Species richness is a fundamental measurement of community and regional diversity, and it underlies many ecological models and conservation strategies. In spite of its importance, ecologists have not always appreciated the effects of abundance and sampling effort on richness measures and comparisons. We survey a series of common pitfalls in quantifying and comparing taxon richness. These pitfalls can be largely avoided by using accumulation and rarefaction curves, which may be based on either individuals or samples. These taxon sampling curves contain the basic information for valid richness comparisons, including category±subcategory ratios (species-to-genus and species-toindividual ratios). Rarefaction methods ± both sample-based and individual-based ± allow for meaningful standardization and comparison of datasets. Standardizing data sets by area or sampling effort may produce very different results compared to standardizing by number of individuals collected, and it is not always clear which measure of diversity is more appropriate. Asymptotic richness estimators provide lower-bound estimates for taxon-rich groups such as tropical arthropods, in which observed richness rarely reaches an asymptote, despite intensive sampling. Recent examples of diversity studies of tropical trees, stream invertebrates, and herbaceous plants emphasize the importance of carefully quantifying species richness using taxon sampling curves.

EstimateS offers statistical tools for analyzing and comparing the diversity and composition of species assemblages, based on sampling data. The latest version computes a wide range of biodiversity statistics for both sample-based and individual-based data, including analytical rarefaction and non-parametric extrapolation, estimators of asymptotic species richness, diversity indices, Hill numbers, and (for sample-based data) measures of compositional similarity among assemblages. In the first 20 yr of its existence, EstimateS has been downloaded more than 70 000 times by users in 140 countries, who have cited it in 5000 publications in studies of taxa from microbes to mammals in every biome.

Quantifying and assessing changes in biological diversity are central aspects of many ecological studies, yet accurate methods of estimating biological diversity from sampling data have been elusive. Hill numbers, or the effective number of species, are increasingly used to characterize the taxonomic, phylogenetic, or functional diversity of an assemblage. However, empirical estimates of Hill numbers, including species richness, tend to be an increasing function of sampling effort and, thus, tend to increase with sample completeness. Integrated curves based on sampling theory that smoothly link rarefaction (interpolation) and prediction (extrapolation) standardize samples on the basis of sample size or sample completeness and facilitate the comparison of biodiversity data. Here we extended previous rarefaction and extrapolation models for species richness (Hill number qD, where q = 0) to measures of taxon diversity incorporating relative abundance (i.e., for any Hill number qD, q > 0) and present a unified approach for both individual‐based (abundance) data and sample‐based (incidence) data. Using this unified sampling framework, we derive both theoretical formulas and analytic estimators for seamless rarefaction and extrapolation based on Hill numbers. Detailed examples are provided for the first three Hill numbers: q = 0 (species richness), q = 1 (the exponential of Shannon's entropy index), and q = 2 (the inverse of Simpson's concentration index). We developed a bootstrap method for constructing confidence intervals around Hill numbers, facilitating the comparison of multiple assemblages of both rarefied and extrapolated samples. The proposed estimators are accurate for both rarefaction and short‐range extrapolation. For long‐range extrapolation, the performance of the estimators depends on both the value of q and on the extrapolation range. We tested our methods on simulated data generated from species abundance models and on data from large species inventories. We also illustrate the formulas and estimators using empirical data sets from biodiversity surveys of temperate forest spiders and tropical ants.

We review the biogeography of microorganisms in light of the biogeography of macroorganisms. A large body of research supports the idea that free-living microbial taxa exhibit biogeographic patterns. Current evidence confirms that, as proposed by the Baas-Becking hypothesis, 'the environment selects' and is, in part, responsible for spatial variation in microbial diversity. However, recent studies also dispute the idea that 'everything is everywhere'. We also consider how the processes that generate and maintain biogeographic patterns in macroorganisms could operate in the microbial world.

Distributions of Earth's species are changing at accelerating rates, increasingly driven by human-mediated climate change. Such changes are already altering the composition of ecological communities, but beyond conservation of natural systems, how and why does this matter? We review evidence that climate-driven species redistribution at regional to global scales affects ecosystem functioning, human well-being, and the dynamics of climate change itself. Production of natural resources required for food security, patterns of disease transmission, and processes of carbon sequestration are all altered by changes in species distribution. Consideration of these effects of biodiversity redistribution is critical yet lacking in most mitigation and adaptation strategies, including the United Nation's Sustainable Development Goals.

The classic Jaccard and Sørensen indices of compositional similarity (and other indices that depend upon the same variables) are notoriously sensitive to sample size, especially for assemblages with numerous rare species. Further, because these indices are based solely on presence-absence data, accurate estimators for them are unattainable. We provide a probabilistic derivation for the classic, incidence-based forms of these indices and extend this approach to formulate new Jaccard-type or Sørensen-type indices based on species abundance data. We then propose estimators for these indices that include the effect of unseen shared species, based on either (replicated) incidence-or abundancebased sample data. In sampling simulations, these new estimators prove to be considerably less biased than classic indices when a substantial proportion of species are missing from samples. Based on species-rich empirical datasets, we show how incorporating the effect of unseen shared species not only increases accuracy but also can change the interpretation of results.

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A general binomial mixture model is proposed for the species accumulation function based on presence–absence (incidence) of species in a sample of quadrats or other sampling units. The model covers interpolation between zero and the observed number of samples, as well as extrapolation beyond the observed sample set. For interpolation (sample‐based rarefaction), easily calculated, closed‐form expressions for both expected richness and its confidence limits are developed (using the method of moments) that completely eliminate the need for resampling methods and permit direct statistical comparison of richness between sample sets. An incidence‐based form of the Coleman (random‐placement) model is developed and compared with the moment‐based interpolation method. For extrapolation beyond the empirical sample set (and simultaneously, as an alternative method of interpolation), a likelihood‐based estimator with a bootstrap confidence interval is described that relies on a sequential, AIC‐guided algorithm to fit the mixture model parameters. Both the moment‐based and likelihood‐based estimators are illustrated with data sets for temperate birds and tropical seeds, ants, and trees. The moment‐based estimator is confidently recommended for interpolation (sample‐based rarefaction). For extrapolation, the likelihood‐based estimator performs well for doubling or tripling the number of empirical samples, but it is not reliable for estimating the richness asymptote. The sensitivity of individual‐based and sample‐based rarefaction to spatial (or temporal) patchiness is discussed.