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.     

Assessing the performance of nonparametric estimators of species richness in meadows

To accurately measure the number of species in a biological community, a complete inventory should be performed, which is generally unfeasible; hopefully, estimators of species richness can help. Our main objectives were (i) to assess the performance of nonparametric estimators of plant species richness with real data from a small set of meadows located in the Basque campiña (northern Spain), and (ii) to apply the best estimator to a larger dataset to test the effects on plant species richness caused by environmental conditions and human practices. Two non-asymptotic and seven asymptotic accumulation functions were fitted to a randomized sample-based rarefaction curve computed with data from three well sampled meadows, and information theoretic methods were used to select the best fitting model; this was the Morgan-Mercer-Flodin, and its asymptote was taken as our best guess of true richness. Then, five nonparametric estimators were computed: ICE, Chao 2, Jackknife 1 and 2, and Bootstrap; MMRuns and MMMeans were also assessed. According to the criteria set for our performance assessment (i.e., bias, precision, and accuracy), the best estimator was Jackknife 1. Finally, Jackknife 1 was applied to assess the effects of terrain slope and soil parent material, and also fertilization, grazing, and mowing, on plant species richness from a larger dataset (20 meadows). Results suggested that grass cutting was causing a loss of richness close to 30%, as compared to unmowed meadows. It is concluded that the use of nonparametric estimators of species richness can improve the evaluation of biodiversity responses to human management practices.     

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.     

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.     

Small‐Sample Estimation of Species Richness Applied to Forest Communities

Summary Many well‐known methods are available for estimating the number of species in a forest community. However, most existing methods result in considerable negative bias in applications, where field surveys typically represent only a small fraction of sampled communities. This article develops a new method based on sampling with replacement to estimate species richness via the generalized jackknife procedure. The proposed estimator yields small bias and reasonably accurate interval estimation even with small samples. The performance of the proposed estimator is compared with several typical estimators via simulation study using two complete census datasets from Panama and Malaysia.     

Evaluating the performance of species richness estimators: sensitivity to sample grain size

1. Fifteen species richness estimators (three asymptotic based on species accumulation curves, 11 nonparametric, and one based in the species-area relationship) were compared by examining their performance in estimating the total species richness of epigean arthropods in the Azorean Laurisilva forests. Data obtained with standardized sampling of 78 transects in natural forest remnants of five islands were aggregated in seven different grains (i.e. ways of defining a single sample): islands, natural areas, transects, pairs of traps, traps, database records and individuals to assess the effect of using different sampling units on species richness estimations. 2. Estimated species richness scores depended both on the estimator considered and on the grain size used to aggregate data. However, several estimators (ACE, Chao 1, Jackknifel and 2 and Bootstrap) were precise in spite of grain variations. Weibull and several recent estimators [proposed by Rosenzweig et al. (Conservation Biology, 2003, 17, 864-874), and Ugland et al. (Journal of Animal Ecology, 2003, 72, 888-897)] performed poorly. 3. Estimations developed using the smaller grain sizes (pair of traps, traps, records and individuals) presented similar scores in a number of estimators (the above-mentioned plus ICE, Chao2, Michaelis-Menten, Negative Exponential and Clench). The estimations from those four sample sizes were also highly correlated. 4. Contrary to other studies, we conclude that most species richness estimators may be useful in biodiversity studies. Owing to their inherent formulas, several nonparametric and asymptotic estimators present insensitivity to differences in the way the samples are aggregated. Thus, they could be used to compare species richness scores obtained from different sampling strategies. Our results also point out that species richness estimations coming from small grain sizes can be directly compared and other estimators could give more precise results in those cases. We propose a decision framework based on our results and on the literature to assess which estimator should be used to compare species richness scores of different sites, depending on the grain size of the original data, and of the kind of data available (species occurrence or abundance data).     

Application of species richness estimators for the assessment of fungal diversity

Species richness and distribution patterns of wood-inhabiting fungi and mycetozoans (slime moulds) were investigated in the canopy of a Central European temperate mixed deciduous forest. Species richness was described with diversity indices and species-accumulation curves. Nonmetrical multidimensional scaling was used to assess fungal species composition on different tree species. Different species richness estimators were used to extrapolate species richness beyond our own data. The reliability of the abundance-based coverage estimator, Chao, Jackknife and other estimators of species richness was evaluated for mycological surveys. While the species-accumulation curve of mycetozoans came close to saturation, that of wood-inhabiting fungi was continuously rising. The Chao 2 richness estimator was considered most appropriate to predict the number of species at the investigation site if sampling were continued. Gray's predictor of species richness should be used if statements of the number of species in larger areas are required. Multivariate analysis revealed the importance of different tree species for the conservation and maintenance of fungal diversity within forests, because each tree species possessed a characteristic fungal community. The described mathematical approaches of estimating species richness possess great potential to address fungal diversity on a regional, national, and global scale.     

