Correspondence site: http://www.respond2articles.com/MEE/

# Comparing measures of species diversity from incomplete inventories: an update

Article first published online: 8 JAN 2010

DOI: 10.1111/j.2041-210X.2009.00003.x

© 2010 The Authors. Journal compilation © 2010 British Ecological Society

Additional Information

#### How to Cite

Beck, J. and Schwanghart, W. (2010), Comparing measures of species diversity from incomplete inventories: an update. Methods in Ecology and Evolution, 1: 38–44. doi: 10.1111/j.2041-210X.2009.00003.x

#### Publication History

- Issue published online: 23 FEB 2010
- Article first published online: 8 JAN 2010
- Received 12 October 2009; accepted 15 November 2009 Handling Editor: Robert P Freckleton

### Keywords:

- alpha diversity;
- effective number of species;
- Shannon’s entropy;
- simulation;
- species richness;
- undersampling bias

### Summary

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

**1.** Measuring biodiversity quantitatively is a key component to its investigation, but many methods are known to be biased by undersampling (i.e. incomplete inventories), a common situation in ecological field studies.

**2.** Following a long tradition of comparing measures of alpha diversity to judge their usefulness, we used simulated data to assess bias of nine diversity measures – some of them proposed fairly recently, such as estimating true species richness depending on the completeness of inventories (Brose, U. & Martinez, N.D. *Oikos* (2004) **105**, 292), bias-corrected Shannon diversity (Chao, A. & Shen, T.-J. Environmental and Ecological Statistics (2003) **10**, 429), while others are commonly applied (e.g. Shannon’s entropy, Fisher’s α) or long known but rarely used (estimating Shannon’s entropy from Fisher’s α).

**3.** We conclude that the ‘effective number of species’ based on bias-corrected Shannon’s entropy is an unbiased estimator of diversity at sample completeness *c. *>0·5, while below that it is still less biased than, e.g., estimated species richness (Brose, U. & Martinez, N.D. *Oikos* (2004) **105**, 292).

**4.** Fisher’s α cannot be tested with the same rigour because it cannot measure the diversity of completely inventoried communities, and we present simulations illustrating this effect when sample completeness approaches high values. However, we can show that Fisher’s α produces relatively stable values at low sample completeness (an effect previously shown only in empirical data), and we tentatively conclude that it may still be considered a good (possibly superior) measure of diversity if completeness is very low.

### Introduction

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

Biodiversity is a multifaceted phenomenon, and the existence of a plethora of methods to measure quantitatively some of its aspects is therefore not surprising (Purvis & Hector 2000). Evaluations of the usefulness and reliability of various measures have a long history in ecological research (e.g. Hurlbert 1971; May 1975; Taylor 1978; Kempton 1979; Wolda 1981; Peterson & Slade 1998; Brose, Martinez, & Williams 2003; Walther & Moore 2005; Beck & Kitching 2007; Chao *et al.* 2009) as they promise to guide ecologists to the most appropriate method for their respective data sets and research questions.

A common problem in many biodiversity studies is the incompleteness of species inventories due to limited field sampling – particularly in very speciose systems such as tropical invertebrates. Only for sessile organisms, relating inventories to the area of (complete inventoried) sampling plots allows circumventing the problem. It is well known that undersampling biases some diversity measures, hereby hindering a correct assessment of biodiversity patterns. Since the 1990′s a variety of undersampling-corrections to species richness counts became popular (Colwell & Coddington 1994; Chazdon *et al.* 1998), aided by easy-to-use and free software (Chao & Shen 2006; Colwell 2006).

Recently, further methods have been made available that address the issue of controlling diversity measures for undersampling bias. Although these new methods (see below) were tested in their original descriptions for achieving the proposed goals, an updated comparison of method performances, including established ones, is lacking up to now.

