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Hill numbers or the effective number of species are increasingly used to quantify species diversity of an assemblage. Hill numbers were recently extended to phylogenetic diversity, which incorporates species evolutionary history, as well as to functional diversity, which considers the differences among species traits. We review these extensions and integrate them into a framework of attribute diversity (the effective number of entities or total attribute value) based on Hill numbers of taxonomic entities (species), phylogenetic entities (branches of unit-length), or functional entities (species-pairs with unit-distance between species). This framework unifies ecologists' measures of species diversity, phylogenetic diversity, and distance-based functional diversity. It also provides a unified method of decomposing these diversities and constructing normalized taxonomic, phylogenetic, and functional similarity and differentiation measures, including N-assemblage phylogenetic or functional generalizations of the classic Jaccard, Sørensen, Horn, and Morisita-Horn indexes. A real example shows how this framework extracts ecological meaning from complex data.
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Download Supplemental Appendix 1: Comparisons with other unified measures (PDF). Download Supplemental Appendix 2: Two simple illustrative examples (PDF). Download Supplemental Appendix 3: Two kinds of data pooling and decomposition formulas (PDF). Download Supplemental Appendix 4: Two classes of species-overlap (or similarity) measures based on partitioning the generalized hill numbers (PDF). Download Supplemental Appendix 5: An illustrative example (PDF).