Performance of nonparametric species richness estimators in a high diversity plant community

Abstract. The efficiency of four nonparametric species richness estimators — first‐order Jackknife, second‐order Jackknife, Chao2 and Bootstrap — was tested using simulated quadrat sampling of two field data sets (a sandy ‘Dune’ and adjacent ‘Swale’) in high diversity shrublands (kwongan) in south‐western Australia. The data sets each comprised > 100 perennial plant species and > 10 000 individuals, and the explicit (x‐y co‐ordinate) location of every individual. We applied two simulated sampling strategies to these data sets based on sampling quadrats of unit sizes 1/400th and 1/100th of total plot area. For each site and sampling strategy we obtained 250 independent sample curves, of 250 quadrats each, and compared the estimators’ performances by using three indices of bias and precision: MRE (mean relative error), MSRE (mean squared relative error) and OVER (percentage overestimation). The analysis presented here is unique in providing sample estimates derived from a complete, field‐based population census for a high diversity plant community. In general the true reference value was approached faster for a comparable area sampled for the smaller quadrat size and for the swale field data set, which was characterized by smaller plant size and higher plant density. Nevertheless, at least 15–30% of the total area needed to be sampled before reasonable estimates of S_t (total species richness) were obtained. In most field surveys, typically less than 1% of the total study domain is likely to be sampled, and at this sampling intensity underestimation is a problem. Results showed that the second‐order Jackknife approached the actual value of S_t more quickly than the other estimators. All four estimators were better than S_obs (observed number of species). However, the behaviour of the tested estimators was not as good as expected, and even with large sample size (number of quadrats sampled) all of them failed to provide reliable estimates. First‐ and second‐order Jackknives were positively biased whereas Chao2 and Bootstrap were negatively biased. The observed limitations in the estimators’ performance suggests that there is still scope for new tools to be developed by statisticians to assist in the estimation of species richness from sample data, especially in communities with high species richness.     

Optimizing performance of nonparametric species richness estimators under constrained sampling

Understanding the functional relationship between the sample size and the performance of species richness estimators is necessary to optimize limited sampling resources against estimation error. Nonparametric estimators such as Chao and Jackknife demonstrate strong performances, but consensus is lacking as to which estimator performs better under constrained sampling. We explore a method to improve the estimators under such scenario. The method we propose involves randomly splitting species‐abundance data from a single sample into two equally sized samples, and using an appropriate incidence‐based estimator to estimate richness. To test this method, we assume a lognormal species‐abundance distribution (SAD) with varying coefficients of variation (CV), generate samples using MCMC simulations, and use the expected mean‐squared error as the performance criterion of the estimators. We test this method for Chao, Jackknife, ICE, and ACE estimators. Between abundance‐based estimators with the single sample, and incidence‐based estimators with the split‐in‐two samples, Chao2 performed the best when CV < 0.65, and incidence‐based Jackknife performed the best when CV > 0.65, given that the ratio of sample size to observed species richness is greater than a critical value given by a power function of CV with respect to abundance of the sampled population. The proposed method increases the performance of the estimators substantially and is more effective when more rare species are in an assemblage. We also show that the splitting method works qualitatively similarly well when the SADs are log series, geometric series, and negative binomial. We demonstrate an application of the proposed method by estimating richness of zooplankton communities in samples of ballast water. The proposed splitting method is an alternative to sampling a large number of individuals to increase the accuracy of richness estimations; therefore, it is appropriate for a wide range of resource‐limited sampling scenarios in ecology.     

Sampled‐based estimation of diversity sensu stricto by transforming Hurlbert diversities into effective number of species

