Copulas reveal complex and informative dependencies propagating throughout ecology

Abstract

All branches of ecology study relationships among and between environmental and biological variables. However, standard approaches to studying such relationships, based on correlation and regression, provide only a small slice of the complex information contained in the relationships. Other statistical approaches exist that provide a complete description of relationships between variables, based on the concept of the copula; they are applied in finance, neuroscience and other fields, but rarely in ecology. We here explore the concepts that underpin copulas and examine the potential for those concepts to improve our understanding of ecology. We find that informative copula structure in dependencies between variables is common across all the environmental, species-trait, phenological, population, community, and ecosystem functioning datasets we considered. Many datasets exhibited asymmetric tail associations, whereby two variables were more strongly related in their left compared to right tails, or vice versa. We describe mechanisms by which observed copula structure and asymmetric tail associations can arise in ecological data, including a Moran-like effect whereby dependence structures between environmental variables are inherited by ecological variables; and asymmetric or nonlinear influences of environments on ecological variables, such as under Liebigs law of the minimum. We also describe consequences of copula structure for ecological phenomena, including impacts on extinction risk, Taylors law, and the stability through time of ecosystem services. By documenting the importance of a complete description of dependence between variables, advancing conceptual frameworks, and demonstrating a powerful approach, we aim to encourage widespread use of copulas in ecology, which we believe can benefit the discipline.