Species richness:
Which data? Which estimator?

Gotelli, N J; R K Colwell. 2001. Quantifying biodiversity: procedures and pitfalls in the measurement and comparison of species richness. Ecological Letters 4:379-391.

The authors make the distinction between individual-based and sample-based protocols (the latter including replicated quadrats, mist-net samples, arthropod trap contents). MacKinnon lists (m-lists) are neither one nor the other and rarefaction is not possible. They also distinguish between an accumulation curve describing the progress of sampling and a rarefaction curve produced by resampling. (Some reserve ‘rarefaction’ for resampling of individual-based data and refer to the sample-based analogue as ‘randomized’ or ‘smoothed’ accumulation curves.)

It is best not to disaggregate samples, as they reflect patchiness: individual-based rarefaction curves would be unrealistically high. However, for comparison, # of species should be plotted against # of individuals (not # of samples), as sample sizes may vary. 

In case that seems counterintuitive, let me repeat that: use sample-based rarefaction, but compare richness with equal numbers of individuals.

Sample sizes must be large enough, as accumulation curves converge at low numbers of individuals, and identical sampling methods must be used. The shape of the curve depends on relative abundances (for individuals) or patchiness (for samples), evenness producing a steeper initial rise. Hence curves for different sites may cross, leading to different rankings for different sample sizes.

Species-per-individual measures do not work, because the relationship is not linear and varies with sample size. The same applies of species-per-genus, which also depends on the sample size.

Species density (= # of species per unit area) is also not linear, and comparison of different areas requires area based rarefaction (eg quadrat-based). Species density is sensitive to the density of individuals; disturbances or seral succession which reduce the individual density will ipso facto reduce species density!

Rarefaction cannot be used for extrapolation. Various estimators have been developed and tested on data sets which have reached an asymptote, where they work well. However, on non-trivial, hyperdiverse data sets they often do not reach an asymptote, rising in parallel with the species accumulation curve.

Longino, J T; J Coddington; R K Colwell. 2002. The ant fauna of a tropical rain forest: estimating species richness three different ways. Ecology 83:689-702

This paper analyses the data from many years of collections at La Selva, including 6 quantitative sampling methods and intensive species-hunting by entomologists. Records of individual ants are far from independent, so they used “occurrence”, the presence of a species in one trap (this is equivalent to incidence data if the trap is the sample unit). The overall species accumulation curve was close to an asymptote and numbers of uniques and duplicates were both declining, but the curves for only one of the methods (fogging) appeared to be nearing an asymptote. 

They tried three estimators of species richness:

• the area under the fitted lognormal distribution (a theoretical parametric method). A plot of frequency vs log species abundance should be bell-shaped, but the left part of the curve is truncated (chopped off) because the rarer species are not detected. Fitting only makes sense if the peak of the bell curve is visible: this was the case for only the fogging and the aggregated data sets.
• the asymptote of the Michaelis-Menten equation fit to the species accumulation curve (an empirical parametric method). 
• the Incidence-based Coverage Estimator (ICE) (a non-parametric method based on the number of rare species in the data set).

They concluded that none of the estimators was particularly useful; they only gave sensible estimates for the data sets which were close to being complete anyway.

Species richness is ultimately determined by rare species. This study only counted worker ants, so there were probably no ‘tourist’ species in the collections (cf Summerville and Crist’s moths below). Some may be genuinely rare throughout their range, but others were termed ‘edge species’ by Longino et al. "Methodological edge species" are possibly abundant at the site but rarely detected with the method used (eg canopy species turning up in a litter sample), and "geographic edge species" are common in habitats or regions outside the study area. The existence of edge species implies that the species pool being sampled has a fuzzy edge: see O’Hara below. 

Hortal, J; P A V Borges; C Gaspar. 2006. Evaluating the performance of species richness estimators: sensitivity to sample grain size. J Animal Ecology 75:274-287.

Working with terrestrial arthropods in the Azores, the authors investigated the effect of aggregating data for analysis at the level of the island, reserve, transect, trap or ‘record’ (= “occurrence” of Longino et al.). They found asymptotic estimators (Michaelis–Menten, negative exponential, Weibull), Rosenzweig’s F3, F5 and F6 estimators and Ugland’s species-area estimator to be useless. 

