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In [[multivariate statistics]], '''exploratory factor analysis''' ('''EFA''') is a statistical method used to uncover the underlying structure of a relatively large set of [[Variable (research)|variables]]. EFA is a technique within [[factor analysis]] whose overarching goal is to identify the underlying relationships between measured variables.<ref name=Norris>{{cite journal|last=Norris|first=Megan|author2=Lecavalier, Luc|title=Evaluating the Use of Exploratory Factor Analysis in Developmental Disability Psychological Research|journal=Journal of Autism and Developmental Disorders|date=17 July 2009|volume=40|issue=1|pages=8–20|doi=10.1007/s10803-009-0816-2|pmid=19609833|s2cid=45751299 }}</ref> It is commonly used by researchers when developing a scale (a ''scale'' is a collection of questions used to measure a particular research topic) and serves to identify a set of [[Latent variable|latent constructs]] underlying a battery of measured variables.<ref name=Fabrigar>{{cite journal|last=Fabrigar|first=Leandre R.|author2=Wegener, Duane T. |author3=MacCallum, Robert C. |author4=Strahan, Erin J. |title=Evaluating the use of exploratory factor analysis in psychological research.|journal=Psychological Methods|date=1 January 1999|volume=4|issue=3|pages=272–299|doi=10.1037/1082-989X.4.3.272|url=http://www.statpower.net/Content/312/Handout/Fabrigar1999.pdf}}</ref> It should be used when the researcher has no ''a priori'' hypothesis about factors or patterns of measured variables.<ref name=Finch>{{cite journal | last1 = Finch | first1 = J. F. | last2 = West | first2 = S. G. | year = 1997 | title = The investigation of personality structure: Statistical models | journal = Journal of Research in Personality | volume = 31 | issue = 4| pages = 439–485 | doi=10.1006/jrpe.1997.2194}}</ref> ''Measured variables'' are any one of several attributes of people that may be observed and measured. Examples of measured variables could be the physical height, weight, and pulse rate of a human being. Usually, researchers would have a large number of measured variables, which are assumed to be related to a smaller number of "unobserved" factors. Researchers must carefully consider the number of measured variables to include in the analysis.<ref name =Fabrigar/> EFA procedures are more accurate when each factor is represented by multiple measured variables in the analysis.
EFA is based on the common factor model.<ref name =Norris/> In this model, manifest variables are expressed as a function of common factors, unique factors, and errors of measurement. Each unique factor influences only one manifest variable, and does not explain correlations between manifest variables. Common factors influence more than one manifest variable and "factor loadings" are measures of the influence of a common factor on a manifest variable.<ref name =Norris/> For the EFA procedure, we are more interested in identifying the common factors and the related manifest variables.
EFA assumes that any indicator/measured variable may be associated with any factor. When developing a scale, researchers should use EFA first before moving on to [[confirmatory factor analysis]].<ref name=worthington>{{cite journal|last=Worthington|first=Roger L.|author2= Whittaker, Tiffany A J. |title=Scale development research: A content analysis and recommendations for best practices.|journal=The Counseling Psychologist|date=1 January 2006|volume=34|issue=6|pages=806–838|doi=10.1177/0011000006288127|s2cid=146284440 }}</ref> EFA is essential to determine underlying factors/constructs for a set of measured variables; while confirmatory factor analysis allows the researcher to test the hypothesis that a relationship between the observed variables and their underlying latent {{Not a typo|factor(s)/construct(s)}} exists.<ref>Suhr, D. D. (2006). Exploratory or confirmatory factor analysis? (pp. 1-17). Cary: SAS Institute.</ref>
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==Selecting the appropriate number of factors==
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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
{{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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{{anchor|Kaiser criterion}}
===Kaiser's (1960) eigenvalue-greater-than-one rule (K1 or Kaiser criterion)===
Compute the eigenvalues for the correlation matrix and determine how many of these eigenvalues are greater than 1. This number is the number of factors to include in the model. A disadvantage of this procedure is that it is quite arbitrary (e.g., an eigenvalue of 1.01 is included whereas an eigenvalue of .99 is not). This procedure often leads to overfactoring and sometimes underfactoring. Therefore, this procedure should not be used.<ref name =Fabrigar /> A variation of the K1 criterion has been created to lessen the severity of the criterion's problems where a researcher calculates [[confidence interval]]s for each eigenvalue and retains only factors which have the entire confidence interval greater than 1.0.<ref>{{cite journal | last1 = Larsen | first1 = R. | last2 = Warne | first2 = R. T. | year = 2010 | title = Estimating confidence intervals for eigenvalues in exploratory factor analysis | journal = Behavior Research Methods | volume = 42 | issue = 3| pages = 871–876 | doi = 10.3758/BRM.42.3.871 | pmid = 20805609 | doi-access = free }}</ref><ref>{{cite journal | last1 = Warne | first1 = R. T. | last2 = Larsen | first2 = R. | year = 2014 | title = Evaluating a proposed modification of the Guttman rule for determining the number of factors in an exploratory factor analysis | journal = Psychological Test and Assessment Modeling | volume = 56 | pages = 104–123 }}</ref>
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===Parallel analysis===
{{Main|Parallel analysis}}
