Parallel analysis#Implementation
{{Short description|Statistical method}}
Parallel analysis, also known as Horn's parallel analysis, is a statistical method used to determine the number of components to keep in a principal component analysis or factors to keep in an exploratory factor analysis. It is named after psychologist John L. Horn, who created the method, publishing it in the journal Psychometrika in 1965.{{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 }} The method compares the eigenvalues generated from the data matrix to the eigenvalues generated from a Monte-Carlo simulated matrix created from random data of the same size.{{cite book|author=Mike Allen|title=The SAGE Encyclopedia of Communication Research Methods|url=https://books.google.com/books?id=4GFCDgAAQBAJ&pg=PA518|date=11 April 2017|publisher=SAGE Publications|isbn=978-1-4833-8142-8|pages=518}}
Evaluation and comparison with alternatives
Parallel analysis is regarded as one of the more accurate methods for determining the number of factors or components to retain. In particular, unlike early approaches to dimensionality estimation (such as examining scree plots), parallel analysis has the virtue of an objective decision criterion.{{cite journal |last1=Zwick |first1=William R. |last2=Velicer |first2=Wayne F. |title=Comparison of five rules for determining the number of components to retain. |journal=Psychological Bulletin |date=1986 |volume=99 |issue=3 |pages=432–442 |doi=10.1037/0033-2909.99.3.432}} Since its original publication, multiple variations of parallel analysis have been proposed.{{cite journal |last1=Glorfeld |first1=Louis W. |title=An Improvement on Horn's Parallel Analysis Methodology for Selecting the Correct Number of Factors to Retain |journal=Educational and Psychological Measurement |date=2 July 2016 |volume=55 |issue=3 |pages=377–393 |doi=10.1177/0013164495055003002|s2cid=123508406 }}{{cite journal |last1=Crawford |first1=Aaron V. |last2=Green |first2=Samuel B. |last3=Levy |first3=Roy |last4=Lo |first4=Wen-Juo |last5=Scott |first5=Lietta |last6=Svetina |first6=Dubravka |last7=Thompson |first7=Marilyn S. |title=Evaluation of Parallel Analysis Methods for Determining the Number of Factors |journal=Educational and Psychological Measurement |date=September 2010 |volume=70 |issue=6 |pages=885–901 |doi=10.1177/0013164410379332|s2cid=63269411 }} Other methods of determining the number of factors or components to retain in an analysis include the scree plot, Kaiser rule, or Velicer's MAP test.{{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}}
Anton Formann provided both theoretical and empirical evidence that parallel analysis's application might not be appropriate in many cases since its performance is influenced by sample size, item discrimination, and type of correlation coefficient.{{cite journal | last1 = Tran | first1 = U. S. | last2 = Formann | first2 = A. K. | year = 2009 | title = Performance of parallel analysis in retrieving unidimensionality in the presence of binary data | journal = Educational and Psychological Measurement | volume = 69 | pages = 50–61 | doi = 10.1177/0013164408318761 | s2cid = 143051337 }}
An extensive 2022 simulation study by Haslbeck and van Bork{{Cite journal |last=Haslbeck |first=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 |language=en |volume=29 |issue=1 |pages=48–64 |doi=10.1037/met0000528 |issn=1939-1463|url-access=subscription }} found that parallel analysis was among the best-performing existing methods, but was slightly outperformed by their proposed prediction error-based approach.
Implementation
Parallel analysis has been implemented in JASP, SPSS, SAS, STATA, and MATLAB{{cite journal |last1=Hayton |first1=James C. |last2=Allen |first2=David G. |last3=Scarpello |first3=Vida |title=Factor Retention Decisions in Exploratory Factor Analysis: a Tutorial on Parallel Analysis |journal=Organizational Research Methods |date=29 June 2016 |volume=7 |issue=2 |pages=191–205 |doi=10.1177/1094428104263675|s2cid=61286653 }}{{cite web |last1=O'Connor |first1=Brian |title=Programs for Number of Components and Factors |url=https://people.ok.ubc.ca/brioconn/nfactors/nfactors.html |website=people.ok.ubc.ca}}{{cite journal |last1=O’connor |first1=Brian P. |title=SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test |journal=Behavior Research Methods, Instruments, & Computers |date=September 2000 |volume=32 |issue=3 |pages=396–402 |doi=10.3758/BF03200807|pmid=11029811 |doi-access=free }} and in multiple packages for the R programming language, including the psych{{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}}{{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}} multicon,{{cite web |last1=Sherman |first1=Ryne A. |title=multicon: Multivariate Constructs |url=https://cran.r-project.org/web/packages/multicon/index.html |date=2 February 2015}} hornpa,{{cite web |last1=Huang |first1=Francis |title=hornpa: Horn's (1965) Test to Determine the Number of Components/Factors |url=https://cran.r-project.org/web/packages/hornpa/index.html |date=3 March 2015}} and paran packages.{{cite journal |last1=Dinno |first1=Alexis |title=Gently Clarifying the Application of Horn's Parallel Analysis to Principal Component Analysis Versus Factor Analysis |url=https://alexisdinno.com/Software/files/PA_for_PCA_vs_FA.pdf}}{{cite journal |last1=Dinno |first1=Alexis |title=paran: Horn's Test of Principal Components/Factors |date=14 October 2018 |url=https://cran.r-project.org/web/packages/paran/}} Parallel analysis can also be conducted in Mplus version 8.0 and forward.https://www.statmodel.com/HTML_UG/chapter16V8.htm