Using simulated data, we compare a number of still infrequently applied methods for the measurement of alpha diversity to established techniques of particular interest (due to either to their widespread use and/or their perceived applicability to undersampled inventories). A similar assessment of measures for beta diversity is under way and will be published separately. Table 1 provides an overview on the measures included in this study.

Type | Reference | Remarks |
---|---|---|

^{}Measures relating to the known, whole community of *N*individuals are indicated by the subscript TRUE (e.g. S_{TRUE}for true species richness). Sample completeness (Comp_{TRUE}) is calculated as Sobs/S_{TRUE}(note the discrepancy of this definition with ‘estimated completeness’ = Sobs/Sest in the Brose & Martinez (2004); termed ‘coverage’ there) decision framework and some other publications. Formulas for calculations, if not given in the table, can be found in the references.
| ||

Sobs: Observed species richness | Species richness in samples; compared with S_{TRUE} | |

Sest: Estimated species richness (coverage-based decision) | Brose & Martinez (2004) | Decision framework to use the ‘best’ estimator of species richness for a data set; compared to S_{TRUE}. |

H: Shannon’s entropy | Ulanowicz (2001) | Compared to H_{TRUE}. |

eH: Effective number of species | Jost (2006) | Exponent of Shannon’s entropy (e^{H}); compared to eH_{TRUE}. |

Hbc: Bias-controlled Shannon’s entropy | Chao & Shen (2003) | Software by A. Chao available at http://www.chao.stat.nthu.edu.tw/softwareCE.html; compared to H_{TRUE}. |

eHbc: Bias-controlled effective number of species | Jost (2006) | Exponent of bias-controlled Shannon’s entropy (e^{Hbc}); compared to eH_{TRUE}. |

Hfa: Estimating Shannon’s entropy from Fisher’s α | Bulmer 1974 | Hfa ≈0·577 + ln(α); compared to H_{TRUE}. |

eHfa: Estimated effective number of species | Jost (2006) | Exponent of estimated Shannon’s entropy (e^{Hfa}), based on Fisher’s α; compared with eH_{TRUE}. |

FA: Fisher’s α | Fisher et al. (1943) | Curvature parameter of the expected species-abundance relationship; see main text for properties. |

Observed species richness is the most straightforward measure of alpha diversity, but it is also most clearly biased in the presence of undersampling. Brose & Martinez (2004) suggested that the performance of estimators of ‘true’ species richness (Chazdon *et al.* 1998; Colwell 2006) for mobile organisms depends on the sample completeness (i.e. the proportion of observed species, related to those actually present at a site). They formulated a decision framework that leads to the ‘best’ species richness estimator for a given data set, based on a preliminary assessment of completeness. Note that Brose & Martinez (2004) used the term ‘coverage’ to describe sample completeness, but this term had earlier been defined differently (Good & Toulmin 1956).

Shannon’s measure of entropy has an excellent foundation in information theory (see Ulanowicz 2001 for an insightful review). Recently, some of its properties have been ‘rediscovered’ within the concept of ‘effective number of species’ (Jost 2006), which also facilitates its application to alternative approaches of measuring beta diversity (Jost 2007). However, as long as its sensitivity to undersampling persists (May 1975), it is unwise to apply Shannon’s measure to many field study-type biodiversity questions. In an attempt to solve this problem, ‘bias-controlled Shannon’s entropy’ was introduced by Chao & Shen (2003).

