Exploratory factor analysis: Difference between revisions

Revision as of 14:50, 11 March 2025

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''Underfactoring'' occurs when too few factors are included in a model. If not enough factors are included in a model, there is likely to be substantial error. Measured variables that load onto a factor not included in the model can falsely load on factors that are included, altering true factor loadings. This can result in rotated solutions in which two factors are combined into a single factor, obscuring the true factor structure.

There are a number of procedures designed to determine the optimal number of factors to retain in EFA. Broadly speaking, most of the existing procedures approach the determination of the appropriate number of factors (1) by inspecting patterns of eigenvalues of the covariance matrix, or (2) treating it as a model selection problem.<ref name=":0">{{Cite journal |last=Haslbeck |first=Jonas M. B. |last2=van Bork |first2=Riet |date=2024-02 |title=Estimating the number of factors in exploratory factor analysis via out-of-sample prediction errors. |url=https://doi.apa.org/doi/10.1037/met0000528 |journal=Psychological Methods |language=en |volume=29 |issue=1 |pages=48–64 |doi=10.1037/met0000528 |issn=1939-1463}}</ref> Existing approaches include: Kaiser's (1960) eigenvalue-greater-than-one rule (or K1 rule),<ref>{{cite journal|last=Kaiser|first=H.F.|title=The application of electronic computers to factor analysis|journal=Educational and Psychological Measurement|year=1960|volume=20|pages=141–151|doi=10.1177/001316446002000116|s2cid=146138712 }}</ref> Cattell's (1966) [[scree plot]],<ref name="Cattell, R. B. 1966">Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research, I, 245-276.</ref> Revelle and Rocklin's (1979) very simple structure criterion,<ref>{{cite journal | last1 = Revelle | first1 = W. | last2 = Rocklin | first2 = T. | year = 1979 | title = Very simple structure-alternative procedure for estimating the optimal number of interpretable factors | journal = Multivariate Behavioral Research | volume = 14 | issue = 4| pages = 403–414 | doi = 10.1207/s15327906mbr1404_2 | pmid = 26804437 }}</ref> model comparison techniques,<ref>{{cite journal | last1 = Fabrigar | first1 = Leandre R. | last2 = Wegener | first2 = Duane T. | last3 = MacCallum | first3 = Robert C. | last4 = Strahan | first4 = Erin J. | year = 1999 | title = Evaluating the use of exploratory factor analysis in psychological research. | journal = Psychological Methods | volume = 4 | issue = 3| pages = 272–299 | doi = 10.1037/1082-989X.4.3.272 }}</ref> Raiche, Roipel, and Blais's (2006) acceleration factor and optimal coordinates,<ref>Raiche, G., Roipel, M., & Blais, J. G.|Non graphical solutions for the Cattell’s scree test. Paper presented at The International Annual Meeting of the Psychometric Society, Montreal|date=2006|Retrieved December 10, 2012 from {{cite web |url=https://ppw.kuleuven.be/okp/_pdf/Raiche2013NGSFC.pdf |title=Archived copy |access-date=2013-05-03 |url-status=live |archive-url=https://web.archive.org/web/20131021052759/https://ppw.kuleuven.be/okp/_pdf/Raiche2013NGSFC.pdf |archive-date=2013-10-21 }}</ref> Velicer's (1976) minimum average partial,<ref name=Velicer>{{cite journal|last=Velicer|first=W.F.|title=Determining the number of components from the matrix of partial correlations|journal=Psychometrika|year=1976|volume=41|issue=3|pages=321–327|doi=10.1007/bf02293557|s2cid=122907389 }}</ref> Horn's (1965) [[parallel analysis]], and Ruscio and Roche's (2012) comparison data.<ref name =Ruscio>{{cite journal|last=Ruscio|first=J.|author2=Roche, B.|title=Determining the number of factors to retain in an exploratory factor analysis using comparison data of a known factorial structure|journal=Psychological Assessment|year=2012|volume=24|issue=2|pages=282–292|doi=10.1037/a0025697|pmid=21966933}}</ref> Recent simulation studies assessing the robustness of such techniques suggest that the latter five can better assist practitioners to judiciously model data.<ref name =Ruscio/> These five modern techniques are now easily accessible through integrated use of IBM SPSS Statistics software (SPSS) and R (R Development Core Team, 2011). See Courtney (2013)<ref name="pareonline.net">Courtney, M. G. R. (2013). Determining the number of factors to retain in EFA: Using the SPSS R-Menu v2.0 to make more judicious estimations. ''Practical Assessment, Research and Evaluation'', 18(8). Available online:

{{cite web |url=http://pareonline.net/getvn.asp?v=18&n=8 |title=Archived copy |access-date=2014-06-08 |url-status=live |archive-url=https://web.archive.org/web/20150317145450/http://pareonline.net/getvn.asp?v=18&n=8 |archive-date=2015-03-17 }}</ref> for guidance on how to carry out these procedures for continuous, ordinal, and heterogenous (continuous and ordinal) data.

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*'''Root mean square error of approximation (RMSEA) fit index:''' RMSEA is an estimate of the discrepancy between the model and the data per degree of freedom for the model. Values less that .05 constitute good fit, values between 0.05 and 0.08 constitute acceptable fit, a values between 0.08 and 0.10 constitute marginal fit and values greater than 0.10 indicate poor fit .<ref name =Browne/><ref>Steiger, J. H. (1989). EzPATH: A supplementary module for SYSTAT andsygraph. Evanston, IL: SYSTAT</ref> An advantage of the RMSEA fit index is that it provides confidence intervals which allow researchers to compare a series of models with varying numbers of factors.

*'''Information Criteria:''' Information criteria such as Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC) <ref>Neath, A. A., & Cavanaugh, J. E. (2012). The Bayesian information criterion: background, derivation, and applications. Wiley Interdisciplinary Reviews: Computational Statistics, 4(2), 199-203.</ref> can be used to trade-off model fit with model complexity and select an optimal number of factors.

*'''Out-of-sample Prediction Errors (PE):''' Using the connection between model-implied covariance matrices and standardized regression weights, the number of factors can be selected using out-of-sample prediction errors.<ref name=":0" />Haslbeck, J.In other words, &the vanPE approach tests the ability of a factor model with ''k'' factors to predict scores on ''p'' items in held-out Borkrespondents, R.using the model-implied covariance structure to derive item-level regressions (2022)e.g., Estimatingpredicting theitem number''i'' as a linear combination of factorsall inother exploratoryitems, factorwith coefficients given by the inverse covariance matrix), selecting the value of ''k'' that analysisbest viapredicts out-of-sample predictionitem errorsscores. Psychological Methods.</ref>

===Optimal Coordinate and Acceleration Factor===