Editing Factor analysis (section)

====Modern criteria====
[[Horn's parallel analysis]] (PA):<ref name="Horn1965">{{cite journal |last1=Horn |first1=John L. |title=A rationale and test for the number of factors in factor analysis |journal=Psychometrika |date=June 1965 |volume=30 |issue=2 |pages=179–185 |doi=10.1007/BF02289447|pmid=14306381 |s2cid=19663974 }}</ref> A Monte-Carlo based simulation method that compares the observed eigenvalues with those obtained from uncorrelated normal variables. A factor or component is retained if the associated eigenvalue is bigger than the 95th percentile of the distribution of eigenvalues derived from the random data. PA is among the more commonly recommended rules for determining the number of components to retain,<ref name="Zwick1986" /><ref>{{Cite arXiv|last=Dobriban|first=Edgar|date=2017-10-02|title=Permutation methods for factor analysis and PCA|class=math.ST|language=en|eprint=1710.00479v2}}</ref> but many programs fail to include this option (a notable exception being [[R (programming language)|R]]).<ref>* {{cite journal | last1 = Ledesma | first1 = R.D. | last2 = Valero-Mora | first2 = P. | year = 2007 | title = Determining the Number of Factors to Retain in EFA: An easy-to-use computer program for carrying out Parallel Analysis | url = http://pareonline.net/getvn.asp?v=12&n=2 | journal = Practical Assessment Research & Evaluation | volume = 12 | issue = 2| pages = 1–11 }}</ref> However, [[Anton Formann|Formann]] provided both theoretical and empirical evidence that its application might not be appropriate in many cases since its performance is considerably influenced by [[sample size]], [[Item response theory#The item response function|item discrimination]], and type of [[correlation coefficient]].<ref>Tran, U. S., & Formann, A. K. (2009). Performance of parallel analysis in retrieving unidimensionality in the presence of binary data. ''Educational and Psychological Measurement, 69,'' 50-61.</ref>

Velicer's (1976) MAP test<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> as described by 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:
http://pareonline.net/getvn.asp?v=18&n=8 {{Webarchive|url=https://web.archive.org/web/20150317145450/http://pareonline.net/getvn.asp?v=18&n=8 |date=2015-03-17 }}</ref> “involves a complete principal components analysis followed by the examination of a series of matrices of partial correlations” (p.&nbsp;397 (though this quote does not occur in Velicer (1976) and the cited page number is outside the pages of the citation). The squared correlation for Step “0” (see Figure 4) is the average squared off-diagonal correlation for the unpartialed correlation matrix. On Step 1, the first principal component and its associated items are partialed out. Thereafter, the average squared off-diagonal correlation for the subsequent correlation matrix is then computed for Step 1. On Step 2, the first two principal components are partialed out and the resultant average squared off-diagonal correlation is again computed. The computations are carried out for k minus one step (k representing the total number of variables in the matrix). Thereafter, all of the average squared correlations for each step are lined up and the step number in the analyses that resulted in the lowest average squared partial correlation determines the number of components or factors to retain.<ref name=Velicer/> By this method, components are maintained as long as the variance in the correlation matrix represents systematic variance, as opposed to residual or error variance. Although methodologically akin to principal components analysis, the MAP technique has been shown to perform quite well in determining the number of factors to retain in multiple simulation studies.<ref name="Zwick1986" /><ref name="Warne, R. T. 2014"/><ref name =Ruscio>{{cite journal|last=Ruscio|first=John|author2=Roche, B.|title=Determining the number of factors to retain in an exploratory factor analysis using comparison data of known factorial structure|journal=Psychological Assessment|year=2012|volume=24|issue=2|pages=282–292|doi=10.1037/a0025697|pmid=21966933}}</ref><ref name=Garrido>Garrido, L. E., & Abad, F. J., & Ponsoda, V. (2012). A new look at Horn's parallel analysis with ordinal variables. Psychological Methods. Advance online publication. {{doi|10.1037/a0030005}}</ref> This procedure is made available through SPSS's user interface,<ref name="pareonline.net"/> as well as the ''psych'' package for the [[R (programming language)|R programming language]].<ref>{{cite journal |last1=Revelle |first1=William |title=Determining the number of factors: the example of the NEO-PI-R |date=2007 |url=http://www.personality-project.org/r/book/numberoffactors.pdf}}</ref><ref>{{cite web |last1=Revelle |first1=William |title=psych: Procedures for Psychological, Psychometric, and PersonalityResearch |url=https://cran.r-project.org/web/packages/psych/ |date=8 January 2020}}</ref>