Coverage-based rarefaction does not quantify species richness

Coverage-based rarefaction does not quantify species richness | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 25 April 2025 V1 Latest version Share on Coverage-based rarefaction does not quantify species richness Author : John Alroy 0000-0002-9882-2111 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.174558077.77764261/v1 597 views 294 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Coverage-based rarefaction (CBR) is a high-profile tool for assessing biodiversity that provides relative species richness estimates. It leverages the Good-Turing index u to interpolate expected richness given a specified level of frequency distribution coverage. In contrast to alternatives such as the Shannon and Simpson indices, CBR’s main appeal is providing values in units of species. CBR is tested against a series of other biodiversity measures. Data are both simulated and empirical, in the latter case drawn from an eclectic global database of terrestrial organisms. First, species counts are simulated under three underlying abundance distributions: the compound exponential-geometric series (CEGS), Poisson log normal, and discretised Weibull. CBR and five other diversity estimators are then computed. Second, diversity estimates are computed for species inventories and then recomputed after excluding the single most common species in each one. Third, randomly selected pairs of inventories are either (1) analysed separately with richness estimates summed, or (2) combined and only then analysed. On average, fitting CEGS in simulation consistently returns an accurate and precise estimate of richness. CBR yields little signal. CEGS returns much the same values regardless of whether empirical data sets include or exclude dominants and regardless of whether they are combined or analysed separately. CBR often overestimates by a large margin when dominants are excluded and underestimates by a large margin when data are combined. CBR does not respond predictably to variation in species richness and cannot reconstruct it when the data have strong internal structure. CBR’s usefulness as a biodiversity indicator is unclear. [1]¿p#1 Coverage-based rarefaction does not quantify species richness Coverage-based rarefaction (CBR) is a high-profile tool for assessing biodiversity that provides relative species richness estimates. It leverages the Good-Turing index u to interpolate expected richness given a specified level of frequency distribution coverage. In contrast to alternatives such as the Shannon and Simpson indices, CBR’s main appeal is providing values in units of species. CBR is tested against a series of other biodiversity measures. Data are both simulated and empirical, in the latter case drawn from an eclectic global database of terrestrial organisms. First, species counts are simulated under three underlying abundance distributions: the compound exponential-geometric series (CEGS), Poisson log normal, and discretised Weibull. CBR and five other diversity estimators are then computed. Second, diversity estimates are computed for species inventories and then recomputed after excluding the single most common species in each one. Third, randomly selected pairs of inventories are either (1) analysed separately with richness estimates summed, or (2) combined and only then analysed. On average, fitting CEGS in simulation consistently returns an accurate and precise estimate of richness. CBR yields little signal. CEGS returns much the same values regardless of whether empirical data sets include or exclude dominants and regardless of whether they are combined or analysed separately. CBR often overestimates by a large margin when dominants are excluded and underestimates by a large margin when data are combined. CBR does not respond predictably to variation in species richness and cannot reconstruct it when the data have strong internal structure. CBR’s usefulness as a biodiversity indicator is unclear. Keywords: compound exponential-geometric series, coverage-based rarefaction, Poisson log normal, shareholder quorum subsampling, species diversity, Weibull distribution Introduction The classical method of rarefaction involves algorithmically or analytically drawing down each species inventory in a study to a fixed number of individuals, which nominally creates a level playing field (Sanders 1968, Hurlbert 1971). This form of rarefaction was long a fundamental method in community ecology, but in recent years it has fallen out of favour amongst field-based researchers. Instead, coverage-based rarefaction (CBR) has largely taken its place as a tool for measuring biodiversity (Roswell et al. 2021). CBR involves estimating the number of species that would be found in a sample with specified frequency distribution coverage sensu Good (1953). A species is covered in a sample if it is represented by at least one individual. The sum of underlying species abundances is overall coverage. These individual abundances are not directly estimable with any accuracy, but the Good-Turing index u is a highly robust estimator of joint coverage (Good 1953). An algorithmic method to compute interpolations to fixed coverage levels was outlined under the name of shareholder quorum subsampling (SQS: Alroy 2010a, 2010b). The name CBR was introduced when an analytical version that is based on combinatorial equations was described (Chao and Jost 