Fisher’s α (Fisher, Corbet, & Williams 1943; sometimes also referred to as William’s α) was often recommended as the most reliable assessment of alpha diversity (e.g. Hayek & Buzas 1997; Southwood & Henderson 2000; Magurran 2004), referring mostly to Taylor (1978) who used empirical data on British moths to show superior discrimination ability between sites and, most relevant to this study, no relationship (above a relatively low threshold) with sample size (*n*) in replicates from the same site. However, due to its inherent assumption of a logarithmic series-type rank abundance structure of communities (often corroborated empirically, but poorly understood; see below) and because it is, first and foremost, a curve-fitting parameter without deeper implications and interpretations, it is disregarded by some biodiversity researchers. Even more important is another limitation hinted in Taylor (1978, p. 10): it is ‘an approximation […] justified by the small sample’) that is often not clearly appreciated. Because α is mainly (positively) affected by species numbers (Sobs) and (negatively) by individual numbers in a sample (*n*), the hypothetical, possibly unrealistic (see Discussion) situation of a complete species inventory in a closed community inherently leads to decreasing α values if more individuals are sampled (and no new species can be found). Therefore, Fisher’s α cannot be used to assess the diversity of a completely sampled community (corroborating the limited theoretical foundation of this measure), and α values of incomplete samples cannot therefore be compared with ‘true’ values (see Discussion).

Over 30 years ago Bulmer (1974) and May (1975) pointed out that, with the assumption of a logseries species-abundance distribution (SAD) in samples, Shannon’s entropy (H) can be approximated from Fisher’s α. As ‘true’ values for H are available, we can test the properties of this simple estimate, and therefore indirectly investigate the behaviour of Fisher’s α.

Using data with known properties allowed us to assess bias (Walther & Moore 2005) of the different measures in simulated, repeated sampling. Our primary aims are to uncover (i) which method provides unbiased estimates of the true diversity and (ii) how this judgement depends on sampling intensity (i.e. the completeness of the samples). This should allow a recommendation of ‘best’ methods at least within the compared parameter ranges (see Methods for details). As we could not explore in depth, and even less present, all possible combinations of modelling assumptions (see Methods for details), we set several parameters to values of realistic magnitude from the perspective of the first authors’ field work experience on tropical moths (cf. Beck & Chey 2006, 2008).

### Methods

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

We created artificial data of ecological communities based on realistic, but highly simplified assumptions. SADs follow a log-normal distribution, with parameters of mean (μ = 0·507) and SD (σ = 1·183) similar to those observed in large light-trapping samples of moths from Borneo (>600 individuals each; see Beck & Chey 2008 for data sources and details). Log-normals are widely used to describe SADs (Nekola & Brown 2007), and while there is a wide discussion on SADs regarding mechanism (e.g. Volkov *et al.* 2007) or effects of habitat type (e.g. May 1975; Beck & Chey 2006), field data are often equivocal regarding the best distribution model representing the data (e.g. log-normal, logarithmic series, and other distributions of the same general shape). In any case, we expect that a good diversity measure is robust to small deviations of SADs, making it unnecessary to model, e.g., logarithmic series distributions for comparison. Note, however, that Fisher’s α cannot be interpreted as a diversity if empirical SADs are far from this assumption. Investigation of sample SADs prior to diversity analysis has been advised as a standard procedure (e.g. Southwood & Henderson 2000).

We defined a local community (all species combined) as consisting of *N* ≈ 100 000 individuals (±few individuals due to rounding to integer values; Wolda 1981 used the same community size in simulations). This is admittedly arbitrary and simplifying (see Discussion). As our simulation is not spatially explicit, we had to assure that – even for very species-rich communities – each species occurs in at least several individuals. We also defined the species richness of the community, using several values for comparisons (50, 100, 200 and 300 species). Under the assumption of a log-normal SAD, we could therefore calculate the absolute abundance of individuals per species.

We randomly sampled a defined number of individuals (100, 500, 1000 and 2000) from the community. As some measures required repeated sampling (i.e. species richness estimators), we split these samples in 20 random, equally large (regarding individuals numbers) subsamples. For each parameter setting sampling was carried out 100 times. For the whole community as well as for each sample, we calculated the diversity measures listed in Table 1, as well as the completeness of samples related to the true species richness of the whole community. Note that the various versions of the ‘effective number of species’ (eH, eHbc and eHfa) and the estimation of Hfa are algebraic transformations of respective measures of entropy and α. These transformations (exponentials or logarithms respectively), however, may affect assessments of bias, which is why we included them separately in our comparison. In summary, thus, we mimicked 100 independent samples (consisting of 20 subsamples each) for each of four communities (differing in diversity), and for each of four sampling efforts (i.e. numbers of sampled individuals). We then compared nine different alpha diversity measures with true values of the respective whole communities.

