Phylogenetic beta diversity, similarity, and differentiation measures based on Hill numbers

Until now, decomposition of abundance‐sensitive gamma (regional) phylogenetic diversity measures into alpha and beta (within‐ and between‐group) components has been based on an additive partitioning of phylogenetic generalized entropies, especially Rao's quadratic entropy. This additive approach led to a phylogenetic measure of differentiation between assemblages: (gamma − alpha)/gamma. We show both empirically and theoretically that this approach inherits all of the problems recently identified in the additive partitioning of non‐phylogenetic generalized entropies. When within‐assemblage (alpha) quadratic entropy is high, the additive beta and the differentiation measure (gamma − alpha)/gamma always tend to zero (implying no differentiation) regardless of phylogenetic structures and differences in species abundances across assemblages. Likewise, the differentiation measure based on the phylogenetic generalization of Shannon entropy always approaches zero whenever gamma phylogenetic entropy is high. Such critical flaws, inherited from their non‐phylogenetic parent measures (Gini‐Simpson index and Shannon entropy respectively), have caused interpretational problems. These flaws arise because phylogenetic generalized entropies do not obey the replication principle, which ensures that the diversity measures are linear with respect to species addition or group pooling. Furthermore, their complete partitioning into independent components is not additive (except for phylogenetic entropy). Just as in the non‐phylogenetic case, these interpretational problems are resolved by using phylogenetic Hill numbers that obey the replication principle. Here we show how to partition the phylogenetic gamma diversity based on Hill numbers into independent alpha and beta components, which turn out to be multiplicative. The resulting phylogenetic beta diversity (ratio of gamma to alpha) measures the effective number of completely phylogenetically distinct assemblages. This beta component measures pure differentiation among assemblages and thus can be used to construct several classes of similarity or differentiation measures normalized onto the range 0, 1. We also propose a normalization to fix the traditional additive phylogenetic similarity and differentiation measures, and we show that this yields the same similarity and differentiation measures we derived from multiplicative phylogenetic diversity partitioning. We thus can achieve a consensus on phylogenetic similarity and differentiation measures, including N ‐assemblage phylogenetic generalizations of the classic Jaccard, Sørensen, Horn, and Morisita‐Horn measures. Hypothetical and real examples are used for illustration.