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

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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 matrix principal component analysis common factor analysis number of principal components number of common factors## List of Symbols

- \({\mathbf {X}}\)
Matrices (bold font, uppercase)

- \(x_{ij}\)
Element of \({\mathbf {X}}\) in the

*i*-th row,*j*-th column- \({\varvec{\Sigma }}\)
Population covariance matrix

- \({\mathbf {C}}\)
Sample covariance matrix

- \(\lambda _k\)
*k*th eigenvalue of the population covariance matrix- \(l_k\)
*k*th eigenvalue of the sample covariance matrix- \(L_k\)
Tracy–Widom statistic for the \(l_k\)

*s*Argument of the Tracy–Widom

*cdf*and*pdf*

## Notes

### Acknowledgments

The authors thank Dick Barelds, Marloes Koster, and Sip Jan Pijl for sharing their data. This work was partly supported by the European Commission-funded FP7 project INFECT (Contract No. 305340).

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