Many more measures of alpha diversity were proposed (see e.g. Hayek & Buzas 1997; Magurran 2004), but we had to restrict our comparison to a handful of methods. We did not, in particular, include rarefaction approaches (e.g. Hurlbert 1971; Simberloff 1978; Mao, Colwell, & Chang 2005), as these, by definition, cannot be compared without relating to sampling effort. Lande (1996) pointed out that Simpson’s index is an unbiased diversity measure with interesting properties for measuring beta diversity, but its use has been discouraged with regards to alpha diversity because of its strong dependency on the few most common species (e.g. Southwood & Henderson 2000).

Although free software is available for most methods employed, we rewrote the calculation of all measures in matlab syntax (http://www.mathworks.com/products/matlab/) and integrated these with sampling simulations, so that the whole routine can be carried out within one run. The *BiodivToolbox* is available as Electronic Supplement in Appendix S1 and can be used, for example, to explore results for other parameter settings. statistica 8 (http://www.statsoft.com) was used for further data analysis.

We calculated the scaled mean error (SME) of sample diversity measures, as compared with true values of the whole community, as a measure of bias (Walther & Moore 2005). Scaled error measures (i.e. relative to true diversities) were applied to facilitate comparison between communities of different diversity. We calculated SDs of the (scaled) error (which is an assessment of precision) and the 95% confidence interval of the mean, i.e. SME (as CI_{M} = SME ± 1·984 × SD/10; 1·984 being the *t*-value associated with 99 degrees of freedom, and 10 being the square root of the 100 samples drawn). We call a diversity measure unbiased if the CI of its SME includes zero.

### Results

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

Table 2 shows all assessments of bias for the different measures. Figure 1 presents SME as a function of completeness for the ‘best candidates’, in our view, of diversity measures. Sobs (not shown) is, naturally, negatively biased over almost the whole range of completeness, whereas Sest loses negative bias at completeness *c. >*0·5, but is positively biased in some data simulations with high levels of sample completeness. Biases and their CIs in entropy estimates (i.e. Shannon’s values H and their estimates; not shown) appear generally smaller than in associated ‘effective numbers of species’ (eH and their estimates), but these are simply transformation effects (entropies are log-transformations of diversities; Jost 2006). Conventional H and eH are negatively biased at almost all levels of sample completeness. Hbc and its associated ‘effective numbers of species’ (eHbc) are unbiased in most simulations with completeness >0·5. Hbc is the measure with lowest bias of all diversity estimates included here (Table 2). Hfa and eHfa show positive bias at low completeness, but strong negative bias at high completeness. The latter may be an effect associated with the peculiarities of Fisher’s α (see below).

S_{TRUE} = 50 | n = 100 | n = 500 | n = 1000 | n = 2000 |
---|---|---|---|---|

^{}Good measures have SME close to zero (no bias). Note that large SME of Fisher’s α were not reported because they are meaningless as true error estimates (see main text).
| ||||

