PCAtest: testing the statistical significance of Principal Component Analysis in R

Principal Component Analysis (PCA) is one of the most broadly used statistical methods for the ordination and dimensionality-reduction of multivariate datasets across many scientific disciplines. Trivial PCs can be estimated from data sets without any correlational structure among the original variables, and traditional criteria for selecting non-trivial PC axes are difficult to implement, partially subjective or based on ad hoc thresholds. PCAtest is an R package that implements permutation-based statistical tests to evaluate the overall significance of a PCA, the significance of each PC axis, and of contributions of each observed variable to the significant axes. Based on simulation and empirical results, I encourage R users to routinely apply PCAtest to test the significance of their PCA before proceeding with the direct interpretation of PC axes and/or the utilization of PC scores in subsequent evolutionary and ecological analyses.

Data reduction prior to inference: Are there consequences of comparing groups using a t-test based on principal component scores?

Researchers often use a two-step process to analyze multivariate data. First, dimensionality is reduced using a technique such as principal component analysis, followed by a group comparison using a t -test or analysis of variance. Although this practice is often discouraged, the statistical properties of this procedure are not well understood, starting with the hypothesis being tested. We suggest that this approach might be considering two distinct hypotheses, one of which is a global test of no differences in the mean vectors, and the other being a focused test of a specific linear combination where the coefficients have been estimated from the data. We study the asymptotic properties of the two-sample t -statistic for these two scenarios, assuming a nonsparse setting. We show that the size of the global test agrees with the presumed level but that the test has poor power. In contrast, the size of the focused test can be arbitrarily distorted with certain mean and covariance structures. A simple method is provided to correct the size of the focused test. Data analyses and simulations are used to illustrate the results. Recommendations on the use of this two-step method and the related use of principal components for prediction are provided.

Fast, Exact Bootstrap Principal Component Analysis for p > 1 million

Many have suggested a bootstrap procedure for estimating the sampling variability of principal component analysis (PCA) results. However, when the number of measurements per subject (p) is much larger than the number of subjects (n), calculating and storing the leading principal components from each bootstrap sample can be computationally infeasible. To address this, we outline methods for fast, exact calculation of bootstrap principal components, eigenvalues, and scores. Our methods leverage the fact that all bootstrap samples occupy the same n-dimensional subspace as the original sample. As a result, all bootstrap principal components are limited to the same n-dimensional subspace and can be efficiently represented by their low dimensional coordinates in that subspace. Several uncertainty metrics can be computed solely based on the bootstrap distribution of these low dimensional coordinates, without calculating or storing the p-dimensional bootstrap components. Fast bootstrap PCA is applied to a dataset of sleep electroencephalogram recordings (p = 900, n = 392), and to a dataset of brain magnetic resonance images (MRIs) (p ≈ 3 million, n = 352). For the MRI dataset, our method allows for standard errors for the first 3 principal components based on 1000 bootstrap samples to be calculated on a standard laptop in 47 minutes, as opposed to approximately 4 days with standard methods.

Early developmental plasticity and integrative responses in arctic charr (Salvelinus alpinus): effects of water velocity on body size and shape

Environmental conditions such as temperature and water velocity may induce changes among alternative developmental pathways, i.e. phenotypic responses, in vertebrates. However, the extent to which the environment induces developmental plasticity and integrated developmental responses during early ontogeny of fishes remains poorly documented. We analyzed the responses of newly hatched Arctic charr (Salvelinus alpinus) to four experimental water velocities during 100 days of development. To our knowledge, this work is the first to analyze developmental plasticity responses of body morphology to an experimental gradient of water velocities during early ontogeny of fish. Arctic charr body size and shape responses show first, that morphometric traits display significant differences between low and high water velocities, thus revealing directional changes in body traits. Secondly, trait variation allows the recognition of critical ontogenetic periods that are most responsive to environmental constraints (40-70 and 80-90 days) and exhibit different levels of developmental plasticity. This is supported by the observation of asynchronous timing of variation peaks among treatments. Third, morphological interaction of traits is developmentally plastic and time-dependent. We suggest that developmental responses of traits plasticity and interaction at critical ontogenetic periods are congruent with specific environmental conditions to maintain the functional integrity of the organism.

Sexually antagonistic coevolution in a mating system: combining experimental and comparative approaches to address evolutionary processes

We combined experimental and comparative techniques to study the evolution of mating behaviors within in a clade of 15 water striders (Gerris spp.). Superfluous multiple mating is costly to females in this group, and consequently there is overt conflict between the sexes over mating. Two alternative hypotheses that could generate interspecific variation in mating behaviors are tested: interspecific variation in optimal female mating rate versus sexually antagonistic coevolution of persistence and resistance traits. These potentially coevolving traits include male grasping and female antigrasping structures that further the interests of one sex over the other during premating struggles. Both processes are known to play a role in observed behavioral variation within species. We used two large sets of experiments to quantify behavioral differences among species, as well as their response to an environmentally (sex-ratio) induced change in optimal female mating rate. Our analysis revealed a large degree of continuous interspecific variation in all 20 quantified behavioral variables. Nevertheless, species shared the same set of behaviors, and each responded in a qualitatively similar fashion to sex-ratio alterations. A remarkably large proportion (> 50%) of all interspecific variation in the magnitude of behaviors, including their response to sex ratio, could be captured by a single multivariate axis. These data suggest tight coevolution of behaviors within a shared mating system. The pattern of correlated evolution was best accounted for by antagonistic coevolution in the relative abilities of each sex to control the outcome of premating struggles. In species where males have a relative advantage, mating activity is high, and the opposite is found in species where females have gained a relative advantage. Our analyses also suggested that evolution has been unconstrained by history, with no consistent evolutionary tendency toward or away from male or female relative advantage.