Analysis of two-state multivariate phenotypic change in ecological studies

Analyses of two-state phenotypic change are common in ecological research. Some examples include phenotypic changes due to phenotypic plasticity between two environments, changes due to predator/non-predator character shifts, character displacement via competitive interactions, and patterns of sexual dimorphism. However, methods for analyzing phenotypic change for multivariate data have not been rigorously developed. Here we outline a method for testing vectors of phenotypic change in terms of two important attributes: the magnitude of change (vector length) and the direction of change described by trait covariation (angular difference between vectors). We describe a permutation procedure for testing these attributes, which allows non-targeted sources of variation to be held constant. We provide examples that illustrate the importance of considering vector attributes of phenotypic change in biological studies, and we demonstrate how greater inference can be made than by evaluating variance components with MANOVA alone. Finally, we consider how our method may be extended to more complex data.