For full access to this pdf, sign in to an existing account, or purchase an annual subscription. In addition, estimates of the additional area required to detect proportion g of the estimated assemblage richness under the Poisson model (Table 3, from Equation 14) are identical to the estimates of the additional number of individuals required to reach proportion g of the estimated assemblage richness under the multinomial model (Table 2, from Equation 11). j Today, rarefaction has grown as a technique not just for measuring species diversity, but of understanding diversity at higher taxonomic levels as well. Sanders, H. L. Marine benthic diversity: a comparative study. 2004, their Table 2). An example of sample-based rarefaction curves. This analytic formula was first derived by Shinozaki (1963) and rediscovered multiple times (Chiarucci et al. This estimator has long been called Mao Tau in the widely used software application EstimateS (Colwell 2011). Two important measures of biodiversity with regards to spatial scale are alpha and beta diversity. 1 The statistical technique or method used to evaluate species richness from the results of sampling is rarefaction. n The total number of species observed in the reference sample is Sobs (only species with Yi > 0 contribute to Sobs). sample-based interpolation, extrapolation and prediction of number of additional sampling units required to reach gSest, under the multinomial product model, for ant samples from five elevations in northeastern Costa Rica (Longino and Colwell 2011). {\displaystyle \sum _{i=1}^{K}N_{i}=N} j Because each sampling unit is a 1-m2 plot, what Fig. Colwell et al. When sample size is not sufficiently large, the unconditional variances tend to overestimate and, thus, produce a conservative confidence interval. individual-based interpolation (rarefaction) and extrapolation from two reference samples (filled black circles) of beetles from southwestern Costa Rica (Janzen 1973a, 1973b), illustrating the computation of estimators from Fig. = In ecology, rarefaction is a technique to assess species richness from the results of sampling. In Fig. i 1 Oxford, 1999. Table 6)for sample-based models. It is often done by subsampling without replacement, which means that each read that is selected and assigned to the normalized sample will not be included in the original pool of samples. 4b. We postpone specification of f^0, which estimates the species present in the assemblage but not observed in the reference sample, for a later section. f 4b) (Longino and Colwell 2011), which spans an elevation gradient from lowland rainforest at 50-m elevation to montane cloud forest at 2000 m, is an excellent example. In general, lack of overlap between 95% confidence intervals (mean plus or minus 1.96 SE) does indeed guarantee significant difference in means at P 0.05, but this condition is overly conservative: samples from normal distributions at the P = 0.05 threshold have substantially overlapping 95% confidence intervals. 2, even though the Osa old-growth site extrapolation for large sample sizes exhibits high variance, the old-growth and second-growth confidence intervals do not overlap for any sample size considered. 1985), although sampling designs are nonetheless critical to avoiding bias from spatial structure (Collins and Simberloff 2009; Chiarucci et al. = "Thus rarefaction generates the expected number of species in a small collection of n individuals (or n samples) drawn at random from the large pool of N samples.". But consider two equally abundant species in the same assemblage, one with a very patchy spatial distribution and the other with all individuals distributed independently and at random. ; DEB-0424767 and DEB-0639393 to R.L.C. (a) individual-based interpolation (rarefaction) and extrapolation from three reference samples (filled black circles) from 1-ha tree plots in northeastern Costa Rica (Norden et al. For extrapolation, the SE values are relatively small up to a doubling of the reference sample, signifying quite accurate extrapolation in this range. 4a. Hurlbert, S. H. The Nonconcept of Species Diversity: A Critique and Alternative Parameters. For this reason, the accuracy of our extrapolation and variance estimators is of course dependent upon the accuracy of the asymptotic richness estimates they rely upon. 4b plots on the species axis are actually estimates of species density, the number of species in multiples of a 1-m2 area. Species richness in the old-growth plot (LEP old growth, shown in red) consistently exceeds the richness in second-growth plot, LEP second growth (29 years old, shown in green) and Lindero Sur second growth (21 years old, shown in blue). [ Maximum species density is found at the 500-m elevation site, consistently exceeding the species density at both higher and lower elevations. A sample-by-species incidence matrix was therefore produced for each of the five sites. We use the term assemblage to refer to the set of all individuals that would be detected with this sampling method in a very large sample. n 1 Due to the prevalence of rare species in old-growth tropical forests and widespread dispersal limitation of large-seeded animal-dispersed species, tree species richness is slow to recover during secondary succession and may require many decades to reach old-growth levels, even under conditions favorable to regeneration. 