A theoretical framework based on Hill numbers has recently been advocated to measure and partition diversity sensu stricto. Hill numbers can be interpreted intuitively as effective number of species (ENS). They conform to the so‐called replication principle allowing a mathematically coherent multiplicative partitioning of diversity. They form a family of ENS defined by the parameter q which controls the weight attributed to rare species. Despite its advantages, this framework was developed without considering its robustness when treating community samples. In this study, we first show that Hurlbert diversity indices (expected number of species among k individuals) can be transformed into ENS that conform asymptotically to the replication principle while controlling the weight given to rare species through parameter k. We investigate the statistical properties of Hill and Hurlbert ENS using simulated communities with contrasted diversity. The properties of multiplicative beta diversity estimators based on ENS are also characterized by simulating communities with different levels of differentiation. We show that Hurlbert ENS provides a better statistical performance than Hill numbers when dealing with small sample sizes. By contrast, Hill numbers and their estimators suffer from substantial bias except when rare species have a low weight (q= 2). An estimator of ENS estimating both Hill numbers for q= 2 and Hurlbert ENS for k= 2 is shown to give the best performance and is recommended for processing real datasets when rare species receive low weight. In order to better take account of rare species, current estimators of Hill numbers are not recommended when sample size is too low while Hurlbert’s ENS performs reliably. In conclusion, while Hill numbers possess some interesting mathematical properties that are not shared by Hurlbert’s ENS, the latter outperforms Hill numbers in terms of statistical properties and is well suited to processing community samples, as illustrated on a real dataset.     

Comparative performance of species-richness estimators using data from a subtropical forest tree community

We used survey data collected from a large plot (20 ha) of sub-tropical forest in the Dinghushan Nature Reserve, Guangdong Province, southern China, in 2005 to test the comparative performance of nine species-richness estimators (number of observed species, three species-individual curve models, five nonparametric estimators). As the true species richness, we used the 210 free-standing shrub and tree species of >1 cm diameter at breast height recorded during the survey. This true species richness was then used to calculate performance measures of bias, accuracy, and precision for each estimator, whereby we distinguished performance for low, medium, and high sampling intensity. Unsurprisingly, all estimators performed better than the number of observed species in terms of bias and accuracy. Surprisingly, however, two curve models (logistic and logarithm) outperformed all other estimators in terms of bias, accuracy, and precision, which is in contrast to most other previous studies, in which nonparametric methods usually outperform curve models. Intriguingly, relative estimator performance changed between low, medium, and high sampling intensity, sometimes dramatically, reinforcing the assertion that the influence of sampling intensity on estimator performance is an important aspect to investigate and to consider when choosing estimators for ecological surveys. Because these results are based on only one dataset, the results should be treated with caution, both because (1) the generality of these results needs to be confirmed with simulated datasets and (2) more work is needed to establish what “true” species richness is extrapolated by each of the tested estimators in both the statistical and the practical sense. Nevertheless, the two curve estimators, namely Logistic and Logarithm, should be considered in future studies of comparative performance of species-richness estimators because of their outstanding performance in this study.     

Rarefaction and extrapolation of species richness using an area‐based Fisher's logseries

Fisher's logseries is widely used to characterize species abundance pattern, and some previous studies used it to predict species richness. However, this model, derived from the negative binomial model, degenerates at the zero‐abundance point (i.e., its probability mass fully concentrates at zero abundance, leading to an odd situation that no species can occur in the studied sample). Moreover, it is not directly related to the sampling area size. In this sense, the original Fisher's alpha (correspondingly, species richness) is incomparable among ecological communities with varying area sizes. To overcome these limitations, we developed a novel area‐based logseries model that can account for the compounding effect of the sampling area. The new model can be used to conduct area‐based rarefaction and extrapolation of species richness, with the advantage of accurately predicting species richness in a large region that has an area size being hundreds or thousands of times larger than that of a locally observed sample, provided that data follow the proposed model. The power of our proposed model has been validated by extensive numerical simulations and empirically tested through tree species richness extrapolation and interpolation in Brazilian Atlantic forests. Our parametric model is data parsimonious as it is still applicable when only the information on species number, community size, or the numbers of singleton and doubleton species in the local sample is available. Notably, in comparison with the original Fisher's method, our area‐based model can provide asymptotically unbiased variance estimation (therefore correct 95% confidence interval) for species richness. In conclusion, the proposed area‐based Fisher's logseries model can be of broad applications with clear and proper statistical background. Particularly, it is very suitable for being applied to hyperdiverse ecological assemblages in which nonparametric richness estimators were found to greatly underestimate species richness.     

