Incidence, or compositional, matrices are generated for a broad range of research applications in biology. Zeta diversity provides a common currency and conceptual framework that links incidence‐based metrics with multiple patterns of interest in biology, ecology, and biodiversity science. It quantifies the variation in species (or OTU) composition of multiple assemblages (or cases) in space or time, to capture the contribution of the full suite of narrow, intermediate, and wide‐ranging species to biotic heterogeneity. Here we provide a conceptual framework for the application and interpretation of patterns of continuous change in compositional diversity using zeta diversity. This includes consideration of the survey design context, and the multiple ways in which zeta diversity decline and decay can be used to examine and test turnover in the identity of elements across space and time. We introduce the zeta ratio–based retention rate curve to quantify rates of compositional change. We illustrate these applications using 11 empirical data sets from a broad range of taxa, scales, and levels of biological organization—from DNA molecules and microbes to communities and interaction networks—including one of the original data sets used to express compositional change and distance decay in ecology. We show (1) how different sample selection schemes used during the calculation of compositional change are appropriate for different data types and questions, (2) how higher orders of zeta may in some cases better detect shifts and transitions, and (3) the relative roles of rare vs. common species in driving patterns of compositional change. By exploring the application of zeta diversity decline and decay, including the retention rate, across this broad range of contexts, we demonstrate its application for understanding continuous turnover in biological systems.

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The PD measure of phylogenetic diversity interprets branch lengths cladistically to make inferences about feature diversity. PD calculations extend conventional species-level ecological indices to the features level. The “phylogenetic beta diversity” framework developed by microbial ecologists calculates PD-dissimilarities between community localities. Interpretation of these PD-dissimilarities at the feature level explains the framework’s success in producing ordinations revealing environmental gradients. An example gradients space using PD-dissimilarities illustrates how evolutionary features form unimodal response patterns to gradients. This features model supports new application of existing species-level methods that are robust to unimodal responses, plus novel applications relating to climate change, commercial products discovery, and community assembly.

Summary1. Traditional species diversity measures do not make distinctions among species. Faith's phylogenetic diversity (PD), which is defined as the sum of the branch lengths of a phylogenetic tree connecting all species, takes into account phylogenetic differences among species and has found many applications in various research fields. In this paper, we extend Faith's PD to represent the total length of a phylogenetic tree from any fixed point on its main trunk. 2. Like species richness, Faith's PD tends to be an increasing function of sampling effort and thus tends to increase with sample completeness. We develop in this paper the 'PD accumulation curve' (an extension of the species accumulation curve) to depict how PD increases with sampling size and sample completeness. 3. To make fair comparisons of Faith's PD among several assemblages based on sampling data from each assemblage, we derive both theoretical formulae and analytic estimators for seamless rarefaction (interpolation) and extrapolation (prediction). We develop a lower bound of the undetected PD for an incomplete sample to guide the extrapolation; the PD estimator for an extrapolated sample is generally reliable up to twice the size of the empirical sample. 4. We propose an integrated curve that smoothly links rarefaction and extrapolation to standardize samples on the basis of sample size or sample completeness. A bootstrap method is used to obtain the unconditional variances of PD estimators and to construct the confidence interval of the expected PD for a fixed sample size or fixed degree of sample completeness. This facilitates comparison of multiple assemblages of both rarefied and extrapolated samples. 5. We illustrate our formulae and estimators using empirical data sets from Australian birds in two sites. We discuss the extension of our approach to the case of multiple incidence data and to incorporate species abundances.

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