2012). Researchers in palaeontology continue to use the name SQS, whereas ecologists almost uniformly refer to CBR. This has generated parallel literature. Examples of deep-time ”SQS” studies of temporal trends at the global scale include Alroy (2010a, 2010b), Close et al. (2018), and Henderson et al. (2022). Ecologists have likewise used ”CBR” to analyse large-scale biogeographic patterns without considering the palaeontological literature (Montes et al. 2021, Kusumoto et al. 2023, Thyrring et al. 2024). CBR is also interesting because the coverage concept had already been used to motivate an important species richness extrapolation index called the abundance coverage estimator (ACE: Chao and Lee 1992). Furthermore, CBR later became the basis of a widespread richness extrapolation method called interpolation and extrapolation (iNEXT: Chao et al. 2014, Hsieh et al. 2016). The purpose of this paper is to assess the performance of CBR using three simple tests. Passing all of them is a necessary if not sufficient condition for accepting any richness metric. Because CBR fails all three times, it is necessary to put aside the notion that CBR is a richness indicator. It may indicate biodiversity in some sense, but the fact that it presents estimates in units of species obfuscates this point. In addition to this matter, relative species richness estimates are less intuitive than absolute estimates. Specifically, calculating relative estimates does not answer a fundamental question many field ecologists want answered – exactly how many species could have been found in their species inventory, but weren’t by chance? Material and methods Abundance distributions Species counts must stem from a sampling process that reflects the structure of real-world abundance patterns. It is self-evident that such data are never completely uniform and that they are also not completely random. Thus, it makes sense that a sample of a given community’s count distribution should mirror system-wide abundances that follow an underlying model allowing for variation. There is a rich literature on such species abundance distributions (McGill et al. 2007, Matthews and Whittaker 2014). Currently, it is thought that two in particular are the most realistic (Baldridge et al. 2016, Antão et al. 2021): the log series (Fisher et al. 1943) and the log normal (Preston 1948), now most often modelled as a Poisson-sampled distribution (the Poisson log normal or PLN: Bulmer 1974, Connolly et al. 2005). However, other models have been discussed in recent years. The Weibull distribution may be a strong alternative (Ulrich et al. 2018). In the current context, the log series is problematic because its underlying equation (Fisher et al. 1943) directly implies that the number of unsampled species is infinite. Thus, it describes a sampling process without any biological reality. Studying the log series is outside of the scope of this paper because the discussion focuses on how richness might be estimated if richness could be estimated. The simulation analyses therefore focus on the PLN, the discretised version of the Weibull (Nakagawa and Osaki 1975), and a third distribution that is of interest because it mirrors the PLN but makes fully distinct assumptions. This distribution is called the compound exponential-geometric series (CEGS). It assumes a geometric sampling process instead of a Poisson process and an exponential underlying abundance distribution instead of a log normal distribution, so it fully complements the PLN. The fact that this model predicts the shape of many ecological count distributions with high accuracy is to be discussed elsewhere. Instead, the focus is on outlining the model, using it to generate simulated data, and using it to estimate richness. Equations A CEGS distribution can be derived as follows. First, note that if underlying abundances are randomly generated, the expected spectrum of their values under an exponential distribution is just E = (–ln U )/λ where U is a random uniform variate ranging from zero to one and λ is a scaling constant. The reason is that –ln U is exponentially distributed by definition. Importantly, the expected mean of E /λ given that U is uniform is just 1/λ. This fact is easily confirmed by trivial simulation. Second, note that the expected mean count under a geometric sampling process is (1 – p )/ p = 1/ p – 1 where p is the governing parameter of the geometric series distribution. This p is nothing other than the unchanging probability that sampling will halt after a given draw, conditional on all previous draws not yet having interrupted the sampling process. The probability mass function (PMF) of the geometric series is therefore (1 – p ) k p where k is the number of successful draws. CEGS assumes that p isn’t fixed: it varies amongst species because each one’s expected count is yielded by a random draw from the exponential distribution. So the mean of the counts tracks an integral of random exponential draws: 1/ p – 1 = ∫ E /λ (1) implying that p = ∫ 1/( E /λ + 1) (2) Any substantive abundance distribution will also allow for non-random variation, with the balance of common and rare species being even in some communities and uneven in others. Thus, distributions like the PLN and Weibull also incorporate what is called a shape parameter. Here, the shape parameter is a power 1/γ to which the term p is raised. Adding it yields this expression, again by algebraic