EstimateS 8.0 User's Guide
Last Revised November 12, 2006
Copyright 2006 by Robert
K. Colwell, Department of Ecology & Evolutionary Biology,
University of Connecticut, Storrs, CT 068693043, USA
Website: http://purl.oclc.org/estimates or http://viceroy.eeb.uconn.edu/estimates
Table of Contents
Introduction
Samples and
Species, Abundance and Incidence
Richness and Species Diversity
(Diversity Menu)
Rarefaction
and Species Accumulation Curves
Random Resampling
for Richness Estimators and Diversity Indexes
Species
Richness Estimators
Estimating
Total Species Richness by Functional Extrapolation
The Shuffle Option for
Exploring the Effects of Patchiness
Option to Export
Results from Individual Randomizations
Selecting and
Testing a Random Number Generator
Shared Species and Similarity
Indexes (Shared Species Menu)
CoverageBased
Estimator of Shared Species
Classic
Similarity Indexes
Chao's
Abundancebased Jaccard and Sørensen Indexes and Their Estimators
What EstimateS 8 Computes
How to Use EstimateS
Preparing the
Input (Data) File: General Specifications
Detailed
Specifications for the Input File
Running EstimateS
What to Do Next (All
Optional)
Things You Should Know Before You
Begin
Caveat Receptor
Citing EstimateS
What You Must Agree To:
Copyright and Fair Use
Sharing EstimateS With
Others
References Cited
Appendix A: SampleBased Rarefaction Curve
(Expected Species Accumulation Curve)
Appendix B: Nonparametric Estimators of
Species Richness
Appendix C: CoverageBased Estimator of
Shared Species
Appendix D: Chao's Abundancebased Jaccard
and Sorensen Similarity Indexes
Introduction
EstimateS 8 is a free software application for Windows and Macintosh operating systems that computes a variety of biodiversity functions, estimators, and indexes based on biotic sampling data. For an overview of major features, click here.
Samples and Species, Abundance and Incidence
In this Guide, the term sample refers to any list of species or OTUs from a locality, site, quadrat, trap, time unit, clone library, or some other entity. EstimateS expects sets of samples that share some or all species.
Some estimators and indexes require species abundance data (counts) for each species in each sample. This is called abundance data. Other estimators and indexes require only presence/absence (occurrence) data for each species in each sample. This is called incidence data. When comparing the biotic (species) composition of two or more localities (or habitats, treatments, seasons, etc.), you can do so either using abundance data, or using summed incidence data (frequencies of occurence, pooled among samples) for each or two or more sample sets.
Richness and Species Diversity (Diversity Menu)
Rarefaction and Species Accumulation Curves
Samplebased Rarefaction (species accumulation curves). EstimateS 7 and later versions (indicated as "EstimateS 7+" in this Guide) computes expected species accumulation curves (samplebased rarefaction curves in the terminology of Gotelli & Colwell 2001), with 95% confidence intervals, using the analytical formulas of Colwell et al. (2004), detailed in Appendix A of this Guide. (See also Mao et al. [2004]. An equivalent formula for expected richness, but not for the variance, was independently developed by Ugland et al. [2003]). This method is a precise equivalent of, and completely replaces the resampling technique used in previous versions of Estimates for computing these curves (the variable Sobs). The expected richness function is called Mao Tau in EstimateS output.
The traditional, resampled Sobs is nonetheless still produced by EstimateS 7+, so you can see for yourself that Mao Tau is yielding the expected values of Sobs, and so you can still produce nonrandomized accumulation curves if you wish.
Because they are computed analytically, Mao Tau and its confidence limits do not require any resampling runs in EstimateS. If expected species accumulation curves (and/or Coleman curves) are all you need, you can set the number of runs to 1 in the Diversity Settings screen. Resampling is still required in Estimates 7+ to compute singletons, doubletons, duplicates, uniques, and all richness estimators for accumulating samples (see below).
To compare datasets in terms of species richness instead of species density, Gotelli & Colwell (2001) recommend rescaling of the expected species accumulation (Mao Tau) curves (and their 95% confidence interval curves) by individuals, instead of leaving them scaled by samples. To allow this rescaling to produce smooth curves, EstimateS 7+ computes the number of individuals for each sampling level, instead of taking the mean for number of individuals, among resampling runs. (If there are N individuals, total, in Q(max) samples, total, the expected number of individuals in Q samples is just [Q / Q(max)] * N; these are the values tabled by EstimateS 7+ for the Individuals column of the output.)
Individualbased Rarefaction. EstimateS also computes classic, individualbased rarefaction curves (in the terminology of Gotelli & Colwell 2001) or Coleman curves for samplebased abundance data. Like samplebased rarefaction curves, these curves are computed analytically, not by resampling.
As discussed by Colwell & Coddington 1994, and independently reported by Brewer & Williamson (1994), Coleman curves and classical rarefaction curves (for sampling without replacement) are identical to three or four decimal places for most datasets. The Coleman curve is orders of magnitude more efficient computationally, so EstimateS computes Coleman curves. See Gotelli & Colwell (2001) and Colwell et al. (2004) for more on Coleman curves.
Because they are computed analytically, Coleman curves do not require any resampling runs in EstimateS. If Coleman (and/or expected species accumulation curves) are all you need, you can set the number or runs to 1 in the Diversity Settings screen. Resampling is still required in Estimates 7 to compute singletons, doubletons, duplicates, uniques, and all richness estimators for accumulating samples (see below).
