Parallel analysis

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.^[1] 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.^[2]

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.^[3] Since its original publication, multiple variations of parallel analysis have been proposed.^[4]^[5] 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.^[6]

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.^[7]

An extensive 2022 simulation study by Haslbeck and van Bork^[8] 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^[9]^[10]^[11] and in multiple packages for the R programming language, including the psych^[12]^[13] multicon,^[14] hornpa,^[15] and paran packages.^[16]^[17] Parallel analysis can also be conducted in Mplus version 8.0 and forward.^[18]

References

^ Horn, John L. (June 1965). "A rationale and test for the number of factors in factor analysis". Psychometrika. 30 (2): 179–185. doi:10.1007/bf02289447. PMID 14306381. S2CID 19663974.
^ Mike Allen (11 April 2017). The SAGE Encyclopedia of Communication Research Methods. SAGE Publications. p. 518. ISBN 978-1-4833-8142-8.
^ Zwick, William R.; Velicer, Wayne F. (1986). "Comparison of five rules for determining the number of components to retain". Psychological Bulletin. 99 (3): 432–442. doi:10.1037/0033-2909.99.3.432.
^ Glorfeld, Louis W. (2 July 2016). "An Improvement on Horn's Parallel Analysis Methodology for Selecting the Correct Number of Factors to Retain". Educational and Psychological Measurement. 55 (3): 377–393. doi:10.1177/0013164495055003002. S2CID 123508406.
^ Crawford, Aaron V.; Green, Samuel B.; Levy, Roy; Lo, Wen-Juo; Scott, Lietta; Svetina, Dubravka; Thompson, Marilyn S. (September 2010). "Evaluation of Parallel Analysis Methods for Determining the Number of Factors". Educational and Psychological Measurement. 70 (6): 885–901. doi:10.1177/0013164410379332. S2CID 63269411.
^ Velicer, W.F. (1976). "Determining the number of components from the matrix of partial correlations". Psychometrika. 41 (3): 321–327. doi:10.1007/bf02293557. S2CID 122907389.
^ 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. doi:10.1177/0013164408318761. S2CID 143051337.
^ Haslbeck, Jonas M. B.; van Bork, Riet (February 2024). "Estimating the number of factors in exploratory factor analysis via out-of-sample prediction errors". Psychological Methods. 29 (1): 48–64. doi:10.1037/met0000528. ISSN 1939-1463.
^ Hayton, James C.; Allen, David G.; Scarpello, Vida (29 June 2016). "Factor Retention Decisions in Exploratory Factor Analysis: a Tutorial on Parallel Analysis". Organizational Research Methods. 7 (2): 191–205. doi:10.1177/1094428104263675. S2CID 61286653.
^ O'Connor, Brian. "Programs for Number of Components and Factors". people.ok.ubc.ca. Archived from the original on 2021-05-23. Retrieved 2020-04-10.
^ O’connor, Brian P. (September 2000). "SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test". Behavior Research Methods, Instruments, & Computers. 32 (3): 396–402. doi:10.3758/BF03200807. PMID 11029811.
^ Revelle, William (2007). "Determining the number of factors: the example of the NEO-PI-R" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
^ Revelle, William (8 January 2020). "psych: Procedures for Psychological, Psychometric, and PersonalityResearch".
^ Sherman, Ryne A. (2 February 2015). "multicon: Multivariate Constructs".
^ Huang, Francis (3 March 2015). "hornpa: Horn's (1965) Test to Determine the Number of Components/Factors".
^ Dinno, Alexis. "Gently Clarifying the Application of Horn's Parallel Analysis to Principal Component Analysis Versus Factor Analysis" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
^ Dinno, Alexis (14 October 2018). "paran: Horn's Test of Principal Components/Factors". {{cite journal}}: Cite journal requires |journal= (help)
^ https://www.statmodel.com/HTML_UG/chapter16V8.htm

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[1] Horn, John L. (June 1965). "A rationale and test for the number of factors in factor analysis". Psychometrika. 30 (2): 179–185. doi:10.1007/bf02289447. PMID 14306381. S2CID 19663974.

[Allen2017-2] Mike Allen (11 April 2017). The SAGE Encyclopedia of Communication Research Methods. SAGE Publications. p. 518. ISBN 978-1-4833-8142-8.

[Zwick1986-3] Zwick, William R.; Velicer, Wayne F. (1986). "Comparison of five rules for determining the number of components to retain". Psychological Bulletin. 99 (3): 432–442. doi:10.1037/0033-2909.99.3.432.

[Glorfeld2016-4] Glorfeld, Louis W. (2 July 2016). "An Improvement on Horn's Parallel Analysis Methodology for Selecting the Correct Number of Factors to Retain". Educational and Psychological Measurement. 55 (3): 377–393. doi:10.1177/0013164495055003002. S2CID 123508406.

[Crawford2010-5] Crawford, Aaron V.; Green, Samuel B.; Levy, Roy; Lo, Wen-Juo; Scott, Lietta; Svetina, Dubravka; Thompson, Marilyn S. (September 2010). "Evaluation of Parallel Analysis Methods for Determining the Number of Factors". Educational and Psychological Measurement. 70 (6): 885–901. doi:10.1177/0013164410379332. S2CID 63269411.

[Velicer-6] Velicer, W.F. (1976). "Determining the number of components from the matrix of partial correlations". Psychometrika. 41 (3): 321–327. doi:10.1007/bf02293557. S2CID 122907389.

[7] 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. doi:10.1177/0013164408318761. S2CID 143051337.

[8] Haslbeck, Jonas M. B.; van Bork, Riet (February 2024). "Estimating the number of factors in exploratory factor analysis via out-of-sample prediction errors". Psychological Methods. 29 (1): 48–64. doi:10.1037/met0000528. ISSN 1939-1463.

[9] Hayton, James C.; Allen, David G.; Scarpello, Vida (29 June 2016). "Factor Retention Decisions in Exploratory Factor Analysis: a Tutorial on Parallel Analysis". Organizational Research Methods. 7 (2): 191–205. doi:10.1177/1094428104263675. S2CID 61286653.

[10] O'Connor, Brian. "Programs for Number of Components and Factors". people.ok.ubc.ca. Archived from the original on 2021-05-23. Retrieved 2020-04-10.

[11] O’connor, Brian P. (September 2000). "SPSS and SAS programs for determining the number of components using parallel analysis and Velicer's MAP test". Behavior Research Methods, Instruments, & Computers. 32 (3): 396–402. doi:10.3758/BF03200807. PMID 11029811.

[12] Revelle, William (2007). "Determining the number of factors: the example of the NEO-PI-R" (PDF). {{cite journal}}: Cite journal requires |journal= (help)

[13] Revelle, William (8 January 2020). "psych: Procedures for Psychological, Psychometric, and PersonalityResearch".

[14] Sherman, Ryne A. (2 February 2015). "multicon: Multivariate Constructs".

[15] Huang, Francis (3 March 2015). "hornpa: Horn's (1965) Test to Determine the Number of Components/Factors".

[16] Dinno, Alexis. "Gently Clarifying the Application of Horn's Parallel Analysis to Principal Component Analysis Versus Factor Analysis" (PDF). {{cite journal}}: Cite journal requires |journal= (help)

[17] Dinno, Alexis (14 October 2018). "paran: Horn's Test of Principal Components/Factors". {{cite journal}}: Cite journal requires |journal= (help)

[18] ttps://www.statmodel.com/HTML_UG/chapter16V8.htm

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