2, with the Poisson variables in Fig. We postpone specification of Sest for a later section. In addition to applying estimators based on the multinomial model, we also analysed the Janzen beetle dataset with estimators based on the Poisson model, including Coleman area-based rarefaction (Equations 6 and 7), area-based extrapolation (Equations 12 and 13), and estimation of the additional area required to detect proportion g of the estimated assemblage richness Sest (Equation 14). Rescaling to incidences can also be useful for any organisms that, like ants, live colonially or that cannot be counted individually (e.g. However, when sample-based rarefaction curves are used to compare taxon richness at comparable levels of sampling effort, the number of taxa should be plotted as a function of the accumulated number of individuals, not accumulated number of samples, because datasets may differ systematically in the mean number of individuals per sample. An example of this phenomenon can be seen in the lower two curves of Fig. For assemblages with many rare species, the incidence-based coverage estimator (ICE; ICE takes into account the frequency counts for rare species (, Estimation of the size of a closed population when capture probabilities vary among animals, Non-parametric estimation of the number of classes in a population, Estimating the population size for capture-recapture data with unequal catchability, Sufficient sampling for asymptotic minimum species richness estimators, Estimating the number of shared species in two communities, Estimating the number of classes via sample coverage, Nonparametric prediction in species sampling, Statistical methods for estimating species richness of woody regeneration in primary and secondary rain forests of NE Costa Rica, Forest Biodiversity Research, Monitoring and Modeling: Conceptual Background and Old World Case Studies. Table 6) retains any information on the spatial structure of the biological populations sampled. (See the Discussion for information on approximating species richness from species density.) estimators based on the Bernoulli product model), using replicated incidence data (or sample-based abundance data converted to incidence), perform better in this regard as they retain some aspects of the spatial (or temporal) structure of assemblages (Colwell et al. 4a, and (ii) the extrapolated estimate S~ind(n+m*), where m* ranges from 0 to 1500, 1200 or 1400 individuals (for the three samples), so that all samples are extrapolated to roughly 2400 individuals, along with the unconditional SE (Equation 10). 1b), and the third based on a Bernoulli product distribution, for incidence frequencies among sampling units (Fig. Janzen (1973a, 1973b) tabulated many data sets on tropical foliage insects from sweep samples in southwestern Costa Rica. (2004, their Equation 5) provide a mathematically equivalent equation in terms of the incidence frequency counts Qk similar to our Equation (4). Please submit a detailed description of your project. {\displaystyle X_{n}} (Each incidence is the occurrence of one species in one sampling unit.). Based solely on information in the reference sample of n individuals or the individuals from area A, counted and identified to species, we have these six complementary objectives for abundance-based data (Fig. All our examples (Tables 2, 3, 5 and 7; Figs 2 and 4) reveal that the unconditional variance increases sharply with sample size for extrapolated curves, and thus, the confidence interval expands accordingly. Rarefaction involves the selection of a certain number of samples which is either equal or less than to the number of samples (in the smallest sample), and then randomly discarding reads from the larger samples until the number of remaining samples is equal to the threshold. The numbers on the ordinate show the magnitude of the multinomial estimate minus the Poisson estimate, in ordinary arithmetic units, scaled logarithmically only to spread out the values vertically so they can be seen. For assemblages with many rare species, the abundance-based coverage estimator (ACE) (Chao and Lee 1992; For the Bernoulli product model (sample-based rarefaction), we need to estimate the expected