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 highorder eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higherorder 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 higherorder 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.
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Abbreviations
 \({\mathbf {X}}\) :

Matrices (bold font, uppercase)
 \(x_{ij}\) :

Element of \({\mathbf {X}}\) in the ith row, jth column
 \({\varvec{\Sigma }}\) :

Population covariance matrix
 \({\mathbf {C}}\) :

Sample covariance matrix
 \(\lambda _k\) :

kth eigenvalue of the population covariance matrix
 \(l_k\) :

kth eigenvalue of the sample covariance matrix
 \(L_k\) :

Tracy–Widom statistic for the \(l_k\)
 s :

Argument of the Tracy–Widom cdf and pdf
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Acknowledgments
The authors thank Dick Barelds, Marloes Koster, and Sip Jan Pijl for sharing their data. This work was partly supported by the European Commissionfunded FP7 project INFECT (Contract No. 305340).
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Appendices
Appendices
Appendix 1: The Tracy–Widom distribution
The Tracy–Widom distribution (Tracy & Widom, 1993, 1994, 1996) is defined as
The function q(x) is the unique Hastings–McLeod solution (Hastings & McLeod, 1980) of the nonlinear Painlevé differential equation
satisfying the boundary condition
where Ai(t) is the Airy function (Airy, 1838) and
This distribution was found to be the limiting law for the largest eigenvalue of Gaussian symmetric \(n \times n\) matrices (the socalled GOE, Gaussian Orthogonal Ensemble). Johnstone’s theorem showed that the same limiting distribution holds for the covariance matrices of rectangular data matrices \(n \times p\) when both n and p are large.
Appendix 2: The Baik–Ben ArousPéché phase transition
Baik et al. (2005) provided the asymptotics of the distribution of the largest eigenvalue(s) of a sample covariance matrix under the spike population model. These results were proved for complex data, but there is strong evidence that they hold true also for real data. For this, the results in Baik et al. (2005) can be stated in form of a conjecture, the socalled Baik–Ben ArousPéché (BBP conjecture).
Conjecture 1
Let \(\lambda _1\) be the leading eigenvalue of the population covariance matrix with \(\lambda _k = 1\) for \(2\le k \le p\) and let \(l_1\) be the largest sample eigenvalue. In the asymptotic regime \((n,p) \rightarrow \infty \) with finite limit ratio p / n, it holds that

(1)
If
$$\begin{aligned} \lambda _1 < 1 + \sqrt{\frac{p}{n}} \end{aligned}$$(17)then \(l_1\) when properly normalized to \(L_1\), will have the same distribution as when \(\lambda _1=1\), that is, it will be Tracy–Widom distributed.

(2)
If
$$\begin{aligned} \lambda _1 \ge 1 + \sqrt{\frac{p}{n}} \end{aligned}$$(18)then \(L_1\) will be almost surely unbounded.
Statement (2) was proved for real data in Baik and Silverstein (2006), and Paul (2007) showed that in the real case, \(L_1\) will exhibit Gaussian fluctuations.
The behavior of \(L_1\) will be different depending on the size of \(\lambda _1\), hence the phasetransition denomination. It is clear that as \((n,p) \rightarrow \infty \) the phase transition become arbitrary sharp. Stated otherwise, if \(\lambda _1\) is below the BBP limit there will be little chance to detect structure in the data, as the eigenvalues are distributed according the Tracy–Widom distribution, that is, as noise eigenvalues. On the contrary, if \(\lambda _1\) is above the BBP limit, detection of structure will be easier (Patterson et al., 2006). This phenomenon has been recently used to explain some problems arising in eigenanalysis when applied to population genetic studies (Patterson et al., 2006) and economics (Harding, 2008). Similar results hold for higherorder eigenvalues: it is enough to replace \(l_1\) and \(\lambda _1\) with \(l_k\) and \(\lambda _k\) in Equations (17) and (18). For more details, see Karoui (2007), Johnstone (2006), Paul (2007), Paul & Aue (2014), Tracy & Widom (2009).
An interesting aspect of the BBP phase transition is that it provides one of the few examples of power analysis in the multivariate setting (Saccenti & Timmerman, 2016). Given a problem and fixed the population size p, if \(\lambda _1\) would have been known, then Equation (18) would give a direct estimate of the sample size needed to be able to detect the presence of structure in the data. From Equation (18) also descends that increasing sample size, rather than variable number, is advantageous for detecting structure above the BBP threshold, but if \(\lambda _1\) is below the BBP threshold there is no gain in increasing the sample size (Patterson et al., 2006).
Appendix 3: A Note on Percentiles Estimation
Here, we give a brief look at the quality of the estimates of the percentiles of the null distribution, as obtained using parallel analysis. Consistently with the rest of the paper we consider the percentiles of the Tracy–Widom distribution as the target against which to compare the estimations obtained using parallel analysis. For the Tracy–Widom distribution, the percentiles can be calculated with almost arbitrary precision using computational toolboxes (see Bornemann, 2009, 2010). In contrast, obtaining reliable estimations of the percentiles of a population density function from an empirical distribution is a known problem even when the parameter of the population density (i.e., mean and variance) are known (Efron & Tibshirani, 1993; Efron, 1994; Rice & Church, 1996; DiCiccio & Efron, 1996). Using simulations we found that the error on the percentiles estimation decreases, as expected, with the number N of realizations used to build the empirical distribution. This is also indicated by the formula
where \(y_{100\alpha } = \mathrm{TW}^{1}(x_{100\alpha })\), \(\sigma _\mathrm{TW}\) is the standard deviation of \(\mathrm{TW}_1(s)\), and P is the proportion \(100\alpha \) (see e.g., Kendall & Yule, 1950; Deming, 1966; Rice & Church, 1996).
Figure 6 shows the results of a simulation, where the 95 and \(99\,\%\) percentiles of the Tracy–Widom distribution were estimated using different numbers of realizations N, using 10 replicates. As anticipated, the error decreases with N. Another aspect to be considered is the sampling variability: given a fixed N, different realizations may lead to (slightly) different estimations of the percentiles and this variability can be really large if the number of realizations is limited. This is clearly shown in Figure 6: estimations tend to be really unstable for small numbers of realizations (\(N<10^3\)) (panel a) and even for size of \(N = 10^5\), we can still observe instability of the estimations (panel d).
With \(N= 10^5\), the relative error is slightly below \(1\,\%\). In literature, some authors have recommended \(N= 300\) in (Ledesma & ValeroMora, 2007) or \(N = 2500\) (Buja & Eyübğolu, 1992) being the latter a number that can still lead to a relative error in the range of \(10\,\%\). Theoretical estimates of the error on the estimation can be obtained using the Yule–Kendall formula. This is outside the scope of this paper, but preliminary simulations seem to indicate that at least \(N>10^7\) are needed to obtain a relative precision lower then \(0.1\,\%\).
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Saccenti, E., Timmerman, M.E. Considering Horn’s Parallel Analysis from a Random Matrix Theory Point of View. Psychometrika 82, 186–209 (2017). https://doi.org/10.1007/s113360169515z
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Keywords
 covariance matrix
 principal component analysis
 common factor analysis
 number of principal components
 number of common factors