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Species Diversity • Incorporate both diversity and their relative abundances • Three important assumptions – (1) all individuals assigned to a specific class are assumed to be equal (e.g. no recognition of age, size, or sex classes; if so, impact disproportionate) – (2) all species or classes are assumed to be equally different from one another (e.g. no keystone species identified) – (3) community structure is assumed to be measure in appropriate units (e.g. for animals, individuals but for plants perhaps biomass or percent cover) Species Diversity • Perhaps the most commonly used index is the Shannon-Wiener diversity index H’ = -∑ pi ln(pi) where pi is the relative abundance of the I th species (∑ pi = 1.0) Problems • Most importantly, cannot be interpreted in a biologically meaningful way • They lack a probabilistic basis and consequently, cannot evaluate the biological or statistical differences between two communities (H’ of 1.7 vs. 2.0) Potential Solutions • Still a need to quantify communities (1) Abandon the idea of incorporating both evenness and species richness into a single index (2) Use explicit null models to estimate the nonbiological effects of sample size on species richness and species evenness (although little work on evenness has been conducted) Problems: Species Richness • Species richness is not only influenced by the number of individuals, the species richness of the surrounding community, and area Example • Unless two communities are censused exhaustively and identically, it is inappropriate to compare simple species counts • Compare communities from two different age pine plantations (Table 2.1) • Young forests: 243 individuals of 31 species vs. 63 individuals of 9 species • Distribution of species (between young and old pine plantations) considered in the carabid beetle study Solution: Rarefaction • Sanders (1968) addressed the problem of comparing the species richness of different habitats in a study of marine communities • Using an algorithm for “rarefying” the large samples (based on random subsamples of individuals) • The rarefied sampled can then be directly compared to the smaller sample • If conducted for a number of different abundances, a rarefaction can be plotted Rarefaction Curves • Because subsampling is used, it also generates a probability distribution (CI) Rarefaction • The correct rarefaction model is based on the hypergeometric distribution, sampling without replacement (e.g. jackknife) from some parent distribution (the observed collection) Assumptions of Rarefaction • 1) sampling has been sufficient to guarantee an adequate characterization of the parent distribution (if too small, all samples converge at the coordinate (1,1)) • 2) the spatial distribution of individuals is random Assumptions of Rarefaction • 3) the samples to be compared are taxonomically “similar” and are drawn from the “same” community type • 4) standardized sampling techniques are used for all collections • 5) rarefaction can be used for interpolation to a smaller sample size, but not for extrapolation to a larger sample size (unless distribution is assumed; but species accumulation curves may be better) Statistical Issues • Rarefaction can be used to ask two different questions 1) For a collection of N individuals of S species, what is the expected number of species in a small sample of n individuals 2) What is the likelihood that two collections of sizes N and n were both drawn from the same parent distribution Statistical Considerations • It is important to calculate rarefaction curves along their full length, that is, for several different values of n • For a collection of S species and N individuals, a minimum of S + 1 subsamples of n, evenly spaced along the x axis, should be used to construct the curve Extrapolation of Rarefaction • Most promising nonparametric estimators have been adapted from mark-and-recapture statistics • For a single collection of species, the best estimator of total species richness is Stot = Sobs + (a2 / 2b) a is the number of species with a single individuals b is the number of species with two individuals Criticisms of Rarefaction • Several criticisms of rarefaction include: – It assumes a random spatial distribution of individuals – It loses information about species identity and relative abundance – Is difficult to calculate by hand However, all but the last are leveled at most diversity indices and computer software has addressed the last