The net result: evaluating species richness extrapolation techniques for littoral pond invertebrates

1. Total species richness for an assemblage or site is a valuable measure in conservation monitoring and assessment, but protocols for sampling and species richness determination in wetland habitats such as ponds, bogs or mires remain largely unrefined. 2. Techniques for estimation of total richness of an assemblage based upon replicated sampling offer the opportunity to derive useful estimates of total richness based upon small numbers of samples, and limit sampling‐derived disturbance which can be particularly problematic in small aquatic habitats. 3. We quantified the performance of eight of the most commonly encountered estimators of species richness for a variety of littoral zone macrofauna from ponds, comparing estimated richness to maximum richness derived from sampling. 4. Estimates using non‐parametric techniques based on species incidence provided the most accurate and precise estimates. The estimators Chao 2 and incidence‐based coverage estimator (ICE) from this category were reliable and consistent slight over‐estimators; the abundance‐based estimator Chao1 also performed well. 5. Species inventory based on relatively small numbers of samples might be significantly improved by use of non‐parametric estimators for quantification of species richness. 6. Use of non‐parametric estimators of species richness can assist biodiversity inventory by preventing erroneous rankings of habitat richness based upon observed species numbers from limited sampling.     

Incorporating functional dissimilarities into sample‐based rarefaction curves: from taxon resampling to functional resampling

Question: Indices of functional diversity have been seen as the key for integrating information on species richness with measures that focus on those components of community composition related to ecosystem functioning. For comparing species richness among habitats on an equal‐effort basis, so‐called sample‐based rarefaction curves may be used. Given a study area that is sampled for species presence and absence in N plots, sample‐based rarefaction generates the expected number of accumulated species as the number of sampled plots increases from 1 to N. Accordingly, the question for this study is: can we construct a ‘functional rarefaction curve’ that summarizes the expected functional dissimilarity between species when n plots are drawn at random from a larger pool of N plots? Methods: In this paper, we propose a parametric measure of functional diversity that is obtained by combining sample‐based rarefaction techniques that are usually applied to species richness with Rao's quadratic diversity. For a given set of N presence/absence plots, the resulting measure summarizes the expected functional dissimilarity at an increasingly larger cumulative number of plots n (n≤N). Results and Conclusions: Due to its parametric nature, the proposed measure is progressively more sensitive to rare species with increasing plot number, thus rendering this measure adequate for comparing the functional diversity of species assemblages that have been sampled with variable effort.     

Upper bound estimators of the population size based on ordinal models for capture‐recapture experiments

Capture‐recapture studies have attracted a lot of attention over the past few decades, especially in applied disciplines where a direct estimate for the size of a population of interest is not available. Epidemiology, ecology, public health, and biodiversity are just a few examples. The estimation of the number of unseen units has been a challenge for theoretical statisticians, and considerable progress has been made in providing lower bound estimators for the population size. In fact, it is well known that consistent estimators for this cannot be provided in the very general case. Considering a case where capture‐recapture studies are summarized by a frequency of frequencies distribution, we derive a simple upper bound of the population size based on the cumulative distribution function. We introduce two estimators of this bound, without any specific parametric assumption on the distribution of the observed frequency counts. The behavior of the proposed estimators is investigated using several benchmark datasets and a large‐scale simulation experiment based on the scheme discussed by Pledger.     

A new statistical approach for assessing similarity of species composition with incidence and abundance data

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 abundance‐based 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.     

Do commonly used indices of β‐diversity measure species turnover?

Abstract. Indices of β‐diversity are of two major types, (1) those that measure among‐plot variability in species composition independently of the position of individual plots on spatial or environmental gradients, and (2) those that measure the extent of change in species composition along predefined gradients, i.e. species turnover. Failure to recognize this distinction can lead to the inappropriate use of some β‐diversity indices to measure species turnover. Several commonly‐used indices of β‐diversity are based on Whittaker's β_W (β_W = γ/α, where γ is the number of species in an entire study area and α is the number of species per plot within the study area). It is demonstrated that these indices do not take into account the distribution of species on spatial or environmental gradients, and should therefore not be used to measure species turnover. The terms ‘β‐diversity’ and ‘species turnover’ should not be used interchangeably. Species turnover can be measured using matrices of compositional similarity and physical or environmental distances among pairs of study plots. The use of indices of β‐diversity and similarity‐distance curves is demonstrated using simulated data sets.     

