Spatially constrained rarefaction: incorporating the autocorrelated structure of biological communities into sample-based rarefaction

Rarefaction is a widely applied technique for comparing the species richness of samples that differ in area, volume or sampling effort. Despite widespread adoption of sample-based rarefaction curves, serious concerns persist. In this paper, we address the issue of the spatial arrangement of sampling units when computing sample-based rarefaction curves. If the spatial arrangement is neglected when building rarefaction curves, a direct comparison of species richness estimates obtained for areas that differ in their spatial extent is not possible, even if they were sampled with a similar intensity. We demonstrate a major effect of the spatial extent of the samples on species richness estimates through the use of data from a temperate forest. We show that the use of Spatially Constrained Rarefaction (SCR) results in species richness estimates that are directly comparable for areas that differ in spatial extent. As expected, standard rarefaction curves tend to overestimate species richness because they ignore the spatial autocorrelation of species composition among sampling units. This spatial autocorrelation is captured by the SCR, thus providing a useful technique for characterizing the spatial structure of biodiversity patterns. Further work is necessary to determine how species richness estimates and the shape of the SCR are affected by the method of spatial constraint and sampling unit density and distribution. Abbreviations: SCR–Spatially Constrained Rarefaction, SAR–Species Area Relationship, CI–Confidence Interval have been used to compare species richness across sites, for example, beetle species richness across two ecoregions of USA (Gering et al. 2003), snake species richness as a function of trap-days (Thompson et al. 2003), plant species richness of the biogeographic regions in Switzerland (Koellner et al. 2004), and plant species richness of different forests or nature reserves in Italy (Chiarucci and Bonini 2005, Chiarucci et al. 2008a). A formula for the variance of such curves was recently proposed (Colwell et al. 2004, Mao et al. 2005), although Fattorini (2007) showed that this formula is based on assumptions that are usually violated in ecological data. Sample-based rarefaction curves are equivalent to some species-area curves when the sampling unit is an area. In particular, these curves correspond to a type III Species-Area Relationship (SAR) in the classification of Scheiner (2003). Such SARs are built from non-contiguous sampling units. The difference between a traditional SAR and a samplebased rarefaction curve is that in the latter the x-axis is the number of sampling units rather than the size of the area sampled, although one can be transformed into the other. By formula (1), all possible combinations of sampling units are considered for each n and, thus, traditional rarefaction curves correspond to the type IIIB curves of Scheiner’s scheme: curves not using a spatially-explicit method for obtaining the combinations of sampling units. Scheiner (2003) noted that the shape of a type III curve is to some extent a function of the spacing between the sampling units, often called the lag. Consequently, the problem of estimating species richness in a collection of n units depends on: (1) how many units are sampled, (2) the area of each unit (which together determine the total sampled area), and (3) the lag, which determines the extent of the samples, i.e., the area over which those units are spread (Condit et al. 1996, Palmer et al. 2002, Scheiner 2003, Chiarucci and Bonini 2005, Fattorini 2007, Hui 2008). To our knowledge, the only previous examination of the effects of spatial autocorrelation on the properties of rarefaction curves was done by Collins and Simberloff (2009). They examined the robustness of individual-based rarefaction curves to violations of random and independent dispersion of individuals and species. This paper complements their efforts by examining spatial effects of sample-based rarefaction curves. In this paper, we (1) demonstrate the bias associated with the use of sample-based rarefaction curves (or type IIIB SARs) and (2) introduce a new technique to deal with this problem: the use of a spatially-constrained rarefaction curve. The problem: spatial arrangement matters In a given area, a sample-based rarefaction curve provides the expected (mean) number of species recorded by a set of n sampling units (Ugland et al. 2003, Fattorini 2007, Chiarucci et al. 2008b). Even if this method offers an elegant solution to the interpolation of the number of species collected as a function of sample size, it is affected by various spatial components as described above. When using sampling units of a given size, the shape of the rarefaction curve is determined by two factors: the number of units and the total extent. When comparing two areas sampled with different number of samples, the rarefaction to a similar number of sampling units should allow a direct comparison of species richness, especially if sampled areas have the same density of sampling units (Chiarucci and Bonini 2005). However, this approach considers only the effects of the first factor, the number of units, but ignores the second, the spatial extent of those units. The following example clarifies this issue. Consider a comparison of species richness from two areas of different sizes: area A is 100 ha, while area B is 10 ha (Fig. 1). The two areas are sampled with the same intensity, e.g., one sampling unit per hectare, resulting in two samples of 100 and 10 units, respectively. To make the example clearer we can even consider that the mean number of species per sample unit is the same. The hypothetical sample-based