Rarefaction vs. richness estimation. Please note that neither samplebased rarefaction curves (MaoTau = expected species accumulation curves) nor individualbased rarefaction curves (Coleman curves) are estimators of species richness in the same sense as the other estimators that EstimateS computes. Whereas Chao2, ICE or Jack1, for example, estimate total species richness, including species not present in any sample, rarefaction curves estimate species richness for a subsample of the pooled total species richness, based on all species actually discovered.
Random Resampling for Richness Estimators and Diversity Indexes
Sampling without replacement. If you specify randomization of sample order, without replacement, EstimateS selects a single sample at random, computes the richness estimators (and diversity indexes, if requested) based on that sample, selects a second sample, recomputes the estimators using the pooled data from both samples, selects a third, recomputes, and so on until all samples in the matrix are included. Samples are added to the analysis in random order, without replacement (each sample is selected exactly once).
Each distinct randomization accumulates the samples in a different order, but all samples are included in each randomization. The final value for the averaged, randomorder species accumulation curve therefore matches, precisely, the total number of observed species. The drawback with this protocol is that the variance, among randomizations, of counts (individuals, singletons, etc.) and of estimators for which no analytical variance is provided, goes goes to zero at the righthand end of the species accumulation curve. (Standard deviations based on variation among randomizations are identified as "runs" in EstimateS 7+ output. Standard deviations computed analytically are identified as "analytical" in EstimateS 7+ output.)
Sampling with replacement. If you specify randomization of sample order, with replacement, EstimateS selects a single sample at random, computes the richness estimators (and diversity indexes, if requested) based on that sample; then selects two samples at random from the entire set of samples, recomputes the estimators using the pooled data from both samples; then selects three samples at random from the entire set of samples, recomputes, and so on until the pooled number of samples is the same as the full sample set. Samples are added to the analysis in random order, with replacement (each sample can appear in any pooled sample, some may appear in none).
Each distinct randomization thus accumulates the samples in a different order, but in general, not all samples will be included, and some are likely to be chosen twice or more. Therefore, the final value for the averaged, randomorder species accumulation curve generally is generally less the total number of observed species, since the missed samples may contain species not found in the samples selected, for any given run. (In fact, the entire species accumulation curve generally lies below the corresponding curve produced by the without replacement option.) The advantage of randomizing samples with replacement is that the variance, among randomizations, of counts (individuals, singletons, etc.) and of estimators for which no analytical variance is provided, remains meaningful at the right hand end of the species accumulation curve, and can thus be used to compare datasets. (Standard deviations based on variation among randomizations are identified as "runs" in EstimateS 7+ output. Standard deviations computed analytically are identified as "analytical" in EstimateS 7+ output.)
The Mao Tau estimator (Colwell et al. 2004), which for most purposes completely replaces resampled Sobs in EstimateS 7+, solves this dilemma for samplebased rarefaction by reproducing the curve expected from the data (the equivalent of sampling without replacement), while yielding the (unconditional) variance by treating the data as a sample from a larger statistical universe. Thus the Mao Tau 95% confidence intervals remain "open" at the righthand end of the samplebased rarefaction (species accumulation) curve.
Sample accumulation. With or without replacement, as the samples accumulate, more and more information is included in the analysis and the richness estimates generally become more accurate (and diversity indexes tend to stabilize). By following the changes in each estimator or index as the samples accumulate, the performance of different estimators or indexes can be compared.
Number of randomizations. You can tell EstimateS to carry out as many different randomizations of sample order as you wish. By randomizing many times, the effect of sample order can be reduced by averaging over randomizations, producing relatively smooth estimator curves or diversity index curves and allowing a comparison of richness estimators or diversity indexes for your data set that does not depend on the particular order that samples were collected or added to the analysis.
As an option, you can tell EstimateS to add samples in the order they appear in your matrix and compute the estimators and indexes only once, for the sample ordering in the input matrix. (You can also use this option if expected species accumulation curves and/or Coleman curves are all you need.)
An additional options allow an analysis of the effect of patchiness on the performance of the estimators and indexes.
Reporting, exporting, and graphing results. Once the randomizations are complete, the mean value of each estimator or diversity index (and in most cases its standard deviation) is computed for each sample accumulation level and summary results are displayed on the screen (mean values among randomizations, for most estimators). For analysis and graphing, you can export these summary statistical results to a text file that can be opened by any competent spreadsheet, statistical, or graphing application. If you wish to analyze the results of individual randomizations, these can be exported to text file as an option. See Option to Export Results from Individual Randomizations.