Montane Andean rain forests are a global diversity hotspot of geometrid moths

Aim Andean forests are known to be a major diversity hotspot for vascular plants and vertebrates, but virtually nothing is known about the diversity of arthropods. We examined whether montane rain forests in southern Ecuador are also a diversity hotspot for arthropods, and chose geometrid moths as a model group. Location The study area in southern Ecuador (Province Zamora‐Chinchipe, 79° W, 04° S) covers c. 40 km², with 39 collecting sites within an elevational range of 1040–2677 m a.s.l. Thirty‐five of the sites were situated in an area c. 2.5 km². Additional qualitative sampling was carried out in the same area and up to an elevation of 3100 m. Methods Nocturnal moths were collected quantitatively and qualitatively using portable light towers consisting of two 15 W fluorescent tubes, and diurnal moths were collected qualitatively using an insect net. Insects were sampled in six fieldwork periods in the years 1999–2003. As diversity measures, Fisher's alpha of the log‐series distribution as well as eight estimators of total species richness were applied. Results A total of 1266 species were recorded, 63% of which were identified to named species, whereas the remainder are likely to include many undescribed species. Quantitative samples at light towers collected 35,238 specimens representing 1223 species. The extrapolated species number for these data is 1420 (incidence coverage estimator). Twenty‐one additional nocturnal species and 22 exclusively diurnal species were sampled qualitatively at elevations between 1040 and 3100 m. The pooled value of Fisher's alpha for all quantitative samples is 246 ± 3. Main conclusions The diversity of Geometridae documented here is much higher than anywhere else in the world, even without the inclusion of additional species from adjacent lowland rain forests. The number of recorded species in this small area corresponds to more than 6% of the known world fauna of geometrid moths. Our study emphasizes the importance of protecting the remaining montane Andean rain forests. For setting priorities in conservation, more studies on insect diversity are urgently required in other regions of the Andes, since montane forests are being destroyed at an alarming rate.     

Describing Ordinal Odds Ratios for Stratified r × c Tables

For an r × ctable with ordinal responses, odds ratios are commonly used to describe the relationship between the row and column variables. This article shows two types of ordinal odds ratios where local‐global odds ratios are used to compare several groups on a c‐category ordinal response and a global odds ratio is used to measure the global association between a pair of ordinal responses. When there is a stratification factor, we consider Mantel‐Haenszel (MH) type estimators of these odds ratios to summarize the association from several strata. Like the ordinary MH estimator of the common odds ratio for several 2 × 2 contingency tables, the estimators are used when the association is not expected to vary drastically among the strata. Also, the estimators are consistent under the ordinary asymptotic framework in which the number of strata is fixed and also under sparse asymptotics in which the number of strata grows with the sample size. Compared to the maximum likelihood estimators, simulations find that the MH type estimators perform better especially when each stratum has few observations. This article provides variances and covariances formulae for the local‐global odds ratios estimators and applies the bootstrap method to obtain a standard error for the global odds ratio estimator. At the end, we discuss possible ways of testing the homogeneity assumption.     

Estimating population size by spatially explicit capture–recapture

The number of animals in a population is conventionally estimated by capture–recapture without modelling the spatial relationships between animals and detectors. Problems arise with non‐spatial estimators when individuals differ in their exposure to traps or the target population is poorly defined. Spatially explicit capture–recapture (SECR) methods devised recently to estimate population density largely avoid these problems. Some applications require estimates of population size rather than density, and population size in a defined area may be obtained as a derived parameter from SECR models. While this use of SECR has potential benefits over conventional capture–recapture, including reduced bias, it is unfamiliar to field biologists and no study has examined the precision and robustness of the estimates. We used simulation to compare the performance of SECR and conventional estimators of population size with respect to bias and confidence interval coverage for several spatial scenarios. Three possible estimators for the sampling variance of realised population size all performed well. The precision of SECR estimates was nearly the same as that of the null‐model conventional population estimator. SECR estimates of population size were nearly unbiased (relative bias 0–10%) in all scenarios, including surveys in randomly generated patchy landscapes. Confidence interval coverage was near the nominal level. We used SECR to estimate the population of a species of skink Oligosoma infrapunctatum from pitfall trapping. The estimated number in the area bounded by the outermost traps differed little between a homogeneous density model and models with a quadratic trend in density or a habitat effect on density, despite evidence that the latter models fitted better. Extrapolation of trend models to a larger plot may be misleading. To avoid extrapolation, a large region of interest should be sampled throughout, either with one continuous trapping grid or with clusters of traps dispersed widely according to a probability‐based and spatially representative sampling design.     

