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Considering Horn’s Parallel Analysis from a Random Matrix Theory Point of View

Published online by Cambridge University Press:  01 January 2025

Edoardo Saccenti  and
Marieke E. Timmerman
Edoardo Saccenti*
Affiliation:
Wageningen University
Marieke E. Timmerman
Affiliation:
University of Groningen
*
Correspondence should be made to Edoardo Saccenti, Laboratory of Systems and Synthetic Biology, Wageningen University, Stippeneng 4, 6708 WE Wageningen, The Netherlands. Email: esaccenti@gmail.com
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Abstract

Horn’s parallel analysis is a widely used method for assessing the number of principal components and common factors. We discuss the theoretical foundations of parallel analysis for principal components based on a covariance matrix by making use of arguments from random matrix theory. In particular, we show that (i) for the first component, parallel analysis is an inferential method equivalent to the Tracy–Widom test, (ii) its use to test high-order eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higher-order components can be obtained based on a Tracy–Widom approximation. We illustrate the performance of the two testing procedures using simulated data generated under both a principal component model and a common factors model. For the principal component model, the Tracy–Widom test performs consistently in all conditions, while parallel analysis shows unpredictable behavior for higher-order components. For the common factor model, including major and minor factors, both procedures are heuristic approaches, with variable performance. We conclude that the Tracy–Widom procedure is preferred over parallel analysis for statistically testing the number of principal components based on a covariance matrix.

Keywords

covariance matrixprincipal component analysiscommon factor analysisnumber of principal componentsnumber of common factors
Type
Original Paper
Information
Psychometrika , Volume 82 , Issue 1 , March 2017 , pp. 186 - 209
Copyright
Copyright © 2016 The Psychometric Society

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