number of species, Instead, we have only the incidence reference sample to work from, with observed species incidence frequencies, For the Bernoulli product model, the extrapolation problem is to estimate the expected number of species, The extrapolation estimators for the Bernoulli product model require either an estimate of. Datasets range from 200 sampling units (with only 270 incidences) at the 2000 m site, up to 599 sampling units (with 5346 incidences) at the 50-m site. The multinomial model assumes that the sampling procedure itself does not substantially alter relative abundances of species (p1,p2,,pS). Equations (18) and (19), above, both require an estimate of Q0, the number of species present in the assemblage but not detected in any sampling units. Chazdon RL, Colwell RK, Denslow JS. 128-131. For interpolation and extrapolation, the difference is always less than one-tenth of one individual (assuming for the Poisson model that individuals are randomly and independently distributed in space, so that a/Am/n). ); the Taiwan National Science Council (97-2118-M007-MY3 to A.C.); and the University of Connecticut Research Foundation (to R.L.C.). A more diverse ecosystem tends to be more productive and has a greater ability to withstand environmental stresses. = If these assumptions are not met, the resulting curves will be greatly skewed.[8]. Rarefaction assumes that the number of occurrences of a species reflects the sampling intensity, but if one taxon is especially common or rare, the number of occurrences will be related to the extremity of the number of individuals of that species, not to the intensity of sampling. 0 To model species aggregation explicitly, the current models could be extended to a negative binomial model (a generalized form of our Poisson model; Kobayashi 1982, 1983) and to a multivariate negative binomial model (a generalized form of our multinomial) model. Raw species richness counts, which are used to create accumulation curves, can only be compared when the species richness has reached a clear asymptote. 1c): (i) to obtain an estimator S~sample(t) for the expected number of species in a random set of t sampling units from the T sampling units defining the reference sample (t < T), (ii) to obtain an estimator S~sample(T+t*) for the expected number of species in an augmented set of T + t* sampling units (t* > 0) from the assemblage, given Sobs, and (iii) to find a predictor t~g* for the number of additional sampling units required to detect proportion g of the estimated assemblage richness Sest. 2004; Gotelli and Colwell 2001; Smith et al. The number of species in the plot of intermediate age, LEP second growth, significantly exceeds the number of species in the youngest plot, Lindero Sur, for sample sizes between 500 and 1600 individuals, based conservatively on non-overlapping confidence intervals. 1998; Colwell and Coddington 1994; Kobayashi 1982). Rarefaction curves are necessary for estimating species richness. Because each sampling unit is a 1-m2 plot, in the ant study, what Fig. Species density drops significantly with each increase in elevation above 500 m, based conservatively on non-overlapping confidence intervals. On the other hand, beta diversity is the ratio between alpha diversity and regional diversity. The extrapolation is extended to 1000 samples for each elevation. In a paper criticizing many methods of assaying biodiversity, Stuart Hurlbert refined the problem that he saw with Sanders' rarefaction method, that it overestimated the number of species based on sample size, and attempted to refine his methods. We postpone specification of Sest for a later section. Sample-based approaches (e.g. This variance is based on an approach similar to that used by Burnham and Overton (1978) for a jackknife estimator of population size in the context of capturerecapture models. (Boussarie, 2018). 2003) and the Poisson model (Chao and Shen 2004), as well as to methods for predicting the number of additional individuals (multinomial model, Chao et al. is less than K whenever at least one group is missing from this subsample. j i ) f K The results for the Osa old-growth beetle sample appear in Table 3a and the results for the Osa second-growth beetle sample in Table 3b. Clearly the old-growth assemblage is richer, based on these samples. We are grateful to Fangliang He and Sun Yat-sen University for the invitation to contribute this paper to a special issue of JPE and to an anonymous reviewer for helpful comments. In a rarefied sample we have chosen a random subsample n from the total N items. This means that rounding to the nearest individual consistently yields precisely the same values under both models. Based solely on information in the incidence reference sample of T sampling units, we have these three