Sobs | −26·1 | −10·9 | −1·9 | −0·1 |

Sest | 10·5 | 6·3 | −0·2 | −0·1 |

H | −7·4 | −1·7 | −0·7 | −0·4 |

eH | −23·3 | −5·5 | −2·3 | −1·3 |

Hbc | −0·3 | −0·3 | −0·3 | −0·3 |

eHbc | −0·7 | −0·9 | −1·2 | −1·1 |

Hfa | −0·5 | −8·7 | −14·8 | −18·7 |

eHfa | −1·1 | −25·1 | −40·0 | −47·4 |

S_{TRUE} = 100 | n = 100 | n = 500 | n = 1000 | n = 2000 |

Sobs | −53·2 | −25·2 | −9·5 | −6·6 |

Sest | −10·1 | 5·3 | 4·9 | 2·8 |

H | −10·9 | −3·0 | −1·2 | −0·6 |

eH | −35·1 | −10·3 | −4·8 | −2·0 |

Hbc | −1·2 | 0·4 | −0·1 | 0·0 |

eHbc | −3·9 | 1·7 | −0·3 | −0·1 |

Hfa | 2·4 | 3·0 | −6·0 | −3·6 |

eHfa | 11·3 | 11·8 | −21·2 | 2·8 |

S_{TRUE} = 200 | n = 100 | n = 500 | n = 1000 | n = 2000 |

Sobs | −72·8 | −33·7 | −21·7 | −9·5 |

Sest | −35·1 | 3·9 | −1·0 | 5·8 |

H | −15·7 | −4·6 | −2·4 | −1·1 |

eH | −49·7 | −19·0 | −9·9 | −5·0 |

Hbc | −4·2 | −0·3 | 0·3 | −0·1 |

eHbc | −15·8 | −1·0 | 1·3 | −0·3 |

Hfa | 1·3 | 1·5 | 4·2 | −4·2 |

eHfa | 8·2 | 7·1 | 20·3 | −17·5 |

S_{TRUE} = 300 | n = 100 | n = 500 | n = 1000 | n = 2000 |

Sobs | −76·5 | −40·3 | −23·9 | −16·8 |

Sest | −41·4 | −3·2 | 1·9 | −1·6 |

H | −19·5 | −5·7 | −3·0 | −1·7 |

eH | −63·2 | −25·4 | −14·3 | −8·0 |

Hbc | −3·9 | −0·4 | 0·1 | 0·0 |

eHbc | −17·2 | −2·1 | 0·4 | 0·2 |

Hfa | 2·3 | 1·2 | −0·6 | 0·4 |

eHfa | 14·3 | 6·5 | −3·2 | 2·0 |

As expected, SME of many diversity measures are positively correlated with sample completeness (irrespectively of community species richness S_{TRUE}), converging to near-zero values at high completeness (Spearman’s rank correlations, *N* = 16, *P* < 0·01: *R* = 0·67 for Sest, *R* > 0·98 for Sobs, H and eH). These measures are also significantly related to sample size (even though S_{TRUE} vary) at *R* > 0·58. However, no relationship was found for Hbc and eHbc (|*R*| < 0·5), while Hfa and eHfa are negatively related to completeness (*R* < −0·83, *P* < 0·01). Sest, Hfa and eHfa are the only measures where we observed (significant) positive bias, i.e. overestimation of true diversity.

Fisher’s α appear highly inaccurate and positively biased if compared in the same manner as the other measures (data not shown), but this is, at least partly, the consequence of its definition (see Introduction). In the absence of a ‘true’α value, we can only assess the criterion of independence from sample completeness. We simulated communities of 50, 100, 200 and 300 species (individual numbers and all other settings as above) and drew 100 replicate samples each of 500 varying sample sizes (*n*). To our knowledge this is the first time α was tested on simulated data (note that with constant _{TRUE}, Comp_{TRUE} is monotonically related to *n*; (cf. Taylor 1978, p. 15; for empirical results). In order to compare the behaviour of this measure to the supposedly ‘best’ measure in our comparison, we calculated eHbc for the same data. Figure 2 shows that Fisher’s α yields a relatively constant value over a large range of sample completeness (with a slight negative trend), but starts to drop increasingly at a completeness *c. >*0·8. eHbc, on the other hand, is positively related to completeness *c. <*0·7 (as indicated in Fig. 1), but produced constant values for sample completeness above that level.