The strategy of randomization and estimator evaluation is explained in more detail by Colwell & Coddington (1994). EstimateS was used to compute estimators for the seedbank example shown in Figures 1 and 2 and Table 1 of that paper. A copy of the seedbank data set (Butler & Chazdon 1998) is included when you download EstimateS. You can use this data file for demonstration and verification of proper installation.
Species Richness Estimators
The literature on species richness estimators continues to grow in several directions. Key reviews in the 1990s include Bunge & Fitzpatrick (1993) and Colwell & Coddington (1994). For a recent review of the field, see Chao (2004), which, like most key papers cited in this User's Guide, can be downloaded as pdf file.
Chao1 and Chao2 Richness Estimators. In EstimateS 7.5 and later, the classic richness estimators Chao1 and Chao 2 are computed along with loglinear 95% confidence intervals, as suggested by Chao (1987). These asymmetrical confidence intervals, which are based on the assumption that log(Sest  Sobs) is normally distributed, have the commonsense property that the lower confidence bound cannot be less than the observed number of species, Sobs. See Appendix B for details.
CoverageBased Richness Estimators ICE and ACE.The species richness estimators, ICE (Incidencebased Coverage Estimator) and ACE (Abundancebase Coverage Estimator) are modifications of the Chao & Lee (1992) estimators discussed by Colwell & Coddington (1994). Chazdon et al. (1998) introduced ICE and ACE to the ecological literature. For that paper, they found it necessary and useful to change the notation for the variables involved in the other estimators, to allow a unified system of notation covering the new estimators. This new notation is referenced in Table 1 and detailed in the Appendix C of this User's Guide, replacing the notation of Colwell & Coddington (1994). See Chazdon et al. (1998), which can be downloaded as pdf file, for details and rationale.
Estimating Total Species Richness by Functional Extrapolation
There are many possible curvilinear functions, asymptotic and nonasymptotic, that might fit a species accumulation curve (Soberón & Llorente 1993, Colwell & Coddington 1994, Colwell et al. 2004). As a richness estimation option, EstimateS computes the asymptotic function most commonly used, the MichaelisMenten function (Colwell & Coddington 1994).
EstimateS computes two different Michaelis Menten (MM) richness estimators. In both, the data the program produces represent the estimated MM asymptote based on one, two, three...QdMax samples (see Colwell & Coddington 1994, Fig. 1). The difference is that the first method (MMRuns) computes estimates for values for each pooling level, for each randomization run, then averages over randomization runs. If you have some samples that are much richer than others, randomization runs that, by chance, add a rich sample early in the curve are likely to produce enormous estimates of richness, since the rich sample "shoots" the fitted MM curve suddenly skyward. Thus, MMRuns data are often rather erratic for small numbers of samples, even when 100 runs are randomized.
The second method (MMMeans) computes the estimates for each sample pooling level just once, based on the species accumulation curve, as computed by MaoTau, in EstimateS 7+. Since this curve is computed analytically, it is quite smooth, thus the MM Means estimates are much less erratic than for the MMRuns method. This method is therefore generally recommended over MMRuns.
Note: Although means of Sobs among resampling runs are no longer used to compute MMMeans in Estimates 7 and later, the name MMMeans has been retained to make clear that is it the same as the estimator of that name in previous version of EstimateS.
The Shuffle Option for Exploring the Effects of Patchiness
This tool allows you to explore the effects of spatial patchiness on species richness estimators, as discussed by Chazdon et al. (1998). If you specify the Shuffle option (Diversity Settings > Other Options panel), EstimateS uses the following algorithm to reassign individuals at random to samples, within species, with a "tunable" degree of aggregation (patchiness).
If the Patchiness parameter (A) is set to zero. Using the species abundance vector (marginal totals) for all samples pooled, each individual is reassigned at random to a sample, within species. In other words, the distribution of individuals among species in the input matrix as a whole and the number of samples are maintained, but sample affiliations of individuals are randomized within species. Any patchiness of the original data is removed. (As expected, the mean of randomized sample accumulation curves is indistinguishable from the Coleman curve, which assumes spatial homogeneity, for this setting.)
If the Patchiness parameter (A) is set to a value greater than zero. In this case, the first individual of each species is assigned to a sample at random. The second (if there is one) is assigned to the same sample as the first with probability A, and to a randomly chosen sample with probability (1A). In other words, the larger you set A, the patchier the pseudodistribution of individuals becomes. By "tuning" the patchiness of the distribution, you can investigate the effect on the performance of the richness estimators, using real relativeabundance distributions. One could also enter madeup data sets that fit some particular relative abundance distribution(s).
Option to Export Results from Individual Randomizations
As an option, EstimateS (beginning with Version 8) records and exports results from n individual randomizations to a text file, allowing computation of precision, accuracy, and other analyses (Walther and Moore 2005), using Excel or other applications. To choose this option, select Diversity Settings from the Diversity menu, click the Other Options tab, then check the "Export results for each run to a text file" checkbox. When you click the Compute button (or choose Compute Diversity from the Diversity menu), EstimateS displays an expanatory message, and asks you to name and place the text file that will contain the exported results when the randomizations are complete. The data for each randomization appear in the same format as the summary Diversity results that EstimateS creates by default. (The summary results appear onscreen as usual, and may be exported as usual.) For large datasets, this option takes time, so be patient.
