Proposing a resolution to debates on diversity partitioning.

There have been intense debates about the decomposition of regional diversity (gamma) into its within-community component (alpha) and between-community component (beta). Although a recent Ecology Forum achieved consensus in the use of "numbers equivalents" (Hill numbers) as the proper choice of diversity measure, three related major issues were still left unresolved. (1) What is the precise meaning of the "independence" or "statistical independence" of alpha diversity and beta diversity? (2) Which partitioning (additive vs. multiplicative) should be used for a given application? (3) What is the proper formula for alpha diversity, as there are two formulas in the literature? This paper proposes a possible resolution to each of these issues. For the first issue, we clarify the definitions of "independence" and "statistical independence" from two perspectives so that confusion about this issue can be cleared up. We also discuss the causes of dependence, so that the dependence relationship between any two diversity components in both partitioning schemes can be rigorously justified by theory and also intuitively understood by simulation. For the second issue, both multiplicative and additive beta diversities based on Hill numbers are useful measures and quantify different aspects of communities. However, neither can be directly applied to compare relative compositional similarity or differentiation across multiple regions with different numbers of communities because multiplicative beta diversity depends on the number of communities, and additive beta diversity additionally depends on alpha (equivalently, on gamma). Such dependences should be removed. We propose transformations to remove these dependences, and we show that the transformed multiplicative beta and additive beta both lead to the same classes of measures, which are always in a range of [0, 1] and thus can be used to compare relative similarity or differentiation among communities across multiple regions. These similarity measures include multiple-community generalizations of the Sørenson, Jaccard, Horn, and Morisita-Horn measures. For the third issue, we present some observations including a finding about which alpha formula produces independent alpha and beta components. These may help to resolve the choice of a proper formula for alpha diversity. Some related issues are also briefly discussed.