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Flora et al.Factor analysis assumptionsFIGURE 6 | Histograms of standardized residuals for each observed variable from three-factor model fitted to perturbed sample data (N = 100).a warning message) 7 . Eigenvalues extremely close to zero may alsobe suggestive of near-collinearity 8 . A commonly used statistic forevaluating near-collinearity in ordinary regression is the conditionindex, which equals the square root of the ratio of the largest tosmallest eigenvalue, with larger values more strongly indicativeof near-collinearity. For the current example, the condition indexequals 6.34, which is well below the value of 30 suggested by Belsleyet al. (1980) as indicative of problematic near-collinearity.In the context of factor analysis, collinearity often implies anill-conceived selection of variables for the analysis. For example,if both total scores and sub-scale scores (or just one sub-scale)from the same instrument are included in a factor analysis, thetotal score is linearly dependent on, or collinear with, the subscalescore. A more subtle example may be the inclusion of both a“positive mood”scale and a“negative mood”scale; although scoresfrom these two scales may not be perfectly related, they will likely7 Polychoric correlation matrices (defined below) are often non-positive definite.Unlike a product-moment R, a non-positive definite polychoric correlation matrixis not necessarily problematic.8 Most EFA software includes eigenvalues of R as default output. However, it isimportant not to confuse the eigenvalues of R with the eigenvalues of the “reduced”correlation matrix, R − ˆΘ. The reduced correlation matrix often has negativeeigenvalues when R does not.have a very strong (negative) correlation,which can be problematicfor factor analysis. In these situations, researchers should carefullyreconsider their choice of variables for the analysis and remove anythat is collinear with one or more of the other observed variables,which in turn redefines the overall research question addressed bythe analysis (see Fox, 2008, p. 342).NON-NORMALITYIn terms of obtaining accurate parameter estimates, the ULS estimationmethod mentioned above makes no assumption regardingobserved variable distributions whereas ML estimation is based onthe multivariate normal distribution (MacCallum, 2009). Specifically,for both estimators, parameter estimates are trustworthy aslong as the sample size is large and the model is properly specified(i.e., the estimator is consistent), even when the normality assumptionfor ML is violated (Bollen, 1989). However, SE estimates andcertain model fit statistics (i.e., the χ 2 fit statistic and statisticsbased on χ 2 such as CFI and RMSEA) are adversely affected bynon-normality, particularly excess multivariate kurtosis. Althoughthey are less commonly used with EFA, SEs and model fit statisticsare equally applicable with EFA as with CFA and are easily obtainedwith modern software (but see Cudeck and O’Dell, 1994, for cautionsand recommendations regarding SEs with EFA). With bothapproaches,SEs convey information about the sampling variabilityFrontiers in Psychology | Quantitative Psychology and Measurement March 2012 | Volume 3 | Article 55 | 110