complementary objectives for sample-based incidence data (Fig. We recommend R = 10 as rule of thumb, with exploration of other values suggested for samples with large coefficients of variation. The sample-based approach accounts for patchiness in the data that results from natural levels of sample heterogeneity. Clearly the number of species at any plotted sample size (beyond very small samples) is significantly greater for LEP old growth than in either of the two samples from second-growth forest. The two samples may be drawn from either the same assemblage or from two different assemblages. Rarefaction curves produce smoother lines that facilitate point-to-point or full dataset comparisons. Rarefaction only works well when no taxon is extremely rare or common[citation needed], or when beta diversity is very high. 1b substituted for the multinomial variables in Fig. , With rescaling to individuals, however, strong among-sample differences in dominance can produce misleading results. (b) Sample-based interpolation (rarefaction) and extrapolation for reference samples (filled black circles) for ground-dwelling ants from five elevations on the Barva Transect in northeastern Costa Rica (Longino and Colwell 2011) under the Bernoulli product model, with 95% unconditional confidence intervals. In a sample-based study of the same assemblage, however, the aggregated species will generally have a lower incidence frequency (since many individuals will end up some samples and none in others) than the randomly distributed species. individual-based interpolation, extrapolation and prediction of additional individuals required to reach gSest, under the multinomial model, for beetle samples from two sites on the Osa Peninsula in southwestern Costa Rica (Janzen 1973a, 1973b). M Also, an ecosystem with greater species richness has higher productivity making it more sustainable and stable and could respond to more catastrophes. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide, This PDF is available to Subscribers Only. is defined as: Individual-based interpolation and extrapolation, under the multinomial model, for tree samples from three forest sites in northeastern Costa Rica (Norden et al. = For the Osa old-growth site (Table 2a; Fig. Burnham and Overton 1978). The fundamental statistics for all these estimators are the abundance frequency counts fkthe number of species each represented by exactly Xi = k individuals in a reference sample (e.g. = K For this reason, we do not plot the results from the Poisson model because the figure would be identical to Fig. Janzens study recorded 976 individuals representing 140 species in the Osa second-growth site and 237 individuals of 112 species in the Osa old-growth site. 2b), the extrapolation is extended only to double the reference sample size (not fully shown in Fig. [5] The issue of overestimation was also dealt with by Daniel Simberloff, while other improvements in rarefaction as a statistical technique were made by Ken Heck in 1975.[6]. This makes the data retained as a count data which allows it to be used for further analyses using other statistical tools. 2c, open point), using the multinomial model (Equation 4), the ordering of the two sites is reversed. K 2009) or the amount of additional area (Poisson model, Chao and Shen 2004) needed to reach a specified proportion of estimated asymptotic richness. 2a and c; (ii) the extrapolated estimate S~ind(n+m*) (Equation 9), where m* ranges from 0 to 1000 individuals, along with the unconditional SE (Equation 10); and (iii) the number of additional individuals m~g* required to detect proportion g of the estimated assemblage richness (Equation 11), for g = 0.3 to 0.9, in increments of 0.1. richness estimated by the multinomial model versus the Poisson model for the Osa old-growth beetle sample (Janzen 1973a, 1973b). f For the sample-based incidence example in this paper, we have used the Chao2 estimator, above, which Chao (1987) showed is a minimum estimator of asymptotic species richness. Species diversity is a measure of biological diversity in a specific ecological community. 2008). It examines the number of species present in a given sample, but does not look at which species are represented across samples. 2009). Thus, two samples that each contain 20 species may have completely different compositions, leading to a skewed estimate of species richness. the number of groups still present in the subsample of "n" items Rarefaction does not provide an estimate of asymptotic richness, so it cannot be used to extrapolate species richness trends in larger samples.