Their recommendations for estimators:

They strongly recommend that data at the lowest level, eg traps, should be recorded for future analysis, even if not used immediately. Use of trap-level data produced the most stable and comparable estimates in their study.

Brose, U; N D Martinez; R J Williams. 2003. Estimating species richness: sensitivity to sample coverage and insensitivity to spatial patterns. Ecology 84:2364–2377.

Computer simulations were used to examine the accuracy of different incidence-based species richness estimators. This showed that the number of species observed (Sobs) was among the worst estimators and non-parametric estimators were less biased and more precise than the extrapolation methods. In most data sets, ICE and Chao2 were less accurate than the jackknife estimators. (Chao2 is imprecise with low coverage and biased with high coverage.)

Species richness estimation depends on the relative abundance distribution and sampling intensity, but not on spatial patterns; true richness has little effect. There is a trade-off between precision and bias: precision is most important for comparisons between unreplicated data sets, lack of bias most important when replications are done. The second-order jackknife (Jack2) appears to be the best overall estimator with respect to accuracy.

Their recommended procedure for choosing an estimator:

1. Use a variety of estimators to get an estimated range for true richness (Sest) (but see comments by O’Hara below!)
2. Use this to calculate a range for coverage (ie Sobs / Sest )
3. Select one of the jackknife estimators or the observed species richness depending on coverage:

O'Hara, R B. 2005. Species richness estimators: how many species can dance on the head of a pin? J Animal Ecology 74:375-386

Bob O'Hara raises the issue of the population sampled: estimation is hopeless unless the population is closed. Hence he questions the logic of extrapolating to infinity, rather than to some plausible estimate of the total population. 

Parametric models are difficult, as the observed data depend on both actual relative abundances and species-specific catchability, both unknown. Empirical models are often a poor fit to the observed data (if indeed anyone bothers to check), so have little hope of providing good extrapolations. The bias of non-parametric estimates is unbounded – and often very large – which is a weakness of Brose et al’s scheme. ACE and ICE depend on the definition of rare species, which is arbitrary.

He concludes that “Estimating species richness therefore seems futile, as it is impossible to know how bad the estimates are.” The only useful measures are Chao1 and ACE, when used as estimates of the minimum number of species.

Díaz-Francés, E; J Soberón. 2005. Statistical estimation and model selection of species-accumulation curves. Conservation Biology 19:569-573.

These authors see estimation purely in terms of fitting a curve to the empirical data. They provide software for fitting and comparing 3 parametric curves: exponential, Clench (= Michaelis-Menten) and logarithmic. This works with one of their example data sets (butterflies) but they cannot distinguish between the models for the other, although the extrapolations are quite different.

• Crist, T O; J A Veech; J C Gering; K S Summerville. 2003. Partitioning species diversity across landscapes and regions: a hierarchical analysis of alpha, beta, and gamma diversity. American Naturalist 162:734-743

• Summerville, K S; T O Crist. 2005. Temporal patterns of species accumulation in a survey of Lepidoptera in a beech-maple forest. Biodiversity and Conservation 14:3393-3406.

The first paper describes the partitioning of diversity, which is applicable to a number of diversity indices, including richness, Shannon and Simpson. If gamma is the species richness of a set of samples and alpha is the mean richness of subsets, then beta is defined as the difference between alpha and gamma. Other indices need to be weighted by sample size when averaging. This can be extended to multiple levels: their example involves collections of canopy beetles from individual trees within stands within woodlands within geographical regions. They propose two hypothesis tests, comparing the observed pattern with null models which either shuffle individuals between samples or shuffle samples between subsets. Richness and Shannon give different results, as Shannon measures evenness as well as richness; Simpson gives similar results to Shannon. [Why didn’t they use Shannon’s evenness index and be done with it?] They also used the Morisita index to investigate difference in species aggregation.