The use of incidence-based species richness estimators,species accumulation curves and similarity measures to appraise ethnobotanical inventories from South Africa

The incorporation of suitable quantitative methods into ethnobotanical studies enhances the value of the research and the interpretation of the results. Prediction of sample species richness and the use of species accumulation functions have been addressed little in applied ethnobotany. In this paper, incidence-based species richness estimators, species accumulation curves and similarity measures are used to compare and predict species richness, evaluate sampling effort and compare the similarity of species inventories for ethnobotanical data sets derived from the trade in traditional medicine in Johannesburg and Mpumalanga, South Africa. EstimateS was used to compute estimators of species richness (e.g. Jackknife), rarefaction curves, species accumulation curves and complimentarity. Results showed that while the Michaelis–Menten Means estimator appeared to be the best estimator because the curve approached a horizontal asymptote, it was not able to accurately predict species richness for one of the data sets when two of its subsamples were individually tested. Instead, the first-order Jackknife estimator best approximated the known richness.     

Estimation of local extinction rates when species detectability covaries with extinction probability: is it a problem?

Estimating the rate of change of the composition of communities is of direct interest to address many fundamental and applied questions in ecology. One methodological problem is that it is hard to detect all the species present in a community. Nichols et al. presented an estimator of the local extinction rate that takes into account species probability of detection, but little information is available on its performance. However, they predicted that if a covariance between species detection probability and local extinction rate exists in a community, the estimator of local extinction rate complement would be positively biased.
Here, we show, using simulations over a wide range of parameters that the estimator performs reasonably well. The bias induced by biological factors appears relatively weak. The most important factor enhancing the performance (bias and precision) of the local extinction rate complement estimator is sampling effort. Interestingly, a potentially important biological bias, such as the covariance effect, improves the estimation for small sampling efforts, without inducing a supplementary overestimation when these sampling efforts are high. In the field, all species are rarely detectable so we recommend the use of such estimators that take into account heterogeneity in species detection probability when estimating vital rates responsible for community changes.     

Estimating the richness of species with variable mobility

The vast majority of species are animals that, unlike most plants and fungi, are variably and often highly mobile. While species' mobility affects species' probabilities of being sampled, effects of movement on the estimation of species richness have yet to be systematically investigated. Information-rich abundance-based estimators may be able to address variably mobile species but the accuracy of these estimators has also yet to be investigated. Here, we address both issues by variably sampling simulated landscapes with up to 250 species and evaluating the performance of ten non-parametric estimators and one species accumulation curve. Our results show that some abundance-based estimators are as accurate as better known and tested incidence-based estimators. Increased movement heterogeneity between the species reduced estimator performance by reducing the sample coverage, which systematically determined which estimator was most accurate. Based on these findings, we present the first decision framework for choosing the most accurate of many available abundance-based species-richness estimators. These decisions, based on data coverage, can significantly improve investigators' ability to estimate faunal species richness.     

Efficient sampling of plant diversity in arid deserts using non-parametric estimators

Surveying plant diversity in arid desert areas is extremely difficult because of the harsh climate, hostile terrain, lack of roads, and insecurity, which is why it is particularly important to improve the sampling efficiency, but few relevant studies have been done. The performance of non-parametric estimators was assessed with first-hand field data to determine (a) the threshold of the proportion of uniques (number of species that occur in exactly one plot divided by the number of species sampled) that involves the least sampling effort and (b) the method of locating plots to obtain a more reliable estimate of species richness. The study area (Gurbantunggut desert, China) was divided into five sub-regions based on variation in physical environment and vegetation. The following common correction factors were selected: ACE, Chao1, Bootstrap, Chao2, ICE, Jack1, and Jack2. The estimates for each sub-region (partition) and for the entire region (without partition), the threshold of proportion of uniques, and the method of determining sampling locations (including prior sampling of plots that show large differences in habitats) were compared in terms of their ability to predict the number of species more accurately. We found that ACE and Chao1 (which use abundance data) showed more biased estimates than the other factors (incidence data), and best estimator is Jack1. Species richness was significantly underestimated for the region, but the non-parametric estimators could estimate the species richness for each sub-region reliably. Sampling locations affected the performance of non-parametric estimators significantly. The threshold of minimum sampling was 15% and that of uniques was 30%; the two were able to limit the bias within 5 and 10%, respectively. It is concluded that the non-parametric estimators can estimate the plant diversity of arid deserts reliably from the data on incidence. The study area (on the scale of a region) should be partitioned to improve the performance of the non-parametric estimators. The plots with larger differences in habitats should be sampled more extensively based on the threshold of the proportion of uniques.     