Selecting and Testing a Random Number Generator
EstimateS offers two random number generators (Diversity menu > Diversity Settings > Randomization tab > Random Number Generator panel). The Strong Hash Encryption generator samples from a 160bit strong hash (SHA) encryption function, seeded from the computer's clock. This procedure, developed by Jason Swain (personal communication), produces a nonrepeating random number series that passes the most demanding tests.
The Difference Equation alternative (Savitch (1992) is based on a seed number that you supply. Thus it permits EstimateS to generate precisely the same results on repeated sets of resampling runs with the same dataset. Unless you require precise repeatability, the Strong Hash Encryption option is recommended.
If you would like to do a visual test of either random number generator, choose Test Random Number Generator from the Special menu.
Shared Species and Similarity
Indexes (Shared Species Menu)
CoverageBased Estimator of Shared Species: V_{jk}
As discussed by Colwell & Coddington (1994), the problem of estimating the true number of species shared by two (or more) sites or biotas based on sample data presents a difficult but important challenge. The first statistical estimator of shared species was developed by Anne Chao and her colleagues (Chen et al. 1995 in Chinese; Chao et. al. 2000 in English), based on the same statistical strategy as ICE and ACE. Like ACE, the shared species estimator V requires abundance data. Just as ACE augments the observed number of species in a sample by a correction term dependent on the relative abundance of the rarest species (those with fewer than 10 individuals) in the sample, V augments the observed number of shared species by a correction term based on the relative abundance of shared, rare species.
EstimateS computes Chao's shared species estimator for all pairs of samples in the data matrix. A brief presentation of the mathematics behind the sharedspecies estimator appears in Appendix C of this Guide.
EstimateS computes the ACE estimate of species richness for each sample, as well as the estimate of shared species for each pair of samples.
Classic Similarity Indexes
EstimateS computes four classic indexes of similarity, based on the raw data from the input file: the Classic Jaccard index, the Classic Sørensen incidencebased (qualitative, presence/absence) index, the BrayCurtis index (= "Sørensen quantitative" index), and the MorisitaHorn index. Dozens of overlap indexes exist in the literature; these were chosen based on the recommendations of Magurran (1998, 2004).
Note: The BrayCurtis (= "Sørensen quantitative") index and the MorisitaHorn index can be used with either integer or decimal (real number) input data. However, since EstimateS requires all data to be integer counts for estimator computation, all decimal data values are rounded to the nearest integer when imported into EstimateS. For this reason, values of the Sørensen Abundancebased index and the MorisitaHorn index computed by EstimateS will differ slightly from the corresponding indexes computed for corresponding decimal data values, including Magurran's (1998) worked examples (Magurran 1988, pp. 165166), which are based on decimal data.
Chao's Abundancebased Jaccard and Sørensen Indexes and Their Estimators
Chao's Abundancebased Jaccard and Sørensen indexes are based on the probability that two randomly chosen individuals, one from each of two samples (quadrats, sites, habitats, collections, etc.), both belong to species shared by both samples (but not necessarily to the same shared species). The estimators for these indexes take into account the contribution to the true value of this probability made by species actually present at both sites, but not detected in one or both samples. This approach has been shown to reduce substantially the negative bias that undermines the usefulness of traditional similarity indices, especially with incomplete sampling of rich communities (Chao et al. 2005).
EstimateS 7.5+ computes the raw Chao Abundancebased Jaccard and Sørensen indexes (not corrected for undersampling bias) as well as the estimators of their true values, so that you can assess the effect of the bias correction on the indexes. In addition, the standard errors of the estimators are computed, allowing (for the first time!) statistically rigorous comparison of two or more similarity index values. (To compute 95% confidence intervals, just add and subtract 1.96*SE to/from the index estimate.) Because the Jaccard and Sørensen indexes depend on exactly the information, which one you use is strictly a matter of taste.
What EstimateS 8.0 Computes
Table 1, below, lists the variables and statistics that EstimateS 8.0 computes from the Diversity menu. Table 2 lists the variable and statistics computed from the Shared Species menu.
Table 1: Accumulated species and individuals, richness estimators, species diversity indexes and related variables computed by EstimateS 8.0. In the output screen (and exported text files), values for accumulated species, richness estimators, and diversity indexes appear for each level of sample accumulation (Qd =1 to QdMax), as expected values, or as mean values for the number of randomizations you specify. Formulas for the estimators appear in Appendix B .
Variable 
Estimator 
Reference 
Samples (Qd) 
Number of samples (Quadrats) accumulated 
m in Chazdon et al.
(1998) 
Individuals (computed) 
[Qd/Qd(max)]*N, where N is the total number of individuals in the Qd(max) samples  
Number of species expected in the pooled Qd samples, given the empirical data 