manipulation: p = ∫ 1/( E /λ + 1) γ (3) The PMF then follows by inserting the right-hand side of eqn. 3 into the usual PMF of the geometric series, as stated above. Eqn. 3 has the important property of limiting on p = 1/(1/λ + 1) as γ approaches 1, implying an expected mean abundance of 1/λ at this point because the geometric series is approximated. As γ approaches zero p approaches 1, meaning that sampling is infinitely poor; and vice versa. The proportion of species that have not been sampled in a geometric distribution is just p because (1 – p ) 0 p = p . Therefore, a valid estimator of total species richness under the CEGS model is just 1 – p divided into observed richness. If a maximum likelihood estimation procedure is favoured, CEGS can be fitted by leveraging the standard likelihood equation that is used in R statistical packages such as poilog (Grøtan and Engen 2008) and sads (Prado et al. 2018). The PMF equation is used to find the probability p x that any one draw will result in any one of the observed counts having a value of x , and the joint likelihood is computed as the product of p x across all individual counts. However, maximum likelihood solutions are unstable when there are ridges in a likelihood surface. Therefore, this paper instead uses likelihood differencing (LD). Briefly, an LD calculation involves gridding the parameter space; computing likelihoods at each grid point; finding the sums of absolute values of differences in likelihoods across each point; dividing the values by their sums; multiplying these weights by the parameter values; and summing the products to obtain best estimates. The calculation is not elaborated because it is very similar to a Bayesian estimation procedure using gridded data. The advantages are twofold: first, LD finds values close to steep sides of ridges or peaks in likelihoods and avoids low valleys, which is an intuitive way to integrate the likelihood information; and second, LD is largely independent of the gridding scheme because adding points to a region subdivides the likelihood differences proportionately, decreasing the weight of the points in the region. The number of points and the sum of differences therefore maintain a balance. Thus, LD dispenses with the need for a Bayesian prior. The grids used here span a range defined by an odds calculation: 52 divided by the series 1 to 51 minus 1, yielding the range 52/1 – 1 = 51 and 52/51 – 1 = 0.0196 for each parameter. The cegsLD function fits count data with this procedure. Diversity statistics Seven metrics were applied to data (1) generated by Monte Carlo simulation and (2) drawn from an available literature compilation of empirical species inventories. All of the metrics were computed using provided functions. They were raw richness; relative CBR richness ( sqs ) with a quorum (= sampling coverage target) of 0.5; richness estimated by fitting the CEGS model (eqn. 3) with the cegsLD function and the PLN model with the pln function; Fisher’s alpha ( fisher ); Shannon’s H ( shannon ); and Simpson’s D ( simpson ). The latter three statistics were computed to emphasise two points: (1) they do not capture richness signals in a reliable way either because they do not measure anything equivalent to richness in terms of units (Fisher et al., 1943) or because they downweight the influence of rare species (Hurlbert 1971, Hill 1973); and (2) they are strongly cross-correlated with CBR, so it has the same issues. These facts suggest that there is no easy and obvious alternative to using CBR other than fitting a distribution or using an index. However, indices are not taken up here because commonly used ones such as Chao 1 (Chao 1984) and ACE (Chao and Lee 1992) consistently provide unacceptably low estimates (O’Hara 2005). The reason is that they implicitly assume the underlying distributions are uniform (Alroy 2017). iNEXT (Chao et al. 2014) is also in this family. Being intended as lower-bound estimators only, these three methods are systematically biased not only in practice but by design. They accordingly have an unclear interpretation. Regardless of their clear shortcomings, all of them are widely popular amongst field ecologists (Moreno et al. 2018). To save space, I also do not present richness results based on fitting the discrete Weibull distribution – although this is possible to do. It is sufficent to show that at least two model-fitting approaches do yield accurate estimates on average when their underlying models are correctly assumed. If the Weibull also did so, this would not affect the argument that model-fitting approaches are best. If it did not, then this unsettling observation would be tangential because CEGS actually does. Simulations During each trial of the simulation, a species pool size was generated by randomly drawing an integer from the geometric series with the built-in R function rgeom . The distribution’s parameter p was set to 0.01, yielding an expected mean of 100. Trials were skipped whenever the resulting pool size was less than five. Counts were then generated using the CEGS, PLN, and Weibull models. For each sample, distributions of underlying model parameters were set randomly. The rules used to generate the parameters were designed to yield great variation in the shapes of the distributions. The random generators for these distributions are trivial in R. For CEGS: rgeom(n,1/(rexp(n)/l + 1)^g) where n is the desired number of