Estimating the number of species in a stochastic abundance model

Consider a stochastic abundance model in which the species arrive in the sample according to independent Poisson processes, where the abundance parameters of the processes follow a gamma distribution. We propose a new estimator of the number of species for this model. The estimator takes the form of the number of duplicated species (i.e., species represented by two or more individuals) divided by an estimated duplication fraction. The duplication fraction is estimated from all frequencies including singleton information. The new estimator is closely related to the sample coverage estimator presented by Chao and Lee (1992, Journal of the American Statistical Association 87, 210-217). We illustrate the procedure using the Malayan butterfly data discussed by Fisher, Corbet, and Williams (1943, Journal of Animal Ecology 12, 42-58) and a 1989 Christmas Bird Count dataset collected in Florida, U.S.A. Simulation studies show that this estimator compares well with maximum likelihood estimators (i.e., empirical Bayes estimators from the Bayesian viewpoint) for which an iterative numerical procedure is needed and may be infeasible.     

The relationship between percentage of singletons and sampling effort: A new approach to reduce the bias of richness estimates

Estimate the richness of a community with accuracy despite differences in sampling effort is a key aspect to monitoring high diverse ecosystems. We compiled a worldwide multitaxa database, comprising 185 communities, in order to study the relationship between the percentage of species represented by one individual (singletons) and the intensity of sampling (number of individuals divided by the number of species sampled). The database was used to empirically adjust a correction factor to improve the performance of non-parametrical estimators under conditions of low sampling effort. The correction factor was tested on seven estimators (Chao1, Chao2, Jack1, Jack2, ACE, ICE and Bootstrap). The correction factor was able to reduce the bias of all estimators tested under conditions of undersampling, while converging to the original uncorrected values at higher intensities. Our findings led us to recommend the threshold of 20 individuals/species, or less than 21% of singletons, as a minimum sampling effort to produce reliable richness estimates of high diverse ecosystems using corrected non-parametric estimators. This threshold rise for 50 individuals/species if non-corrected estimators are used which implies in an economy of 60% of sampling effort if the correction factor is used.     

Species richness in soil bacterial communities: a proposed approach to overcome sample size bias

Estimates of species richness based on 16S rRNA gene clone libraries are increasingly utilized to gauge the level of bacterial diversity within various ecosystems. However, previous studies have indicated that regardless of the utilized approach, species richness estimates obtained are dependent on the size of the analyzed clone libraries. We here propose an approach to overcome sample size bias in species richness estimates in complex microbial communities. Parametric (Maximum likelihood-based and rarefaction curve-based) and non-parametric approaches were used to estimate species richness in a library of 13,001 near full-length 16S rRNA clones derived from soil, as well as in multiple subsets of the original library. Species richness estimates obtained increased with the increase in library size. To obtain a sample size-unbiased estimate of species richness, we calculated the theoretical clone library sizes required to encounter the estimated species richness at various clone library sizes, used curve fitting to determine the theoretical clone library size required to encounter the "true" species richness, and subsequently determined the corresponding sample size-unbiased species richness value. Using this approach, sample size-unbiased estimates of 17,230, 15,571, and 33,912 were obtained for the ML-based, rarefaction curve-based, and ACE-1 estimators, respectively, compared to bias-uncorrected values of 15,009, 11,913, and 20,909.     

A Class of Ratio-Cum-Product-Type Estimator

The problem of estimating the population mean using an auxiliary information has been dealt with in literature quite extensively. Ratio, product, linear regression and ratio-type estimators are well known. A class of ratio-cum-product-type estimator is proposed in this paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and good range of α-values, it is found that the proposed estimator is superior than a set of estimators (i.e., sample mean, usual ratio and product estimators, SRIVASTAVA's (1967) estimator, CHAKRABARTY's (1979) estimator and a product-type estimator) which are, in fact, the particular cases of it. At optimum value of α, the proposed estimator is as efficient as linear regression estimator.     

Model misspecification and bias for inverse probability weighting estimators of average causal effects