Lower bound of 95% Confidence Interval for Sobs (Mao Tau) 

Upper bound of 95% Confidence Interval for Sobs (Mao Tau) 

Standard deviation of Sobs (Mao Tau, analytical) 
Colwell et al. (2004)  
Sobs Mean 
Number of species in the pooled Qd samples (mean among runs) 
= Sobs in earlier versions of EstimateS 
Singletons Mean 
Number of singletons (species with only one individual) in the pooled Qd samples (mean among runs) 
a in Colwell & Coddington (1994) F1 in Chazdon et al. (1998) 
Singletons SD (runs) 
Standard deviation of Singletons, among randomizations of sample order 

Doubletons Mean 
Number of doubletons (species with only two individuals) in the pooled Qd samples (mean among runs) 
b in Colwell & Coddington (1994) F2 in Chazdon et al. (1998) 
Doubletons SD (runs) 
Standard deviation of Doubletons, among randomizations of sample order 

Uniques Mean 
Number of uniques (species that occur in a only one sample) among the Qd samples (mean among runs) 
L in Colwell & Coddington (1994) Q1 in Chazdon et al. (1998) 
Uniques SD (runs) 
Standard deviation of Uniques, among randomizations of sample order 

Duplicates Mean 
Number of duplicates (species that occur in a only two samples) among the Qd samples (mean among runs) 
M in Colwell & Coddington (1994) Q2 in Chazdon et al. (1998) 
Duplicates SD (runs) 
Standard deviation of Duplicates, among randomizations of sample order 

ACE Mean 
Abundancebased Coverage Estimator of species richness (mean among runs) 

ACE SD (runs) 
Standard deviation of ACE, among randomizations of sample order 

ICE Mean 
Incidencebased Coverage Estimator of species richness (mean among runs) 

ICE SD (runs) 
Standard deviation of ICE, among randomizations of sample order 

Chao1 Mean 
Chao 1 richness estimator (mean among runs) 
Chao (1984), with special cases as detailed in Appendix B. 
Chao1 95% CI Lower Bound 
Chao 1 loglinear confidence interval lower bound (mean among runs) 
Chao (1987), see Appendix B. 
Chao1 95% CI Upper Bound 
Chao 1 loglinear confidence interval upper bound (mean among runs) 
Chao (1987), see Appendix B. 
Chao1 SD (analytical) 
Chao 1 standard deviation (by Chao's formulas) 
Chao (1987) (not Chao 1984). Note: The formula in Colwell & Coddington is incorrect. See Appendix B for the correct formula and for special cases. 
Chao2 Mean 
Chao 2 richness estimator (mean among runs) 
Chao (1984, 1987), with special cases as detailed in Appendix B. 
Chao2 95% CI Lower Bound 
Chao 2 loglinear confidence interval lower bound (mean among runs) 
Chao (1987), see Appendix B. 
Chao2 95% CI Upper Bound 
Chao 2 loglinear confidence interval upper bound (mean among runs) 
Chao (1987), see Appendix B. 
Chao2 SD (analytical) 
Chao 2 standard deviation (by Chao's formula) 
Chao (1987) Note: The formula in Colwell & Coddington is incorrect. See Appendix B for the correct formula and for special cases. 
Jack1 Mean 
Firstorder Jackknife richness estimator (mean among runs) 
Burnham & Overton(1978, 1979), Smith & van Belle (1984), Heltshe & Forrester (1983) 
Jack1 SD (runs) 
Firstorder Jackknife std. deviation 

Jack2 Mean 
Secondorder Jackknife richness estimator (mean among runs) 
Burnham & Overton(1978, 1979), Smith & van Belle (1984), Palmer (1991) 
Jack2 SD (runs) 
Standard deviation of Jack2, among randomizations of sample order 

Bootstrap Mean 
Bootstrap richness estimator (mean among runs) 

Bootstrap SD (runs) 
Standard deviation of Bootstrap, among randomizations of sample order 

MMRuns Mean 
MichaelisMenten richness estimator: estimators averaged over randomizations (mean among runs) 

MMMeans (1 run) 
MichaelisMenten richness estimator: estimators computed once for Mao Tau species accumulation curve 

Cole Rarefaction 
Coleman rarefaction (number of species expected in the pooled Qd samples, assuming individuals distributed at random among samples) 

Cole SD 
Coleman standard deviation (analytical) 

Alpha Mean 
Fisher's alpha diversity index 

Alpha SD (analytical) 
Fisher's alpha standard deviation 

Shannon diversity index (mean among runs) 

Shannon SD (runs) 
Standard deviation of Shannon index among randomizations of sample order 

Simpson Mean 
Simpson (inverse) diversity index (mean among runs) 
Magurran (1988, eq. 2.27), Magurran (2004, p. 115), Hayek & Buzas (1996) 
Simpson SD (runs) 
Standard deviation of Simpson index among randomizations of sample order 

Table 2: Shared Species estimators, classic similarity indexes, Chao's abundancebased Jaccard and Sorensen similarity indexes and their estimators, and related variables computed by EstimateS 7.5. In the output screen (and exported text files), values for these statistics and variables appear for each possible pair of samples. The formula for the shared species estimator appears in Appendix C , and the formulas for Chao's abundancebased Jaccard and Sorensen similarity indexes, and their estimators and variances appears in Appendix D .
Variable 
Estimator 
Reference 
First Sample 

j in Appendix C 
Second Sample 

k in Appendix C 
Sobs First Sample 
Observed number of species in the First Sample 

Sobs Second Sample 
Observed number of species in the Second Sample 

Shared Spp Observed 
Observed number of species shared by First and Second samples 

ACE First 
Estimated number of species in the First Sample: ACE 

ACE Second 
Estimated number of species in the Second Sample: ACE 

Estimated number of species shared by First and Second samples: V(est) 

