Resolving the relationship between biodiversity and ecosystem functioning has been one of the central goals of modern ecology. Early debates about the relationship were finally resolved with the advent of a statistical partitioning scheme that decomposed the biodiversity effect into a "selection" effect and a "complementarity" effect. We prove that both the biodiversity effect and its statistical decomposition into selection and complementarity are fundamentally flawed because these methods use a na\"ive null expectation based on neutrality, likely leading to an overestimate of the net biodiversity effect, and they fail to account for the nonlinear abundance-ecosystem functioning relationships observed in nature. Furthermore, under such nonlinearity no statistical scheme can be devised to partition the biodiversity effects. We also present an alternative metric providing a more reasonable estimate of biodiversity effect. Our results suggest that all studies conducted since the early 1990s likely overestimated the positive effects of biodiversity on ecosystem functioning.