Commonly used semiparametric estimators of causal effects specify parametric models for the propensity score (PS) and the conditional outcome. An example is an augmented inverse probability weighting (IPW) estimator, frequently referred to as a doubly robust estimator, because it is consistent if at least one of the two models is correctly specified. However, in many observational studies, the role of the parametric models is often not to provide a representation of the data-generating process but rather to facilitate the adjustment for confounding, making the assumption of at least one true model unlikely to hold. In this paper, we propose a crude analytical approach to study the large-sample bias of estimators when the models are assumed to be approximations of the data-generating process, namely, when all models are misspecified. We apply our approach to three prototypical estimators of the average causal effect, two IPW estimators, using a misspecified PS model, and an augmented IPW (AIPW) estimator, using misspecified models for the outcome regression (OR) and the PS. For the two IPW estimators, we show that normalization, in addition to having a smaller variance, also offers some protection against bias due to model misspecification. To analyze the question of when the use of two misspecified models is better than one we derive necessary and sufficient conditions for when the AIPW estimator has a smaller bias than a simple IPW estimator and when it has a smaller bias than an IPW estimator with normalized weights. If the misspecification of the outcome model is moderate, the comparisons of the biases of the IPW and AIPW estimators show that the AIPW estimator has a smaller bias than the IPW estimators. However, all biases include a scaling with the PS-model error and we suggest caution in modeling the PS whenever such a model is involved. For numerical and finite sample illustrations, we include three simulation studies and corresponding approximations of the large-sample biases. In a dataset from the National Health and Nutrition Examination Survey, we estimate the effect of smoking on blood lead levels.     

Effects of topographic variability on the scaling of plant species richness in gradient dominated landscapes

It is commonly assumed that variation in abiotic site conditions influences the number of niches, which in turn affects the potential species richness in an area. Based on theoretical considerations, abiotic variation is often used as an estimator of species richness at broad scales, while at finer landscape scales the diversity of habitat types is used. However, habitat estimators assume the landscape to be composed of discrete, homogeneous patches with sharp boundaries, and such a concept is hard to apply in gradient-dominated landscapes. The aim of this study was therefore to investigate the influence of topographic variability (TV) on species richness at the landscape level (gamma (γ) diversity) and on its components (alpha (α) and beta (β) diversity) at microsite and habitat group levels. Using floristic data from 12 "landscapes" of 1 km² we investigated the influence on diversity components of two simple and one complex measures of TV. While the standard deviation (SD) of altitude explained a high proportion of the variation in γ diversity (linear regression model, R²=0.63), the complex measure, SD of solar radiation explained it even better (R²=0.82). There were strong effects of TV on α and β diversity components at the microsite level, but only marginal increases of the diversity components at the habitat level. Further analyses revealed that the missing increase of the habitat level components was caused by differences between habitat groups and that only grassland diversity components increased significantly with TV. We conclude that TV at a landscape scale has strong effects on niche or microsite diversity and is an appropriate estimator of relative species richness in landscapes that are topographically heterogeneous and gradient dominated.     

Temporal Patterns of Species Accumulation in a Survey of Lepidoptera in a Beech-Maple Forest

One of the most significant challenges to insect conservation is lack of information concerning species diversity and distribution. Because a complete inventory of all species in an area is virtually impossible, interest has turned to developing statistical techniques to guide sampling design and to estimate total species richness within a site. We used two such techniques, diversity partitioning and non-parametric richness estimation, to determine how variation in sampling effort over time affected species accumulation for a survey of Lepidoptera in an old-growth beech-maple forest. Temporal scaling of sampling effort had significant effects on two measures of species diversity. Increases in species richness were primarily driven by changes in species occurrences with season, while Shannon diversity was largely determined at the scale of individual sampling units (i.e. by spatial effects). Variation in sampling effort affected the values of the two most widely regarded richness estimators (ICE and Chao 2); neither diversity estimator achieved stable values across a range of sampling efforts. Even after 52 trap-nights and accounting for seasonality, rare species (singletons and uniques) remained a significant component of the moth community. To the extent that moth communities in other forest systems are similarly comprised of many rare species, non-parametric richness estimators should be expected to yield variable estimates with increased effort and should only be used to provide a minimum benchmark for predicting the number of species remaining to be sampled. Our results suggest the best strategy for a short-term survey of forest Lepidoptera should emphasize spreading sampling intervals throughout a given year rather than focusing on intensive sampling during a short time period or prolonged sampling over many years.     

Reduction of bias in estimating the frequency of recessive genes.

The standard approach to estimating the frequency of a completely recessive autosomal gene is to use the maximum-likelihood estimator (MLE), q = square root q2. Since the expectation oof Q using MLE is systematically less than the true value, this estimator always gives a negatively biased estimate of q. Here we describe the bias associated the MLE over a range of q and N values, explore some of the properties of this estimator, and propose new estimators which reduce the bias. We also describe some of the new estimators' properties, as well as the remaining bias associated with them for varying q and N values. We further propose one of these estimators as the one which most effectively reduces bias over a specific q value range of approximately .005 to .05, and which is less biased than JLE over essentially all q and N values. The proposed estimator also is directly compared with MLE in calculating various available estimates of q, demonstrating the percentage of reduction in bias achieved. This reduction varies from negligible for estimates of q above .3 and N greater than 100, to a 23% reduction in bias for a q value of .09 and an N value of 215.