draws and g and l are respectively γ and λ in eqn. 3. In simulation, λ was the square root of a random exponential variate divided by four; γ was the square root of a random exponential variate. For the PLN: rpois(n,exp(rnorm(n,mean=x,sd=s))) The mean x was set to a random normal variate plus 1, and the standard deviation s was set to a random exponential variate. Shapes again varied greatly. The expected geometric mean abundance was exp(1) = 2.718 and the mean standard deviation of the logged counts was 1. For the Weibull: sample(0:x,n,replace=T,prob=-diff(a^(0:(x + 1))^b)) where x is a large number (here 2 18 = 262,144); a is the scale constant (here set to a random uniform variate ranging from 0.5 to 1); and b is the shape constant (set to a random exponential variate). After being generated, zeroes were removed from each of the count vectors. Statistics were computed only if the data included at least four distinct counts plus at least two individuals falling within at least one count class. Three hundred simulation trials are presented because differences among methods are large. Empirical data The actual species inventories used in the two empirical analyses are drawn from a freely available literature compilation (Alroy 2024) of 3257 tree and terrestrial animal surveys called the Ecological Register (Alroy 2015, 2017). This global data set is unusual for two reasons. First, every survey is matched with direct counts of individuals, making it possible to compute all of the statistics mentioned in this paper. Second, the data include roughly balanced subsets of at least 50 inventories representing each of 13 major ecological groups: trees; large mammal communities dominated by carnivores; small mammal communities dominated by rodents; bats; birds; frogs; lizards; ants; butterflies; dung beetles; mosquitoes; odonates; and orthopterans. Minor groups ranging from turtles to spiders are also represented. Subdominant species richness This paper’s second major test follows from simple logic: if a species richness estimator is any good, it should return the same result regardless of whether the frequency of the most common species (= dominant) is considered. Here, the requirement is tested by computing each diversity value for each species inventory; excluding the dominant; recomputing the values; and multiplying the second set of values by S /( S – 1) where S is the raw number of species. The last step is needed because excluding a species decreases the size of the potentially estimated species pool by definition. A good estimator should be unaffected regardless of which species is excluded. So if raw richness is 11 species, the subdominant count of 10 species should be multiplied by 11/(11 – 1) to become 11. As with the simulation analysis, 300 inventories were randomly selected to illustrate what turn out to be obvious differences. All of the inventories were required to include at least two species with the same count and at least four distinct count classes. The reason is that fitting a non-trivial species abundance distribution model is impossible if a data set includes only three count classes, much less only singletons or doubletons. Therefore, the dominant will at least be a quadrupleton. As a result, not only raw richness but Chao 1 are virtually required to be unaffected by the excluding the dominant species. The reason is that Chao 1’s calculations rest entirely on singleton counts, doubleton counts, and overall raw richness counts. Thus, there is no need to illustrate raw species counts or Chao 1 values. Summed and combined species richness Any good richness indicator must pass a third test: it must show no bias when a combined data set is analysed and when the same data set is split up into two parts, each part is analysed independently, and the resulting two estimates are summed (Alroy 2020). Likewise, it shouldn’t matter in conducting a political poll whether all local government areas are polled separately in proportion to their population sizes or all of them are polled at once at random. The overall percentage of voting preferences should be the same across both treatments. Here, 300 artificial ecological communities were created by randomly drawing two species inventories at a time from the above-mentioned database. In each case, first, all methods were used to obtain species diversity figures for the two inventories and these figures were summed. Second, the inventories were merged and overall figures were computed. The pairs of inventories are almost guaranteed to share no particular species because the data are so eclectic, with global coverage and balanced representation of all the cited groups. At the same time, the test is strong because the likelihood of drawing two inventories with highly similar underying richness levels combined with highly similar count distributions is low. Simulations The simulations allow for intense enough sampling that the median ratios between true and raw richness are not very high: respectively 1.50, 1.19, and 1.38 given the CEGS, PLN, and Weibull distributions (Fig. 1, first column). In other words, estimating richness should be fairly easy with such data. CBR yields a variable signal of richness regardless of this fact and regardless of the underlying abundance model (Fig. 1, second column). The Spearman’s rank-order correlation ρ between true richness and CBR richness is 0.940 given the CEGS model, but only 