Selecting the number of factors in principal component analysis by permutation testingNumerical and practical aspects

Selecting the correct number of factors in principal component analysis (PCA) is a critical step to achieve a reasonable data modelling, where the optimal strategy strictly depends on the objective PCA is applied for. In the last decades, much work has been devoted to methods like Kaiser's eigenvalue greater than 1 rule, Velicer's minimum average partial rule, Cattell's scree test, Bartlett's chi-square test, Horn's parallel analysis, and cross-validation. However, limited attention has been paid to the possibility of assessing the significance of the calculated components via permutation testing. That may represent a feasible approach in case the focus of the study is discriminating relevant from nonsystematic sources of variation and/or the aforementioned methodologies cannot be resorted to (eg, when the analysed matrices do not fulfill specific properties or statistical assumptions). The main aim of this article is to provide practical insights for an improved understanding of permutation testing, highlighting its pros and cons, mathematically formalising the numerical procedure to be abided by when applying it for PCA factor selection by the description of a novel algorithm developed to this end, and proposing ad hoc solutions for optimising computational time and efficiency.