Jaccard Classic 
Classic Jaccard sample similarity index 
Chao et al. (2005, eq. 1) 
Sørensen Classic 
Classic Sørensen incidencebased (qualitative) sample similarity index 
Chao et al. (2005, eq. 2) 
ChaoJaccRaw Abundancebased 
Chao's Jaccard Raw (uncorrected for unseen species) Abundancebased similarity index 
Chao et al. (2005, eq. 5) 
ChaoJaccEst Abundancebased 
Chao's estimator (corrected for unseen species) for Chao's Jaccard Abundancebased similarity index 
Chao et al. (2005, eq. 9) 
ChaoJaccEstSD Abundancebased 
Standard Deviation of Chao's estimator (corrected for unseen species) for Chao's Jaccard Abundancebased similarity index 

ChaoJaccEst Incidencebased 
Chao's estimator (corrected for unseen species) for Chao's Jaccard similarity index for replicated Incidencebased data 
Chao et al. (2005, eq. 13) 
ChaoSorEstSD Indidencebased 
Standard Deviation of Chao's estimator (corrected for unseen species) for Chao's Jaccard similarity index for replicated Incidencebased data 

ChaoSorRaw Abundancebased 
Chao's Sørensen Raw (uncorrected for unseen species) Abundancebased similarity index 
Chao et al. (2005, eq. 6) 
ChaoSorEst Abundancebased 
Chao's estimator (corrected for unseen species) for Chao's Sørensen Abundancebased similarity index 
Chao et al. (2005, eq. 10) 
ChaoSorEstSD Abundancebased 
Standard Deviation of Chao's estimator (corrected for unseen species) for Chao's Sørensen Abundancebased similarity index 

ChaoSorEst Incidencebased 
Chao's estimator (corrected for unseen species) for Chao's Sørensen similarity index for replicated Incidencebased data 
Chao et al. (2005, eq. 14) 
ChaoSorEstSD Indidencebased 
Standard Deviation of Chao's estimator (corrected for unseen species) for Chao's Sørensen similarity index for replicated Incidencebased data 