[9]. The ant dataset (Fig. Deriving Rarefaction: For both samples, the unconditional variance, and thus the 95% confidence interval, increased with sample size. All rights reserved. For small samples, we suggest estimating variance by non-parametric bootstrapping. Analytical methods (classical rarefaction and Coleman rarefaction) have existed for decades for estimating the number of species in a subset of samples from an individual-based dataset. For most assemblages, no sampling method is completely unbiased in its ability to detect individuals of all species (e.g. n (2009) compared species composition of trees, saplings and seedlings in six 1-ha forest plots spanning three successional stages in lowland forests of northeastern Costa Rica. From this it follows that 0 f(n) K. ( Payton et al. This implies that beetle species richness for any sample size is significantly greater in the old-growth site than that in the second-growth site for sample size up to at least 1200 individuals. Even though the mathematical derivations for interpolation and extrapolation are fundamentally different, the interpolation and extrapolation curves join smoothly at the single data point of the reference sample. = We assume that, in most biological applications, the biological populations in the assemblage being sampled are sufficiently large that this assumption is met. = Under the Poisson model, individual-based rarefaction curves and species accumulation curves, because they rely on area, assume that individuals are randomly distributed in space, within and between species. This can also be used to infer whether a group of samples are from the same community. 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Accuracy of microbial community diversity estimated by closed- and open-reference OTUs. For interpolation to samples smaller than the reference sample, these correspond to classical rarefaction (Hurlbert 1971), Coleman rarefaction (Coleman 1981) and sample-based rarefaction (Colwell et al. Most commonly, the number of species is sampled to predict the number of genera in a particular community; similar techniques had been used to determine this level of diversity in studies several years before Sanders quantified his individual to species determination of rarefaction. We postpone specification of an estimator for Q0 for the next section. Rarefaction is unrealistic in its assumption of random spatial distribution of individuals. = Under all three of the models we discuss, all our estimators for extrapolated richness, as well as all our unconditional variance estimators, require an estimate of asymptotic species richness for the assemblage sampled. ; DEB-0541936 to N.J.G. One can plot the number of species as a function of either the number of individuals sampled or the number of samples taken. 2b) yielding a quite accurate extrapolated estimate with a narrow confidence interval. j In Fig. Therefore the rarefaction curve, Comparison of the results for the Poisson model estimators (Table 3) with the corresponding results for the multinomial model estimators (Table 2) reveals a remarkable similarity that makes sense mathematically because the distribution for the Poisson model (Equation 2), conditional on the total number of individuals, is just the multinomial model (Equation 1). Consider a species assemblage consisting of S different species, each of which may or may not be found in each of T independent sampling units (quadrats, plots, traps, microbial culture plates, etc.) The number of species present in the assemblage but not detected in the reference sample is thus represented as f0. The species frequency counts for the three plots appear in Table 4. Copyright 2022 CD Genomics. multiple stems of stem-sprouting plants or cover-based vegetation data). Extra parameters that describe spatial aggregation would need to be introduced in the generalized model, and thus, statistical inference would become more complicated. j For these datasets, abundances can first be converted to incidences (presence or absence) before applying incidence-based rarefaction. (a) Osa old-growth forest sample. {\displaystyle \sum _{j=1}^{\infty }M_{j}=K} We selected two beetle data sets (Osa primary and Osa secondary) to compare beetle species richness between old-growth forest and second-growth vegetation on the Osa Peninsula. ) {\displaystyle f(0)=0,f(1)=1,f(N)=K} 1a and b): (i) to obtain an estimator S~ind(m) for the expected number of species in a random sample of m individuals from the assemblage (m < n) or (ii) an estimator S~area(a) for the expected number of species in a random area of size a within the reference area of size A (a < A); (iii) to obtain an estimator S~ind(n+m*) for the expected number of species in an augmented sample of n + m* individuals from the assemblage (m* > 0), given Sobs, or (iv) an estimator S~area(A+a*) for the expected number of species in an augmented area A + a* (a* > 0), given Sobs; and (v) to find an predictor m~g* for the number of