The Effect of Sample Size on the Stability of Principal Components Analysis of Truss-Based Fish Morphometrics

Abstract

Multivariate analysis of fish morphometric truss elements for stock identification, description of new species, assessment of condition, and other applications is frequently conducted on data sets that have sample sizes smaller than those recommended in the literature. Minimum sample size recommendations are rarely accompanied by empirical support, and we know of no previous assessment of minimum sample sizes for multivariate analysis of fish truss elements. We examined the stability of outcomes of principal components analysis (PCA) of truss elements, a commonly applied method of morphometric analysis for fishes, by conducting PCA on 1,000 resamples for each of 24 different sample sizes (N; each sample drawn without replacement) from collections of yellow perch Perca flavescens (397 fish), white perch Morone americana (208 fish), and siscowet lake trout Salvelinus namaycush (560 fish). Eigenvalues were inflated and loadings on eigenvectors were highly unstable for the first three principal components (PCs) whenever N was smaller than the number of truss elements (P). Stability of eigenvalues and eigenvectors increased as the N:P ratio increased for all three species, but the N:P ratio at which stable results were achieved varied by species. Our results suggest that an N:P ratio of 3.5-8.0 was required for stability of PC2 and PC3, which is required for analysis of fish shape. Because some of our results varied among the species we examined, we recommend similar evaluations for other species. Results from past work that used PCA of truss elements and where N was less than P may require re-evaluation.