On The Correct Mathematical Derivation And Ecological Application Of Unbiased Estimators In Biodiversity Research

Clark et al. (2019) sought to extend the Loreau-Hector partitioning scheme by showing how to estimate selection and complementarity effects from an incomplete sample of species. We demonstrate that their approach suffers from serious conceptual and mathematical errors. Instead of finding unbiased estimators for a finite population, they inserted ad hoc correction factors into unbiased parameter estimators for an infinite population without any mathematical justification in order to force the sample estimators of an infinite population to converge to the true finite population parameter values as sample sizenapproached population sizeN. In doing so, they confused the unbiasedness of a sample estimator with its equivalence to the true population parameter value whenn=N. Additionally, we show that their estimators of complementarity, selection and the net biodiversity effect are incorrect. We then derive the correct unbiased estimators but caution that, contrary to what Clark et al. claim, these quantities will not approximate the corresponding population parameters without significant repeated random sampling, something that would likely be unfeasible in most if not all biodiversity experiments. Clark et al. also state that their method can be used to compare distinct experiments characterized by different species and diversity levels, or extrapolate from biodiversity experiments to natural systems. This is incorrect because relative yields are not a property of individual species like monoculture yields but an emergent and specific feature of an experimental community. As such, two experimental communities, even when overlapping significantly in species, are incommensurable for the purpose of predicting relative yields. In other words, different experimental communities are not equivalent to different samples taken from the same statistical population. Finally, Clark et al. incorrectly claim that both the original Loreau-Hector partitioning scheme and their extension work for any baseline despite the fact that recent research has shown that a nonlinear relationship between monoculture density and ecosystem functioning will likely inflate the net biodiversity effect in plant systems, and will always lead to spurious measurements of complementarity and selection.