0.637 and 0.664 for the PLN and Weibull data. By contrast, the same figures are 0.976, 0.902, and 0.885 for the CEGS estimator (Fig. 1, third column). CEGS values are slightly too high when the PLN model is assumed (Fig. 1b), and likewise PLN estimates are too low when CEGS is assumed (Fig. 1a). Both fitting methods are broadly accurate if conservative under a Weibull model. Regardless, offsets are not nearly as extreme as the scatter in the CBR data. Fisher’s α, Shannon’s H , and Simpson’s D (Figs. 1, last three columns) provide decreasingly strong richness signals, as expected given that none of them were formulated as richness indicators per se. This result is a caution to those who might not realise they have such a property. Futhermore, it emphasises that CBR is more in their family than the family of actual richness estimators. All three of these non-richness statistics plus CBR yield values closer to the horizontal line of unity when confronted with PLN data (Fig. 1b) instead of CEGS data (Fig. 1a), albeit with high scatter. The reason is that the standard deviation of underlying abundances in the simulation model is a simple random exponential variate, so the mean is only 1. Meanwhile, the shape parameters assumed in the CEGS simulation are narrow enough to yield tigher distributions than those seen in the PLN data. Subdominant species richness Putting side the most common species in each species inventory greatly increases CBR estimates (Fig. 2a). Bias runs in the same direction for Fisher’s α, Shannon’s H , and Simpson’s D (Figs. 2d – f). PLN fits tend to underestimate richness and show high scatter (Fig. 2c). Finally, there is only a subtle effect when the CEGS model is fitted to the data (Figs. 2b). It is very important to stress that CBR, α, H , and D are biased in a counter-intuitive direction: excluding common species inflates their estimates. The more common the dominant species, the greater the effect is likely to be. By implication, if an inventory happens at random to include high counts of common species (= is stochastically uneven), then all four of these indicators should be biased downward. Fisher’s α is the least affected in this regard: CBR and Simpson’s D are about equally influenced. The sensitivity of CBR to dominant species counts is easy to illustrate. Suppose two inventories each include 100 individuals and the first one includes 99 of the dominant. Meanwhile, the second is an even split amongst 10 species. Obviously, CBR will draw down to 1 individual in the first case almost regardless of the target coverage level (quorum). In the second case, with a quorum of 0.5 it will on average need to draw eight individuals and return about six species (this can be shown using the sqs function). After combining the data, a quorum of 0.5 actually yields a lower total draw and a lower total richness value: respectively, about five and about three. Given almost any inhomogeneity in the evenness of data compartments, CBR will always yield this kind of a result. Summed and combined richness When pairs of inventories inventories are either (1) used to provide separate and summed estimates or (2) analysed after being combined, CBR is again catastrophically biased (Fig. 3a). The high variance mirrors the way it fails to recover the underlying relative richness of almost any simulated inventory (Fig. 1). The reason that CBR summations are almost always too low is that CBR responds strongly to the abundance of the dominant few species (Fig. 2A). Fitting CEGS or the PLN recovers values without substantial bias, despite high variance in the PLN data (Fig. 3b, c). Finally, Fisher’s α, Shannon’s H , and Simpson’s D are not designed to measure species richness sensu stricto. But even if this definitional problem is put aside, they still underestimate whatever it is they are trying to measure when confronted with paired samples (Figs. 3d – f). At the same time, they do actually capture richness signals, which is problematic because it means they also can’t be interpreted as pure indicators of distribution evenness. This explains why it has been proposed that ratios of Hill numbers such as H and D , which cancel out richness signals, should be used to quantify evenness (Hill 1973). Discussion Coverage-based rarefaction strongly appeals to many ecologists. It is easy to think that coverage is a rock-bottom basic measure of sampling quality, so it is easy to think that fair sampling constitutes fixed-coverage sampling. Indeed, Chao et al. (2020) premised a highly-cited paper on the idea that sampling completeness is nothing other than coverage. This paradigm has found acceptance (e.g., Callaghan et al. 2022). But it has two issues. First, it is a definition and not a deduction from any observed fact. Other definitions are just as intuitive. For example, one could infer sampling completeness from the arithmetic mean number of individuals drawn across all species, or better yet the geometric mean. Completeness in any sense is virtually required to increase as the mean increases, so why not? Second, defining coverage as completeness conflates sampling most species well with sampling some species very well. For example, if just one species is sampled indeed very well and if its observed and underlying frequencies are similar, then coverage is ”good”. As the frequency of the dominant species asymptotes on 100%, coverage asymptotes on ”perfect” regardless of how well