Measurement and Analysis of Biodiversity Thus, despite the greater effort involved in collecting abundance data, it may be useful to record species abundances so that abundance-based estimators may be used as well observations of species i, written Xi, has probability N xi a xi NÀa PfXi ¼ ag ¼ 1À n n a Other Sampling Issues: Surrogates for Quadrats and Species If we sample without replacement n À xi xi a NÀa PfXi ¼ ag ¼ n Two other sampling issues bear mention here First, it is important to note that the use of areal units (quadrats) for sampling units is not required Indeed, although quadrats are often appropriate for plants and other sessile organisms, they are less so for mobile organisms For mobile taxa, it is useful to substitute a more practical measureFsuch as observer-hours, trap-days, or volume of substrateFas a sampling unit Additionally, all is not lost if sampling units are of unequal sizes Equal-sized samples will meet the assumptions for a larger number of estimators However, some estimators exist which may be applied when it is not possible to keep sample sizes approximately constant over a census Second, techniques for estimating species richness are potentially applicable to taxonomic levels other than species For any of a variety of reasons, it is often impractical or impossible to key all organisms censused to the species level For example, in studies of paleontological data, it may only be possible to place an individual to the generic level In other cases, the presence of undescribed species or other lack of knowledge of species status may necessitate the use of a morphospecies concept In any case, it is important to note that richness estimators will estimate the number of classes (genera and morphospecies in the previous examples) at whatever taxonomic level was used in sampling 181 ð1Þ ð2Þ N Under certain conditions, both of these distributions can be approximated by the Poisson distribution according to P fXi ẳ ag ẳ Nli ịa Nli e a! 3ị where li ẳ xi/n Strictly speaking, the approximation of Eq (1) is exact only for communities containing infinite species richness and infinite abundance Practically speaking, the Poisson approximation should work best on large, species-rich communities A similar argument holds for the approximation of Eq (2) The requirement that n be considered infinite is sufficient to get to Eq (3), but it is not necessary For example, if the successive times between discoveries of an individual of species i are exponentially distributed and the number of individuals found in disjoint samples are independent, then we will get Eq (3) There is a natural way in which this can happen First, detections of individuals must be independent events Second, once we know the current value of Xi, the distribution of future values Xi must be completely determined Then, we can expect that Xi can be modeled with a Poisson process Estimators Based on Sampling Theory Species Observed (Sobs) Species richness estimators are all alike in the obvious way: They all estimate species richness However, SR estimators take different approaches to estimation This section presents approaches involving fine-scale, detailed models of the sampling process Estimators Based on Extrapolation discusses extrapolative methods based on coarse-scale, global models of species accumulation As a starting point, we consider the number of observed species, Sobs, as our first estimate of s Our presentation of Sobs serves as the template for the more complicated estimators to follow First, we consider the type of data we record Then, we state the assumptions made in deriving the estimator and its properties We next turn to theoretical properties of the estimator, for example, its bias and variance Finally, we discuss issues related to the application of the estimator to real data Estimators may be applied to two types of data: species abundance and species incidence If we record abundance, denoted Xi,j, our data set will be a matrix of the number of individuals of species i captured on sampling occasion j We define the abundance statistics, Ri, to be the number of species containing exactly i individuals Then, Models of the Sampling Process: General Concepts For the fixed collection from which we sample, (r0,y, rn) specifies the distribution of abundances among the species Sampling introduces randomness into our data so that when we census the area, we record (X0,y, XSobs) from which we compute (R0,y, RN) When we sample a set of t quadrats, we may keep track of the abundance of the ith species within the j th quadrat, Xi, j Then, we P compute the abundance of species i by Xi ¼ tj¼1 Xi;j However, in many cases, we may be unable to distinguish individuals In these cases, we record Yi,j from which we compute the incidence of P species i by Yi ¼ tj¼1 Yi;j The distribution of Xi,j depends on how we sample individuals Suppose we sample from a region (an urn) containing n individuals (marbles) distributed among s species (colors) Write xi for the number of individuals contained in species i We collect N individuals out of n and Sobs species out of s If we sample with replacement, we find that the number of Ri ẳ s X jẳ1 IXj ẳ iị ð4Þ We define the incidence statistics, Fi, to be the number of species occurring in exactly i samples in a census Then, Fi ẳ s X kẳ1 IYk ẳ iị ð5Þ When we wish to refer to the vector of Ri or Fi values for a given data set we will write R and F, respectively The number