1 . Thus, each common factor mayinfluence more than one observed variable while each unique factor(i.e., residual) influences only one observed variable. As withthe standard regression model, the residuals are assumed to beindependent of the explanatory variables; that is, all unique factorsare uncorrelated with the common factors. Additionally, theunique factors are usually assumed uncorrelated with each other(although this assumption may be tested and relaxed in CFA).Given Eq. 1, it is straightforward to show that the covariancesamong the observed variables can be written as a function of modelparameters (factor loadings, common factor variances and covariances,and unique factor variances). Thus, in CFA, the parametersare typically estimated from their covariance structure:Σ = ΛΨΛ ′ + Θ, (2)where Σ is the p × p population covariance matrix for the observedvariables, Ψ is the m × m interfactor covariance matrix, and Θ isthe p × p matrix unique factor covariance matrix that often containsonly diagonal elements, i.e., the unique factor variances. Thecovariance structure model shows that the observed covariances1 For traditional factor analysis models, the means of the observed variables arearbitrary and unstructured by the model, which allows omission of an interceptterm in Eq. 1 by assuming the observed variables are mean-deviated, or centered(MacCallum, 2009).are a function of the parameters but not the unobservable scoreson the common or unique factors; hence, it is not necessary toobserve scores on the latent variables to estimate the model parameters.In EFA, the parameters are most commonly estimatedfrom the correlation structureP = Λ ∗ Ψ ∗ Λ ∗′ + Θ ∗ (3)where P is the population correlation matrix and, in that P issimply a re-scaled version of Σ, we can view Λ ∗ , Ψ ∗ , and Θ ∗as re-scaled versions of Λ, Ψ, and Θ, respectively. This tendencyto conduct EFA using correlations is mainly a result of historicaltradition, and it is possible to conduct EFA using covariances orCFA using correlations. For simplicity, we focus on the analysisof correlations from this point forward, noting that the principleswe discuss apply equivalently to the analysis of both correlationand covariance matrices (MacCallum, 2009; but see Cudeck, 1989;Bentler, 2007; Bentler and Savalei, 2010 for discussions of theanalysis of correlations vs. covariances). We also drop the asteriskswhen referring to the parameter matrices in Eq. 3.Jöreskog (1969) showed how the traditional EFA model, oran “unrestricted solution” for the general factor model describedabove, can be constrained to produce the “restricted solution”that is commonly understood as today’s CFA model and is wellintegratedin the structural equation modeling (SEM) literature.Specifically, in the EFA model, the elements of Λ are all freely estimated;that is, each of the m factors has an estimated relationship(i.e., factor loading) with every observed variable; factor rotationis then used to aid interpretation by making some values in Λlarge and others small. But in the CFA model, depending on theresearcher’s hypothesized model, many of the elements of Λ arerestricted,or constrained,to equal zero,often so that each observedvariable is determined by one and only one factor (i.e., so that thereare no “cross-loadings”). Because the common factors are unobservedvariables and thus have an arbitrary scale, it is conventionalto define them as standardized (i.e., with variance equal to one);thus Ψ is the interfactor correlation matrix 2 . This convention is nota testable assumption of the model, but rather imposes necessaryidentification restrictions that allow the model parameters to beestimated (although alternative identification constraints are possible,such as the marker variable approach often used with CFA).In addition to constraining the factor variances, EFA requires adiagonal matrix for Θ, with the unique factor variances along thediagonal.Exploratory factor analysis and CFA therefore share the goalof using the common factor model to represent the relationshipsamong a set of observed variables using a small number of factors.Hence, EFA and CFA should not be viewed as disparate methods,despite that their implementation with conventional softwaremight seem quite different. Instead, they are two approaches to2 In EFA, the model is typically estimated by first setting Ψ to be an identity matrix,which implies that the factors are uncorrelated, or orthogonal, leading to the initialunrotated factor loadings in Λ. Applying an oblique factor rotation obtains a new setof factor loadings along with non-zero interfactor correlations. Although rotationis not a focus of the current paper, we recommend that researchers always use anoblique rotation.Frontiers in Psychology | Quantitative Psychology and Measurement March 2012 | Volume 3 | Article 55 | 102