other species are sampled. The strong response of CBR to such variation in evenness has been noted before on theoretical grounds (Shimadzu 2018) and empirical grounds (Alroy 2020, Henderson et al. 2022), provoking the argument that species richness should instead be extrapolated on the basis of distribution modelling (O’Hara 2005, Willis 2019). Because iNEXT depends on CBR calculations (Chao et al. 2014; Hsieh et al. 2016), the evenness problem also calls iNEXT into question. None of this would be an issue if CBR still consistently returned a richness signal instead of a dominance signal. After all, being a little off depending on the level of dominance is not by itself a fatal flaw – and CBR has considerable appeal because it is sample-size independent both by definition and in practice (Close et al. 2018). But there is no evidence that CBR returns strong richness signals if there are any complications, such as happening to include or exclude one very common species (Fig. 2a), representing a composite abundance distribution (Fig. 3a), or even stemming from any of several realistic abundance distributions (Fig. 1). CBR does replicate (Hill 1973), so it will double if the size of a species pool doubles in a consistent way (see Alroy 2010b for an example). But replicating does not mean reflecting true richness. If ecologists need to have a replicating statistic that also strongly reflects dominance, they have at least two palatable options: Shannon’s H and Simpson’s D . It may be unclear as to why anyone would want to use a statistic like these that blends two unrelated properties when richness, at least, can be measured independently (Figs. 1, 2b, 3b). But if one does, then these metrics are at least widely used throughout the sciences, and they are interesting because of their membership in a single equation family (the Hill numbers: Hill 1973). The true appeal of CBR over H and D is exactly that it provides values in units of species richness. Ecologists find richness easy to understand and intuitive. But in this case, they have been sold a bill of goods: CBR does not consistently recover even the relative magnitude of a community’s species pool. References Adler, D., Kelly, S. T., Elliott, T. and Adamson, J. 2025. vioplot: violin plot. – R package version 0.5.1. – https://CRAN.R-project.org/package=vioplot. Alroy, J. 2010a. The shifting balance of diversity among major marine animal groups. – Science 329: 1191-1194. Alroy, J. 2010b. Geographical, environmental and intrinsic biotic controls on Phanerozoic marine diversification. – Palaeontology 53: 1211-1235. Alroy, J. 2015. The shape of terrestrial abundance distributions. – Sci. 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Violin plots (Adler et al. 2025) showing diversity estimates yielded by seven methods when applied to data simulated under three distributions. True richness and estimates are recorded across 300 simulation trials (see text). y-axes are logged and show ratios of estimates to true richness. In each panel, grey = raw counts of species; red = coverage-based rarefaction estimates based on a target coverage level (quorum) of 0.5; blue = estimates based on fitting the CEGS model; gold = estimates based on fitting the PLN model; purple = Fisher’s α; pink = Shannon’s H ; and orange = Simpson’s D . (a) Data simulated from the compound exponential-geometric series (CEGS). (b) Data simulated from the Poisson log normal (PLN). (c) Data simulated from the discretised Weibull. Figure 2. Diversity estimates yielded by six methods when applied to counts that include or exclude the dominant (most common) species. x-axes show data for 300 randomly selected full species inventories including dominants; y-axes show values recomputed after excluding dominants. (a) Coverage-based rarefaction estimates based on a target coverage level (quorum) of 0.5. (b) Estimates based on fitting the CEGS model. (c) Estimates based on fitting the PLN model. (d) Fisher’s α. (e) Shannon’s H . (f) Simpson’s D . Figure 3. Diversity estimates yielded by six methods when applied to 300 randomly matched pairs of empirical species inventories drawn from the Ecological Register (Alroy 2015, 2024). Summed values (x-axes) are based on computing estimates for each inventory separately and then adding them; combined values (y-axes) are based on merging each pair of inventories and computing one overall estimate. (a) Coverage-based rarefaction estimates based on a target coverage level (quorum) of 0.5. (b) Estimates based on fitting the CEGS model. (c) Estimates based on fitting the PLN model. (d) Fisher’s α. (e) Shannon’s H . (f) Simpson’s D . Supplementary Material File (image1.emf) Download 89.49 KB File (image2.emf) Download 884.92 KB File (image3.emf) Download 877.07 KB Information & Authors Information Version history V1 Version 1 25 April 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords compound exponential-geometric series coverage-based rarefaction poisson log normal shareholder quorum subsampling species diversity weibull distribution Authors Affiliations John Alroy 0000-0002-9882-2111 [email protected] Macquarie University View all articles by this author Metrics & Citations Metrics Article Usage 597 views 294 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation John Alroy. Coverage-based rarefaction does not quantify species richness. Authorea . 25 April 2025. 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