To carry out the PA test, users compute the eigenvalues for the correlation matrix and plot the values from largest to smallest and then plot a set of random eigenvalues. The number of eigenvalues before the intersection points indicates how many factors to include in your model.<ref name=Humphreys>{{cite journal | last1 = Humphreys | first1 = L. G. | last2 = Montanelli | first2 = R. G. Jr | year = 1975 | title = An investigation of the parallel analysis criterion for determining the number of common factors | journal = Multivariate Behavioral Research | volume = 10 | issue = 2| pages = 193–205 | doi = 10.1207/s15327906mbr1002_5 }}</ref><ref>{{cite journal|last=Horn|first=John L.|title=A rationale and test for the number of factors in factor analysis|journal=Psychometrika|date=1 June 1965|volume=30|issue=2|pages=179–185|doi=10.1007/BF02289447|pmid=14306381|s2cid=19663974 }}</ref><ref>{{cite journal|last=Humphreys|first=L. G.|author2=Ilgen, D. R.|title=Note On a Criterion for the Number of Common Factors|journal=Educational and Psychological Measurement|date=1 October 1969|volume=29|issue=3|pages=571–578|doi=10.1177/001316446902900303|s2cid=145258601 }}</ref> This procedure can be somewhat arbitrary (i.e. a factor just meeting the cutoff will be included and one just below will not).<ref name =Fabrigar/> Moreover, the method is very sensitive to sample size, with PA suggesting more factors in datasets with larger sample sizes.<ref>{{cite journal | last1 = Warne | first1 = R. G. | last2 = Larsen | first2 = R. | year = 2014 | title = Evaluating a proposed modification of the Guttman rule for determining the number of factors in an exploratory factor analysis | journal = Psychological Test and Assessment Modeling | volume = 56 | pages = 104–123 }}</ref> Despite its shortcomings, this procedure performs very well in simulation studies and is one of Courtney's recommended procedures.<ref name="pareonline.net"/> PA has been [[
===Ruscio and Roche's comparison data===
In 2012 Ruscio and Roche<ref name =Ruscio/> introduced the comparative data (CD) procedure in an attempt improve to upon the PA method. The authors state that "rather than generating random datasets, which only take into account sampling error, multiple datasets with known factorial structures are analyzed to determine which best reproduces the profile of eigenvalues for the actual data" (p. 258). The strength of the procedure is its ability to not only incorporate sampling error, but also the factorial structure and multivariate distribution of the items. Ruscio and Roche's (2012) simulation study<ref name =Ruscio/> determined that the CD procedure outperformed many other methods aimed at determining the correct number of factors to retain. In that study, the CD technique, making use of Pearson correlations accurately predicted the correct number of factors 87.14% of the time. However, the simulated study never involved more than five factors. Therefore, the applicability of the CD procedure to estimate factorial structures beyond five factors is yet to be tested. Courtney includes this procedure in his recommended list and gives guidelines showing how it can be easily carried out from within SPSS's user interface.<ref name="pareonline.net"/>
In 2023, Goretzko and Ruscio proposed the Comparison Data Forest as an extension of the CD approach.<ref>{{Cite journal |last1=Goretzko |first1=David |last2=Ruscio |first2=John |date=2023-06-15 |title=The comparison data forest: A new comparison data approach to determine the number of factors in exploratory factor analysis |journal=Behavior Research Methods
===Convergence of multiple tests===
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===Orthogonal rotation===
Orthogonal rotations constrain factors to be [[perpendicular]] to each other and hence [[
[[Varimax rotation]] is an orthogonal rotation of the factor axes to maximize the variance of the squared loadings of a factor (column) on all the variables (rows) in a factor matrix, which has the effect of differentiating the original variables by extracted factor. Each factor will tend to have either large or small loadings of any particular variable. A varimax solution yields results which make it as easy as possible to identify each variable with a single factor. This is the most common orthogonal rotation option.<ref name =Fabrigar/>
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===Oblique rotation===
Oblique rotations permit correlations among factors. An advantage of oblique rotation is that it produces solutions with better simple structure when factors are expected to correlate, and it produces estimates of correlations among factors.<ref name=Fabrigar/> These rotations may produce solutions similar to orthogonal rotation if the factors do not correlate with each other.
Several oblique rotation procedures are commonly used.
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===Unrotated solution===
Common factor analysis software is capable of producing an unrotated solution. This refers to the result of a [[#Principal_axis_factoring_(PAF)|principal axis factoring]] with no further rotation. The so-called unrotated solution is in fact an orthogonal rotation that maximizes the variance of the first factors. The unrotated solution tends to give a general factor with loadings for most of the variables. This may be useful if many variables are correlated with each other, as revealed by one or a few dominating [[eigenvalue
The usefulness of an unrotated solution was emphasized by a [[meta analysis]] of studies of cultural differences. This revealed that many published studies of cultural differences have given similar factor analysis results, but rotated differently. Factor rotation has obscured the similarity between the results of different studies and the existence of a strong general factor, while the unrotated solutions were much more similar.<ref name="Fog2020">{{cite journal|last=Fog|first=A. |title=A Test of the Reproducibility of the Clustering of Cultural Variables |journal=Cross-Cultural Research |year=2020 |volume=55 |pages=29–57 |doi=10.1177/1069397120956948|s2cid=224909443 }}</ref><ref>{{Cite journal|title=Examining Factors in 2015 TIMSS Australian Grade 4 Student Questionnaire Regarding Attitudes Towards Science Using Exploratory Factor Analysis (EFA)|url=https://twasp.info/journal/gi93583P/examining-factors-in-2015-timss-australian-grade-4-student-questionnaire-regarding-attitudes-towards-science-using-exploratory-factor-analysis-efa|journal=North American Academic Research|volume=3}}</ref>
==Factor interpretation==
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
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}}</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>
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==See also==
*[[Confirmatory factor analysis]]
*[[
*[[v:Exploratory factor analysis|Exploratory factor analysis]] (Wikiversity)
*[[Factor analysis]]
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