Comparison of Correction Factors and Sample Size Required to Test the Equality of the Smallest Eigenvalues in Principal Component Analysis

Abstract

In the inferential process of Principal Component Analysis (PCA), one of the main challenges for researchers is establishing the correct number of components to represent the sample. For that purpose, heuristic and statistical strategies have been proposed. One statistical approach consists in testing the hypothesis of the equality of the smallest eigenvalues in the covariance or correlation matrix using a Likelihood-Ratio Test (LRT) that follows a χ2 limit distribution. Different correction factors have been proposed to improve the approximation of the sampling distribution of the statistic. We use simulation to study the significance level and power of the test under the use of these different factors and analyze the sample size required for an dequate approximation. The results indicate that for covariance matrix, the factor proposed by Bartlett offers the best balance between the objectives of low probability of Type I Error and high Power. If the correlation matrix is used, the factors W ∗ and cχ2 are the most recommended. Empirically, we can observe that most factors require sample sizes 10 or 20 times the number of variables if covariance or correlationmatrices, respectively, are implemented.