### Discussion

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

For a comprehensive understanding of biodiversity, including its importance in conservation and ecosystem management, it is important that erroneous assessments are avoided along the whole chain of evidence – from choice of taxon, sampling site and sampling technique, to assurance of quality in taxonomy (Bortulus 2008), to an unbiased measurement of diversity and a sound interpretation of what can be inferred from it. Our comparison has been strictly limited to the question of undersampling biases in measuring alpha diversity, whereas other properties of diversity measures may also be relevant to judge their usefulness in relation to the research question at hand. Choosing measures of diversity, as well as other quantitative metrics in ecology, must be guided by both the information required and the suitability of the metric to the available data. Various authors have pointed out that different measures actually refer to different definitions of what we mean by ‘diversity’ (e.g. Hurlbert 1971; May 1975; Southwood & Henderson 2000; Magurran 2004), and even more so if functional and phylogenetic aspects or the nonquantitative, colloquial meanings of the term are included (e.g. Purvis & Hector 2000). Variation in the diversity rank order of communities may be simply a consequence of this.

Concentrating on those measures relating to a characterization of the SADs (i.e. ‘evenness’), some authors concluded that indices that are mostly affected by species of medium abundance (such as Fisher’s α) are preferential over those affected mostly by common species (e.g. Shannon’s entropy) as they tend to produce stable values in repeated sampling of the same community, but high discrimination between different communities (Kempton & Taylor 1974; Taylor 1978; Kempton 1979; using empirical data), and many studies published since found good discrimination of site diversities by this index. Beck & Chey (2008, p. 1455) reported that in their empirical data Fisher’s α were not closely related to sample size, whereas this was the case for other diversity measures such as Sobs and several species richness estimators. Such dependency could be indication for undersampling bias in the latter (cf. Taylor 1978), but may, theoretically, also be a true property of the ecosystems.

Our comparison confirmed the (well known) undersampling bias in observed species richness and Shannon’s entropy (H), and, not surprisingly, also in the ‘effective number of species’ (e^{H}). Measures designed to reduce this bias, such as estimating species richness (following the framework of Brose & Martinez 2004), and Chao & Shen’s (2003) bias correction for Shannon’s entropy, were corroborated to have only a minimum of bias (if any) at sample completeness *c. >*0·5 (i.e. if at least half of the species were sampled). Furthermore, for the first time to our knowledge, we show that ‘effective number of species’ calculated from bias-controlled entropy (eHbc) provides an unbiased measure above this level of sample completeness. This encourages, in combination with other preferential properties (e.g. Jost 2006, 2007, 2009), the use of this measure in biodiversity studies.

The low to intermediate bias of the simple estimates of Hfa and eHfa, particularly at low to intermediate completeness, indirectly support the reliability of α (despite using a log-normal, not a logarithmic series, as underlying SAD) at completeness well below 1, but with the availability of bias-controlled Shannon’s entropy there may be no need to use this measure except if very low completeness is suspected. In such cases, eHfa may provide a useful ‘second opinion’ to eHbc.

As pointed out in the Introduction, we cannot subject Fisher’s α to the same rigour of comparison with ‘true’ values because these cannot, by definition, be calculated under the simplifying assumption that a ‘local community’ contains more than one individual even of the rarest species. Simulations in Fig. 2 show that already before approaching full sample completeness, α values begin to loose their stability displayed at low levels of completeness. However, even in very large sampling programmes (e.g. Woiwod & Harrington 1994; Novotny & Basset 2000; Plotkin *et al.* 2000; Volkov *et al.* 2007) such data have not been found empirically, and the spatially structured nature of real ecological communities (i.e. migration of individuals between neighbouring communities; Leibold *et al.* 2004) may indeed make our simulation unrealistic in this respect. Full sample completeness (i.e. 1 or anywhere near) is certainly not likely in those systems where Fisher’s α is commonly applied, i.e. tropical invertebrate studies. Figure 2 confirms the stability of Fisher’s α through a range of incomplete sampling, and we found, upon visual inspection, that value stability is equal or higher at low levels of completeness (i.e. <0·5–0·7) that those of eHbc. Thus, while our comparison cannot confirm the unbiased character of Fisher’s α for technical reasons, we can neither reject it as long as we assume that samples are clearly incomplete.