additional individuals or (vi) the additional area a~g* required to detect proportion g of the estimated assemblage richness Sest. As a consequence, confidence intervals that do not overlap at moderate sample sizes may do so at larger sample sizes, even if the extrapolated curves are not converging. The proposed unconditional variances perform satisfactorily when sample size is relatively large because they were derived by an asymptotic approach (i.e. ); the US Department of Energy (022821 to N.J.G. N n From the unstandardized raw data (the reference samples), one might conclude that the second-growth site has more beetle species than the old-growth site (140 vs. 112; Fig. The row sum of the incidence matrix, Yi=j=1TWij, denotes the incidence-based frequency of species i, for i = 1,2, , S. The frequencies Yi represent the incidence reference sample to be rarefied or extrapolated. Longino and Colwell (2011) sampled ants at several elevations on the Barva Transect, a 30-km continuous gradient of wet forest on Costa Rica's Atlantic slope. We see little reason, for individual-based data, to recommend computing estimators based on one model over the other (although Coleman curves are computationally less demanding than classical rarefaction), and no reason whatsoever to compute both. 2b. 1 (c) Lindero Sur younger (21 years) second growth, Copyright 2022 IBCAS and the Botanical Society of China, Copyright 2022 Oxford University Press. assuming the sample size is large). N Rarefaction can be used to determine whether a specific sample has been sufficiently sequenced to represent its identity. The ability to link rarefaction curves with their corresponding extrapolated richness curves, complete with unconditional confidence intervals, helps to solve one of most frustrating limitations of traditional rarefaction: throwing away much of the information content of larger samples, in order to standardize comparisons with the smallest sample in a group of samples being compared. N This curve is created by randomly re-sampling the pool of N samples several times and then plotting the average number of species found on each sample. N Published by Oxford University Press on behalf of the Institute of Botany, Chinese Academy of Sciences and the Botanical Society of China. 1 However, if and when better estimators of assemblage richness become available, they can simply be plugged into our equations wherever Sest, f^0, or Q^0 appear in our equations. Individual-based rarefaction of abundance data, like the interpolation analysis above, has been carried out in this way for decades. For the first time, we have linked these well-known interpolation approaches with recent sampling-theoretic extrapolation approaches, under both the multinomial model (Shen et al. {\displaystyle X_{n}=} Ni = the number of items in group i (i = 1, , K). (2004, their Equation 6) developed an estimator for the unconditional variance in terms of the frequency counts Qk, similar to our Equation (5), that requires an incidence-based estimator Sest for assembly richness S. We postpone specification of Sest for a later section. The incidence frequency counts for the five sites appear in Table 6. species incidence frequency counts for ant samples from five elevations in northeastern Costa Rica (Longino and Colwell 2011). Colwell et al. 2004, their Equation 6) that are used to construct the 95% confidence intervals shown in Fig. Norden et al. From, If we assume that individuals are randomly and independently distributed in space, then, For the multinomial model, the extrapolation problem is to estimate the expected number of species, For the Poisson model, the objective is to estimate the expected number of species, Several estimators in the previous two sections require either an estimate of. 2011). Here, we apply individual-based rarefaction and extrapolation to the same reference sample for the first time. Smith and Grassle (1977) provide an unconditional variance formula of S~ind(m), but their expression for the variance is difficult to compute. i On the other hand, the distribution or evenness of the species present in that area is termed as species evenness. 2. . Because the MVUE is the same for the hypergeometric and the multinomial models, we can relax our assumption about sampling effects on assemblage abundances. In microbial ecology, a common initial approach to assess the difference between environments is through the analysis of alpha diversity of amplicon sequencing data. [2] Rarefaction techniques are used to quantify species diversity of newly studied ecosystems, including human microbiomes, as well as in applied studies in community ecology, such as understanding pollution impacts on communities and other management applications.
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