Species–area relationships of red-listed species in old boreal forests: a large-scale data analysis

Aim Species–area relationships are often applied, but not generally approved, to guide practical conservation planning. The specific species group analysed may affect their applicability. We asked if species–area curves constructed from extensive databases of various sectors of natural resource administration can provide insights into large‐scale conservation of boreal forest biodiversity if the analyses are restricted only to red‐listed species. Location Finland, northern Europe. Methods Our data included 12,645 records of 219 red‐listed Coleoptera and Fungi from the whole of Finland. The forest data also covered the entire country, 202,761 km². The units of species–area analyses were 224 municipalities where the red‐listed forest species have been observed. We performed a hierarchical partitioning analysis to reveal the relative importance of different potential explanatory variables. Based on the results, for all red‐listed species, species associated with coniferous trees and for Fungi, the area of economically over‐aged forests explained the best the variation in data. For species associated with deciduous trees and Coleoptera, the forest area explained better variation in data than the area of old forests. In the subsequent log–log species–area regression analyses, we used the best variables as the explanatory variable for each species group. Results There was a strong relationship between the number of all red‐listed species and the area of old forests remaining, with a z‐value of 0.45. The area explained better the number of species associated with conifer trees and Fungi than the number of species associated with deciduous trees and Coleoptera. Main conclusions The high z‐values of species–area curves indicate that the remaining old‐growth patches constitute a real archipelago for the conifer‐associated red‐listed species, since lower values had been expected if the surrounding habitat matrix were a suitable habitat for the species analysed.     

Outlier robust model‐assisted small area estimation

Small area estimation with M‐quantile models was proposed by Chambers and Tzavidis ( 2006 ). The key target of this approach to small area estimation is to obtain reliable and outlier robust estimates avoiding at the same time the need for strong parametric assumptions. This approach, however, does not allow for the use of unit level survey weights, making questionable the design consistency of the estimators unless the sampling design is self‐weighting within small areas. In this paper, we adopt a model‐assisted approach and construct design consistent small area estimators that are based on the M‐quantile small area model. Analytic and bootstrap estimators of the design‐based variance are discussed. The proposed estimators are empirically evaluated in the presence of complex sampling designs.     

Estimating similarity of communities: a parametric approach to spatio‐temporal analysis of species diversity

Several stochastic models with environmental noise generate spatio‐temporal Gaussian fields of log densities for the species in a community. Combinations of such models for many species often lead to lognormal species abundance distributions. In spatio‐temporal analysis it is often realistic to assume that the same species are expected to occur at different times and/or locations because extinctions are rare events. Spatial and temporal β‐diversity can then be analyzed by studying pairs of communities at different times or locations defined by a bivariate lognormal species abundance model in which a single correlation occurs. This correlation, which is a measure of similarity between two communities, can be estimated from samples even if the sampling intensities vary and are unknown, using the bivariate Poisson lognormal distribution. The estimators are approximately unbiased, although each specific correlation may be rather uncertain when the sampling effort is low with only a small fraction of the species represented in the samples. An important characteristic of this community correlation is that it relates to the classical Jaccard‐ or the Sørensen‐indices of similarity based on the number of species present or absent in two communities. However, these indices calculated from samples of species in a community do not necessarily reflect similarity of the communities because the observed number of species depends strongly on the sampling intensities. Thus, we propose that our community correlation should be considered as an alternative to these indices when comparing similarity of communities. We illustrate the application of the correlation method by computing the similarity between temperate bird communities.     

Assessing the vulnerability of riparian vegetation to invasion by Mimulus guttatus: relative importance of biotic and abiotic variables in determining species occurrence and abundance