### Conclusion

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

We conclude from our results that the ‘effective number of species’ based on bias-controlled Shannon’s entropy (eHbc; Fig. 1) is the most suitable measures of diversity if undersampling is suspected. This is particularly true when sample completeness is above 0·5. Estimated species richness (Sest) was also found reliable above that threshold, but it exhibited stronger bias at low levels of sample completeness than eHbc. We cannot provide clear evidence for or against reversing the positive assessment of Fisher’s α from earlier publications (based on empirical data), but our results (under the conditions of SADs similar to those often found empirically) suggest that it is probably a reliable (i.e. stable) estimator as long as sample completeness is well below 1, and it may even be superior to eHbc at very low levels of sample completeness (Fig. 2). However, various aspects discussed above, for example problems in defining the ‘true’ value, make Fisher’s α a somewhat diffuse metric of diversity.

We found that all measures except Sobs (which is clearly biased) and entropies (due to their log-transformed character) had high variation in random samples (cf. ±2 SD-lines in Figs 1 and 2), particularly at low levels of sample completeness. This low precision prevents error prediction for any single case of sampling. Unexpected results in biodiversity studies with limited sample replication may, thus, often be a consequence of random sampling and are not necessarily in need of any ecological *ad hoc* explanation.

### Acknowledgements

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

This study profited from earlier, insightful discussions with Jakob Fahr, Konrad Fiedler and Mike Curran, and we are particularly grateful to them for alerting us of relevant literature. Comments from two anonymous reviewers contributed to the presentation of the study.