Morisita Horn 
MorisitaHorn sample similarity index 
Magurran (1988, eq. 5.10), Magurran (2004, page ) 
BrayCurtis 
BrayCurtis (=Sørensen quantitative) sample similarity index 
Magurran (1988, eq. 5.9), Magurran (2004, page ) 
How to Use EstimateS
Preparing the Input (Data) File: General Specifications
Sample Input File: A sample Input File named Seedbank is installed in the EstimateS program folder. Open this file in Excel or a text editor to examine it as you read this section. The Seedbank file is in Format 1. Be sure not to save any changes to this file so it will remain a correct model.
EstimateS was used to compute the species richness estimators for the Seedbank dataset (Butler & Chazdon 1998) that appear in Figures 1 and 2 and Table 1 of Colwell & Coddington (1994). You can use the Seedbank data file for demonstration and verification of proper installation of EstimateS.
File Type, Name and Location: The Input File must be plain text, tabdelimited. The Input File may have any name and may be located in any folder (directory).
Title Record: The first line of the Input File must contain a title, any text will do.
Parameter Record: The second line must contain two obligatory control parameters: the number of species and the number of samples, separated by a TAB character. Additional control parameters are optional, and can be more easily recorded by exporting a new copy of the input file after setting the parameters in EstimateS' Settings screens.
The rest of the Input File contains the input data, which can appear in any one of five alternative formats. When you run the program, you will be asked to specify which of the following formats you used:
Format 1. Species (rows) by Samples (columns) abundance or incidence matrix ("samples" may be collections, quadrats, etc.): one row per species, one column per sample. This is how the demo file (Seedbank) is formatted. The input file may contain any number of initial rows of column labels and/or initial columns of row labels, in which case you must tell EstimateS how many of each there are. (EstimateS simply skips over label rows and columns.)
Note on Format 1: If your file includes one or more rows of column labels, they must follow the required Title and Parameter records and precede the data. If your file includes one or more columns of row labels, the required Title and Parameter records nonetheless begin in the first column.
Format 1 Example: Below is a simple example of an EstimateS Input File in Format 1, for a dataset called "My Input File" that includes data for 8 species (rows) in 10 samples (columns). The data are exactly the same as in the examples, below, for Formats 2 and 3.
Format 2. Samples (rows) by Species (columns) abundance or incidence matrix: one row per sample, one column per species. The input file may contain any number of initial rows of column labels and/or initial columns of row labels, in which case you must tell EstimateS how many of each there are. (EstimateS simply skips over label rows and columns.)
Note on Format 2: If your file includes one or more rows of column labels, they must follow the required Title and Parameter records and precede the data. If your file includes one or more columns of row labels, the required Title and Parameter records nonetheless begin in the first column.
Format 2 Example: Below is a simple example of an EstimateS Input File in Format 2, for a dataset called "My Input File" that includes data for 8 species (columns) in 10 samples (rows). The data are exactly the same as in the example, above, for Format 1, and the example, below, for Format 3.
Format 3. Species, Sample, Abundance triplets: the first column contains the species number, the second the sample number, and the third the number of individuals (abundance) of that species in that sample. A final (extra) record with "1" in each column indicates end of input. This "triplet" format a common input format for statistical programs (e.g. SYSTAT.) You can list one row for every sample/species combination, or rows for only those combinations that have nonzero abundances. (The rest are automatically set to zero.) Using the triplet format and storing only nonzero abundance values requires far less file space than storing the full matrix. In fact, this may be the most practical way to store files larger than your spreadsheet will accept. As an option (see below), EstimateS can export a data matrix in this format, after reading it in using one of the other four formats listed here.
Notes on Format 3: EstimateS expects no more than one record for each species x sample combination. If you have more than one, only the first is read. A special record must terminate triplet files, as detailed in the next section.
Format 3 Example: Below is a simple example of an EstimateS Input File in Format 3, for a dataset called "My Input File" that includes data for 8 species (columns) in 10 samples (rows). The data are exactly the same as in the examples, above, for Formats 1 and 2.
Format 4. Sample, Species, Abundance triplets: format as for (3), but the columns are ordered Sample, Species, Abundance.
Note on Format 4: EstimateS expects no more than one record for each species x sample combination. If you have more than one, only the first is read. A special record must terminate triplet files, as detailed in the next section.
Format 5. This format is output automatically by Biota, with appropriate row and column labels. For other input files that include column or row labels, use Formats 1 or 2
Detailed Specifications for the Input File
The required layout for the input (data) file is detailed below. Each [element], shown in square brackets, must be separated from the next by a single TAB character (do not include the line numbers or the brackets in the actual file).
LINE 1: Title Record
[Title]: Any alphanumeric title for the data input file
LINE 2: Parameter Record (all this on one line, separated by TABs)
Required: [SpMax]: Number of speciesRequired: [QdMax]: Number of samples
Note: The remaining, optional parameters are intended to be used for batch processing or repeated analyses. It is much easier to set these options from graphical query screens during input, or in the graphical Settings screens, once your data have been input to EstimateS.
Optional: [AbMax]: This parameter is ignored in EstimateS 7+; it is retained only for backwards compatibility.
Optional: [Runs]: Number of randomizations to perform.
Optional: [Memory]: If this parameter is blank or zero, the SHA random number generator is used (seeded from the clock). An integer value > 0 in this field is interpreted as the "seed" for the difference equation random number generator. It must an integer, any value between 1 and 700.
Optional: [RareInfreqCut]: The number of abundance classes (singletons, doubletons, tripletons, etc.) or the number of incidence classes (uniques, duplicates, triplicates, etc.) to be included in the calculation of the CV estimates used in ICE, ACE, and shared species estimator V. Anne Chao (pers. comm.) recommends using 10 for this parameter. If this parameter is blank or zero, EstimateS set it to 10.
Optional: [DivIndexFlag]: If this flag is 1, EstimateS computes Fisher's alpha and the Shannon and Simpson indexes. If this flag is blank or zero, these indexes are not computed.
Optional: [RandFlag]: If this flag is set to 1, EstimateS does not randomize sample order and the Runs parameter is set automatically to 1. If this flag is blank or zero, Runs randomizations are carried out.
Optional: [Shuffle]: If this flag is set to 1, EstimateS randomizes the placement of individuals among samples, within species (Chazdon et al. 1998), using the Patchiness parameter to set aggregation. If this flag is blank or zero, no shuffling is done.
