Bridging the variance and diversity decomposition approaches to beta diversity via similarity and differentiation measures

Summary There are many concepts and measures of beta diversity and related similarity/differentiation indices. The variance framework (derived from the total variance of a community species abundance matrix) and diversity decomposition (based on partitioning gamma diversity into alpha and beta components) are two major approaches. There have been no bridges/links between the two approaches. Here, we establish a bridge by extending and modifying each approach so that both lead to the same classes of similarity/differentiation measures, which range in the interval [0, 1] and which can be compared across multiple sets of communities. Our extension/modification in each approach is based on the following major differences between the two approaches. (i) In the decomposition approach, a diversity order q that controls sensitivity to species abundances is used, whereas there is no such order involved in the variance approach. (ii) Transformations of raw abundances are typically used in the variance approach, whereas abundances are not transformed in diversity decomposition. (iii) The variance‐based beta for non‐transformed data is implicitly related to (and constrained by) alpha, gamma and total abundance. Namely, the attained maximum value of this beta when communities are completely distinct (no shared species) is not a fixed constant; the maximum varies with alpha, gamma and total abundance. By contrast, the beta component obtained from the multiplicative decomposition is not constrained by alpha, gamma and total abundance. To construct the bridge, we extend the variance of community data to a class of divergence measures (parameterized by an order q ) and use normalization to remove these measures' constraints by alpha, gamma and total abundance. The resulting normalized divergence measures are legitimate differentiation measures. In the decomposition approach, we adopt a modified multiplicative decomposition scheme; the resulting beta component can be transformed to quantify compositional similarity/differentiation among communities. Then, the similarity/differentiation measures obtained from the extended variance framework turn out to be identical to those from the modified diversity decomposition, establishing the bridge. Other types of similarity/differentiation measures (e.g. N ‐community Bray–Curtis type) and extension to phylogenetic and functional versions are discussed. A real example using corals is given for illustration.