Comparing two classes of alpha diversities and their corresponding beta and (dis)similarity measures, with an application to the Formosan sika deer Cervus nippon taiouanus reintroduction programme

Diversity partitioning, which decomposes gamma diversity into alpha and beta components, is commonly used to obtain measures that quantify spatial/temporal diversity and compositional similarity or dissimilarity among assemblages. We focus on the decomposition of diversity as measured by Hill numbers (parameterized by a diversity order q >= 0). At least three diversity-partitioning schemes based on Hill numbers have been proposed. These schemes differ in the way they formulate alpha diversity. We focus on comparing two classes of alpha diversities, developed, respectively, by Routledge (1979) and Chiu et al. (2014). Both are defined for all diversity orders q >= 0. Because these two approaches to quantifying alpha have not been compared in the literature; it has been unclear how to choose a proper alpha formulation for practical applications. We review the two classes of alpha diversities and discuss the properties of their corresponding beta and (dis)similarity measures. Our research offers clear guidelines regarding the choice of an alpha formula: (a) If the goal is to assess compositional (dis)similarity among (unweighted) species relative abundance datasets, then the two alpha formulas are identical, leading to the same beta and (dis)similarity measures. (b) If the goal is to assess compositional (dis)similarity among (unweighted) species raw abundance datasets, then Chiu et al.'s approach should be used. Their beta can be monotonically transformed to various (dis)similarity measures in the range [0, 1]. (c) If each assemblage is weighted by its absolute, total abundance (i.e. assemblage size), but the goal is to assess compositional (dis)similarity among species relative abundance datasets, then Routledge's approach should be used. In this case, construction of legitimate (dis)similarity measures among species relative abundance datasets for unequal assemblage sizes/weights, for any order q >= 0, has not been addressed in the literature. Here we propose non-monotonic transformations of Routledge's beta to fill this gap. The extension of our analysis to phylogenetic diversity partitioning is generally parallel. We apply various species/taxonomic and phylogenetic dissimilarity measures to Taiwan's plant data; the results provide insights into the assessment of a reintroduction programme of Formosan sika deer into a forest area. Pertinent sampling and related issues are also discussed.