Riparian habitats are particularly susceptible to invasion by non‐native plants. At present, attempts to build consensus as to what the primary drivers of plant invasion in riparian ecosystems might be is hindered by the absence of common standards for data collected on plant species (e.g. occurrence, or relative abundance). Mimulus guttatus L., a non‐native riparian plant species, was used as a model to determine how environmental drivers influence two aspects of invasibility: species occurrence and abundance (assessed in relation to three variables number of patches, patch area and number of stems per patch). Mimulus occurrence and abundance, together with 20 environmental variables, were surveyed in almost 700 contiguous 50‐m‐long riverbank segments within a catchment in north‐east Scotland. More than half of the segments had been colonized by Mimulus. Occurrence and number of patches responded to similar environmental gradients, particularly bare sediment, boulders, high soil moisture, short‐statured ruderal communities, and open canopies, and tended to be highest downstream where the river was widest. In contrast to occurrence and patch number, patch area and stem number per patch were higher in the upper reaches of the catchment and were positively associated with low tree canopy and vegetation dominated by light‐demanding species and smaller‐statured species. Patch area and stem number per patch were also positively related to grazing. This study has highlighted the importance of assessing more than one measure of invasion success (occurrence or patch number and either patch area or stem number per patch), as they are each determined by a different suite of environmental variables. Abiotic factors, such as sediment availability and presence of boulders, appeared to be the major determinants of occurrence and patch number, whereas biotic factors, such as interspecific competition and grazing, were more important ecological determinants underlying area and stem number per patch.     

Double sampling to assess the accuracy of ground‐based surveys of tree hollows in eucalypt woodlands

Ground‐based surveys of tree hollows often give poor estimates of hollow abundance in forests. Woodlands have shorter trees and a more open structure than forests, which may make hollows easier to detect. Therefore, one would expect ground‐based surveys of tree hollows to be more accurate in woodlands than in forests. We compared hollow counts from ground‐based and climbing surveys (double sampling) for four species of Eucalyptus trees in woodlands of central‐western New South Wales, Australia: E. camaldulensis Dehnh, E. melliodora A. Cunn. ex Schauer, E. microcarpa Maiden and E. populnea F. Muell. ssp. bimbil L.A.S. Johnson & K.D. Hill and E. melliodora A. Cunn. ex Schauer. Overall, 83% of hollow‐bearing trees and 93% of trees without hollows were correctly classified by ground‐based surveys. Mean difference in hollow counts of ground‐based surveys to climbed surveys was 1.7 hollows ± 0.2 SE (all species combined) with 91% of ground‐based hollow counts being within five of the actual number of hollows. The error in ground‐based counts of hollows in E. microcarpa was larger than for the other three species. Errors in all species resulted from both overestimation and underestimation of hollow abundance by ground‐based surveys. A larger error was associated with the detection of hollows located in branches compared with hollows located in the main trunk(s). Total number of hollows in the tree (based on climbing surveys), crown area or maximum trunk diameter were significant predictors of ground‐based survey accuracy. Overall, the accuracy associated with ground surveys was relatively high and generally error rates were lower than those published for forests.     

A comparison of single‐sample estimators of effective population sizes from genetic marker data

In molecular ecology and conservation genetics studies, the important parameter of effective population size (N_e) is increasingly estimated from a single sample of individuals taken at random from a population and genotyped at a number of marker loci. Several estimators are developed, based on the information of linkage disequilibrium (LD), heterozygote excess (HE), molecular coancestry (MC) and sibship frequency (SF) in marker data. The most popular is the LD estimator, because it is more accurate than HE and MC estimators and is simpler to calculate than SF estimator. However, little is known about the accuracy of LD estimator relative to that of SF and about the robustness of all single‐sample estimators when some simplifying assumptions (e.g. random mating, no linkage, no genotyping errors) are violated. This study fills the gaps and uses extensive simulations to compare the biases and accuracies of the four estimators for different population properties (e.g. bottlenecks, nonrandom mating, haplodiploid), marker properties (e.g. linkage, polymorphisms) and sample properties (e.g. numbers of individuals and markers) and to compare the robustness of the four estimators when marker data are imperfect (with allelic dropouts). Extensive simulations show that SF estimator is more accurate, has a much wider application scope (e.g. suitable to nonrandom mating such as selfing, haplodiploid species, dominant markers) and is more robust (e.g. to the presence of linkage and genotyping errors of markers) than the other estimators. An empirical data set from a Yellowstone grizzly bear population was analysed to demonstrate the use of the SF estimator in practice.     

Limits to the estimation of species richness: The use of relative abundance distributions

The present study demonstrates the possibility of estimating species numbers of animal or plant communities from samples using relative abundance distributions. We use log‐abundance–species‐rank order plots and derive two new estimators that are based on log‐series and lognormal distributions. At small to moderate sample sizes these estimators appear to be more precise than previous parametric and nonparametric estimators. We test our estimators using samples from 171 published medium‐sized to large animal and plant communities taken from the literature. By this we show that our new estimators define also limits of precision.