Exploratory factor analysis: Difference between revisions

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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 |last1=Haslbeck |first1=Jonas M. B. |last2=van Bork |first2=Riet |date=February 2024 |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 |volume=29 |issue=1 |pages=48–64 |doi=10.1037/met0000528 |pmid=36326634 |issn=1939-1463|url-access=subscription |doi-access=free }}</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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Factor loadings are numerical values that indicate the strength and direction of a factor on a measured variable. Factor loadings indicate how strongly the factor influences the measured variable. In order to label the factors in the model, researchers should examine the factor pattern to see which items load highly on which factors and then determine what those items have in common.<ref name =Fabrigar/> Whatever the items have in common will indicate the meaning of the factor. Interpretation has long been noted as an important, but difficult, part of the analytic process.<ref>{{Cite journal |last=Copeland |first=Herman A. |date=March 1935 |title=A note on "The Vectors of Mind." |url=https://doi.apa.org/doi/10.1037/h0057026 |journal=Psychological Review |volume=42 |issue=2 |pages=216–218 |doi=10.1037/h0057026 |issn=1939-1471|url-access=subscription }}</ref>

However, while exploratory factor analysis is a powerful tool for uncovering underlying structures among variables, it is crucial to avoid reliance on it without adequate theorizing. Armstrong's<ref>{{cite journal |last1=Armstrong |first1=J. Scott |title=Derivation of Theory by Means of Factor Analysis or Tom Swift and His Electric Factor Analysis Machine |journal=The American Statistician |date=December 1967 |volume=21 |issue=5 |pages=17–21 |doi=10.1080/00031305.1967.10479849|hdl=1721.1/47256 |hdl-access=free }}</ref> critique highlights that EFA, when conducted without a theoretical framework, can lead to misleading interpretations. For instance, in a hypothetical case study involving the analysis of various physical properties of metals, the results of EFA failed to identify the true underlying factors, instead producing an "over-factored" model that obscured the simplicity of the relationships amongst the observed variables. Similarly, poorly designed survey items can lead to spurious factor structures.<ref>{{cite journal |last1=Maul |first1=Andrew |title=Rethinking Traditional Methods of Survey Validation |journal=Measurement: Interdisciplinary Research and Perspectives |date=3 April 2017 |volume=15 |issue=2 |pages=51–69 |doi=10.1080/15366367.2017.1348108}}</ref>

==See also==