Be careful with your principal components

Abstract

Principal components analysis (PCA) is a common method to summarize a larger set of correlated variables into a smaller and more easily interpretable axes of variation. However, the different components need to be distinct from each other to be interpretable otherwise they only represent random directions. This is a fundamental assumption of PCA and, thus, needs to be tested every time. Sample correlation matrices will always result in a pattern of decreasing eigenvalues even if there is no structure. Tests are, therefore, needed to discern real patterns from illusionary ones. Furthermore, the loadings of the vectors need to be larger than expected by random data to be useful in the calculation of PC‐scores. PC‐scores calculated from nondistinct PC's have very large standard errors and cannot be used for biological interpretations. I give a number of examples to illustrate the potential problems with PCA. Robustness of the PC's increases with increasing sample size but not with the number of traits. I review a few simple test statistics appropriate for testing PC's and use a real‐world example to illustrate how this can be done using randomization tests. PCA can be very useful but great care is needed to avoid spurious results.

How general is cognitive ability in non-human animals? A meta-analytical and multi-level reanalysis approach

“…Our analysis suggests that the use and test of PCA in most studies on g in animals (1993-2019) is inadequate given current statistical standards for these tests. This conclusion concurs with recent statistical discussions in the fields of evolutionary biology and animal behaviour [56][57][58]. Guidelines for the use of dimensionality reduction methods can be found in the statistic literature and are outside the scope of this study (e.g.…”

“…A similar problem is likely to arise in other studies where loadings differ widely between tasks; it is thus crucial that authors report factor loadings, assess their statistical significance, and provide information on (co)variances of unreduced factors if they choose to compute g using dimension reduction techniques [53,56]. It is only appropriate to use dimension reduction methods when there is sufficient evidence to do so [56]. Multivariate analysis of performance across all tasks showed weak associations between the five cognitive traits ( figure 4).…”

Detecting European Aspen (Populus tremula L.) in Boreal Forests Using Airborne Hyperspectral and Airborne Laser Scanning Data

“…Although most of the spectral variation was concentrated to the first three principal components (91%), the best OA and Kappa values required 38 components when classification was performed using PCA features only. This emphasizes the importance of components containing a very small spectral variation (38 principal components had a spectral variance of 0.0000894%) in tree species classification with narrow band hyperspectral data and supports the suggestion made in Björklund [82] to avoid simple rules of thumb such as eigenvalues larger than 1.0 and loadings larger than 0.5 when selecting the optimal number of principal components for the model.…”

Recent introgression between Taiga Bean Goose and Tundra Bean Goose results in a largely homogeneous landscape of genetic differentiation

“…1 ), it does not automatically follow that the taxa are genetically distinct. Indeed, PCAs tend to overemphasize differences (Björklund 2019 ) and ADMIXTURE -analyses are sensitive to filtering criteria applied to the SNPs (Lawson et al 2018 ). These biases were also apparent in our analyses.…”