### References

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

- 2006) Are rank-abundance distributions a useful tool of assemblage discrimination in tropical moths? Acta Zoological Sinica, 52, 1148–1154. & (
- 2008) Explaining the elevational diversity pattern of geometrid moths from Borneo: a test of five hypotheses. Journal of Biogeography, 35, 1452–1464. & (
- 2007) Estimating regional species richness of tropical insects from museum data: a comparison of a geography-based and sample-based methods. Journal of Applied Ecology, 44, 672–681. & (
- 2008) Error cascades in the biological sciences: the unwanted consequences of using bad taxonomy in ecology. Ambio, 37, 114–118. (
- 2004) Estimating the richness of species with variable mobility. Oikos, 105, 292–300. & (
- 2003) Estimating species richness: sensitivity to sample coverage and insensitivity to spatial patterns. Ecology, 84, 2364–2377. , & (
- 1974) On fitting the Poisson lognormal distribution to species-abundance data. Biometrics, 30, 101–110. (
- 2003) Nonparametric estimation of Shannon’s index of diversity when there are unseen species in sample. Environmental and Ecological Statistics, 10, 429–443. & (
- 2006) Species Prediction and Diversity Estimation (SPADE) [Software and User’s Guide]. URL http://www.chao.stat.nthu.edu.tw/softwareCE.html [accessed 18 November 2009]. & (
- 2009) Sufficient sampling for asymptotic minimum species richness estimators. Ecology, 90, 1125–1133. , , & (
- 1998) Statistical methods for estimating species richness of woody regeneration in primary and secondary rain forests of northeastern Costa Rica. Forest Biodiversity Research and Modeling – Conceptual Backgrounds and old World Case Studies (eds F.Dallmeier & J.A.Corniskey), pp. 285–309. Smithsonian Institution, Washington, DC. , , & (
- 2006) Estimate S: Statistical Estimation of Species Richness and Shared Species From Samples. URL http://viceroy.eeb.uconn.edu/estimates [accessed 18 November 2009]. (
- 1994) Estimating terrestrial biodiversity through extrapolation. Philosophical Transactions of the Royal Society (B), 345, 101–118. & (
- 1943) The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12, 42–58. , & (
- 1956) The number of new species, and the increase in population coverage, when a sample is increased. Biometrika, 43, 45–63. & (
- 1997) Surveying Natural Populations. Columbia University Press, New York. & (
- 1971) The nonconcept of species diversity: a critique and alternative parameters. Ecology, 52, 577–586. (
- 2006) Entropy and diversity. Oikos, 113, 363–375. (
- 2007) Partitioning diversity into independent alpha and beta components. Ecology, 88, 2427–2439. (
- 2009) Mismeasuring biological diversity. Ecological Economic, 68, 925–928. (
- 1979) The structure of species abundance and measurement of diversity. Biometrics, 35, 307–321. (
- 1974) Log-series and log-normal parameters as diversity discriminants for the Lepidoptera. Journal of Animal Ecology, 43, 381–399. & (
- 1996) Statistics and partitioning of species diversity, and similarity among multiple communities. Oikos, 76, 5–13. (
- 2004) The metacommunity concept: a framework for multi-scale community ecology. Ecology Letters, 7, 601–613. , , , , , , , , , , & (
- 2004) Measuring Biological Diversity. Blackwell, Malden. (
- 2005) Estimating the species accumulation curve using mixtures. Biometrics, 61, 433–441. , & (
- 1975) Patterns of species abundance and distribution. Ecology and Evolution of Communities (eds M.C.Cody & J.M.Diamond), pp. 81–120. Belknap Press, Cambridge. (
- 2007) The wealth of species: ecological communities, complex systems and the legacy of Frank Preston. Ecology Letters, 10, 188–196. & (
- 2000) Rare species in communities of tropical insect herbivores: pondering the mystery of singletons. Oikos, 89, 564–572. & (
- 1998) Extrapolating inventory results into biodiversity estimates and the importance of stopping rules. Diversity and Distribution, 4, 95–105. & (
- 2000) Predicting species diversity in tropical forests. Proceedings of the National Academy of Science, 97, 10850–10854. , , , , , , , , , , , & (
- 2000) Getting the measure of biodiversity. Nature, 405, 212–219. & (
- 1978) Use of rarefaction and related methods in ecology. Biological Data in Water Pollution Assessment: Quantitative and Statistical Analyses (eds K.L.Dickson, J.CairnsJr & R.J.Livingston), pp. 150–165. ASTM STP 652, American Society for Testing and Materials, Philadelphia. (
- 2000) Ecological Methods, 3rd edn. Blackwell Science, Oxford. & (
- 1978). Bates, Williams, Hutchinson – a variety of diversities. Diversity of Insect Faunas (eds L.A.Mound & N.Waloff), pp 1–18. Symposium of the Royal Entomological Society 9, London. (
- 2001) Information theory in ecology. Computers and Chemistry, 25, 393–399. (
- 2007) Patterns of relative species abundance in rainforests and coral reefs. Nature, 450, 45–49. , , & (
- 2005) The concepts of bias, precision and accuracy, and their use in testing the performance of species richness estimators, with a literature review of estimator performance. Ecography, 28, 815–829. & (
- 1994) Flying in the face of change: the Rothamsted Insect Survey. Long-Term Experiments in Agricultural and Ecological Sciences (eds R.A.Leigh & A.E.Johnston), pp. 321–342. CAB International, Wallingford. & (
- 1981) Similarity indices, sample size and diversity. Oecologia, 50, 296–302. (

### Supporting Information

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Acknowledgements
- References
- Supporting Information

**Appendix S1.** User Guide to BiodivToolbox.

As a service to our authors and readers, this journal provides support ing information supplied by the authors. Such materials may be re-organized for online delivery, but are not copy-edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.

Filename | Format | Size | Description |
---|---|---|---|

MEE3_3_sm_Appendix_S1.zip | 32K | Supporting info item |

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.