The second paper applies this concept to moths collected at light traps in spring and fall (as well as in different locations) and showed a component of beta diversity due to season. Their study underlines the importance of collecting in different seasons and the inadequacy of ‘Bioblitz’ snapshots. Their species accumulation curves were not approaching an asymptote but the number of singletons and uniques did! As a result, ICE and Chao2 were still rising. There is clearly a huge reservoir of rare species: most of these were “edge species” sensu Longino, ie species not attracted to lights or not normally found in these habitats. This reflects O’Hara’s worry about closed species pools, implying that Summerville and Crist’s accumulation curves will not reach an asymptote until they have collected all the moths of N America (if not the planet).

• Kéry, M; H Schmid. 2006. Estimating species richness: calibrating a large avian monitoring programme. J Applied Ecology 43:101-110

• Dennis, R L H; T G Shreeve; N J B Isaac; D B Roy; P B Hardy; R Fox; J Asher. 2006. The effects of visual apparency on bias in butterfly recording and monitoring. Biological Conservation 128:486-492.

Kéry and Schmid checked the detection of birds by observers involved in a Swiss national bird survey and found that there was very little bias due to differences in detectability between species or between observers.

Dennis et al. showed that records of butterflies in UK surveys were biased, with the more obvious butterflies – as defined by size, coloration, height, etc – more likely to be recorded. An exception to this was inconspicuous butterflies which were famously rare: observers appear to keep an eye out for these!

O'Dea, N; R J Whittaker; K I Ugland. 2006. Using spatial heterogeneity to extrapolate species richness: a new method tested on Ecuadorian cloud forest birds. J Applied Ecology 43:189-198.

The so-called Total Species method was devised to estimate the diversity of the benthic fauna of the Norwegian continental shelf on the basis of a few scoops of mud from the sea bed. The differences in species composition of the different local samples is used to estimate beta diversity and this is turn is used to estimate the total species richness of the region.

O’Dea et al tried this for montane birds in reserves in NW Ecuador, with highly heterogeneous habitat and some species confined to individual valleys. They found that distributing sampling sites over the area was sufficient to capture the heterogeneity involved and that use of the Total Species method resulted in overestimates of species richness.

Golicher, D J; R B O'Hara; L Ruíz-Montoya; L Cayuela. 2006. Lifting a veil on diversity: a Bayesian approach to fitting relative abundance models. Ecological Applications 16:202-212.

The authors were interested in frugivorous butterflies in parts of the Montebello National Park, Mexico, some pristine, others affected by fire and disturbance. They tried fitting 2 theoretical curves – based on Preston’s lognormal and Fisher’s log-series – together with a Poisson sampling function. Maximum likelihood estimates of parameters were obtained using Bayesian methods and WinBUGS software. The species pool was defined, and species not occurring in a sample were scored as zero (and fitted scores were calculated) rather than being ignored. The fitted values were summarized as Shannon and inverse Simpson diversity indices. (Getting a figure for ‘species richness’ is problematic, as the fitted abundances for rare species are fractional rather than zero.)

In spite of the limited and patchy data available (many of the traps had been stolen or vandalized), the analysis showed clearly that the most disturbed parts of the park had impoverished butterfly faunas. 

Rennolls, K; Y Laumonier. 2006. A new local estimator of regional species diversity, in terms of ‘shadow species’, with a case study from Sumatra. J Tropical Ecology 22:321-329.

The non-parametric estimators developed by Anne Chao and colleagues use the number of singletons and doubletons (F1 and F2) in the sample to estimate the number of "zerotons" (F0), species not represented in the sample. Rennolls and Laumonier point out that the distinction applies at the sample level, not the population level - if you take another sample from the same population, different species will appear as singletons. The population contains a pool of rare species, a proportion of which will turn up in a sample. They provide a method for estimating the number of 'shadow' singletons, the species that are (approximately) as rare as the singletons in this sample but weren't recorded this time. The same argument is applied to doubletons and other "k-tons", although the number of shadow species quickly declines to zero. Adding the shadow species to Sobs gives Sest.

Applied to tree data for a tropical forest, with many species and a high proportion of singletons, the results are plausible, and very close to those for Chao1 and Jack1. Unlike other estimators, the approach can be used to estimate the distribution of species in the population and hence estimates of diversity indices (Shannon, Simpson, etc) for the population, instead of just for the sample.

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