Optional: [Patchiness]: This variable must be between 0 and 1, inclusive. See details on the Patchiness parameter earlier in this Guide. The recommended default is zero.
Optional: [SimIndexFlag]: If this flag is set to 1, EstimateS computes the Jaccard, Sørensen, and MorisitaHorn indexes. If this flag is blank or zero, the indexes are not computed.
Optional: [FormatKey]: This variable specifies the input file format, and must be an integer between 0 and 5. EstimateS always allows you to specify the file format during data input, so you need not include this parameter. (It is set automatically to 3 in Format 3 files exported from EstimateS, and is set to 5 when reading Biota to EstimateS input files.)
Optional: [ChaoClassic]: If this flag is blank or zero, EstimateS uses the biascorrected form of the Chao1 and Chao2 richness estimators in all cases (the recommended default). If this flag is set to 1, EstimateS uses the the biascorrected form only when doubletons (Chao1) or duplicates (Chao2) are zero, and uses the approximate ("classic") formulas otherwise.
Optional: [Replace]: If this flag is blank or zero, EstimateS randomizes sample order without replacement. If this flag is 1, samples are selected for accumulation with replacement.
Optional: [SkipRows]: If this parameter is blank or zero, EstimateS assumes the input file contains no label rows. If set to N, EstimateS will skip N rows after reading the Title Record and the Parameter Record, then begin reading the incidence or abundance rows.
Optional: [SkipColumns]: If this parameter is blank or zero, EstimateS assumes the input file contains no label columns. If set to N, EstimateS will skip the first N columns when reading each incidence or abundance row.
Optional: [ExportRuns]: If this parameter is blank or zero, EstimateS does not export the Diversity results for individual randomizations (runs). If set to 1, Diversity results for each randomization are exported. See Option to Export Results from Individual Randomizations.
For data Format 1: Species (i) by Samples (j) abundance or incidence (Nij) matrix
LINES 3 TO (SpMax+2)
[N1,1] [N1,2] ... [N1,j] ... [N1,QdMax][N2,1] [N2,2] ... [N2,j] ... [N2,QdMax]
.
.
.
[Ni,1] [Ni,2] ... [Ni,j] ... [Ni,QdMax]
.
.
.
[NSpMax,1] [NSpMax,2] ... [NSpMax,j] ... [NSpMax,QdMax]
For data Format 2: Samples (i) by Species (j) abundance or incidence (Nij) matrix
LINES 3 TO (QdMax+2)
[N1,1] [N1,2] ... [N1,j] ... [N1,SpMax][N2,1] [N2,2] ... [N2,j] ... [N2,SpMax]
.
.
.
[Ni,1] [Ni,2] ... [Ni,j] ... [Ni,SpMax]
.
.
.
[NQdMax,1] [NQdMax,2] ... [NQdMax,j] ... [NQdMax,SpMax]
For data Format 3: Species (i), Sample (j), Abundance/Incidence (Nij) triplets
LINES 3 and beyond
1 1 [N1,1]
.
.
.
i j [Ni,j]
.
.
.
until all nonzero abundances or incidences are included, then a special filetermination record:
1 1 1
For data Format 4: Sample (i), Species (j), Abundance (Nij) triplets
LINES 3 and beyond
1 1 [N1,1]
.
.
.
i j [Ni,j]
.
.
.
until all nonzero abundances or incidences are included, then a special filetermination record:
1 1 1
For data Format 5: Samples (i) by Species (j) abundance or incidence (Nij) matrix
LINES 3 TO (QdMax+6)
Line 3: IgnoredLine 4: Ignored
Line 5: Ignored
Line 6: Ignored
Field 1 ignored; [N1,1] [N1,2] ... [N1,j] ... [N1,SpMax]
Field 1 ignored; [N2,1] [N2,2] ... [N2,j] ... [N2,SpMax]
.
.
.
Field 1 ignored; [Ni,1] [Ni,2] ... [Ni,j] ... [Ni,SpMax]
.
.
.
Field 1 ignored; [NQdMax,1] [NQdMax,2] ... [NQdMax,j] ... [NQdMax,SpMax]
Running EstimateS
1. Launch EstimateS by doubleclicking the EstimateS icon or application name or (in Windows) by launching EstimateS from the Programs section of the Start menu.
2. If a file navigation window appears asking you to select a "Data File," choose the file called Statistics.4DD (Windows) or Statistics.data (Mac OS). This default file records the statistical output of Biota.
Note 1: Do not try to load your input file at this point. If you cannot find the Statistic Data file, click the New button to create a new data output file. You can name it anything you wish, using the extension .data (Macintosh) or .4DD (Windows).
Note 2: If you want to create a new output data file or find a different existing one, you can force the navigation window to appear as follows:
Windows: Select the EstimateS icon or application name, then choose open from the Windows File menu, while holding down the Alt key.
Macintosh: Click and hold the Option key while launching EstimateS
2. From the File menu in EstimateS, choose Load Input File. The open file window appears.
3. Find the input file and open it. EstimateS presents a confirmation window showing the data set name and the input parameters. (If you want, you can try loading the example Input File called Seedbank.)
4. If the parameters are correct, click the OK button in the dialog. EstimateS loads the file. (Various input data errors will be flagged if they occur. Follow the onscreen instructions if this happens.)
5. To set, change, or check run parameters for the loaded file, choose Diversity Settings from the Diversity menu (richness estimators and diversity indexes), or Shared Species Settings from the Shared Species menu (shared species estimator and similarity indexes). See the description of parameters in the preceding section for details.
6. Launch the computations by clicking the Compute button in a Settings screen, or by choosing Compute Diversity Stats from the Diversity menu (richness estimators and diversity indexes), or Compute Shared Spp Stats from the Shared Species menu (shared species estimator and similarity indexes). EstimateS does the computations and displays the results onscreen. (Clicking a Compute button saves the settings as well as launching the computations.)
7. When computations are completed, click the Export button to export the results (see below), or the Done button to dismiss the results screen. (You can always export the results later from a menu command.) The records are saved in the statistics data file (Statistics or another name you have given it), and can be redisplayed at any time by choosing Show Diversity Stats from the Diversity menu (richness estimators and diversity indexes), or Show Shared Spp Stats from the Shared Species menu (shared species estimator and similarity indexes).
Note: If you relaunch EstimateS using the same statistics (output) data file, the last set of results will still be available by choosing Show Diversity Stats from the Diversity menu or Show Shared Spp Stats from the Shared Species menu.
What to Do Next (All Optional)
1. Export the results to a tabdelimited text file by choosing Export Diversity Stats from the Diversity menu (richness estimators and diversity indexes), or Export Shared Spp Stats from the Shared Species menu (shared species estimator and similarity indexes). You can then open the text file in Excel, a graphing application, or a statistical application to further analyze or graph the data.
2. Export the input data and all current parameter settings to a tabdelimited text file by choosing Export Input File as Triplets from the File menu. EstimateS creates a Format 3 input file, recording all parameter settings. You can reload this file at any time. Triplet files load more quickly than full matrix files (Formats 1, 2, and 5).
3. Save the results in an EstimateS Statistics Data File. The results (statistics) displayed onscreen by EstimateS are actually 4D database records in a file called (initially) Statistics.4DD (Windows) or Statistics.data (Macintosh). When you launch EstimateS, this file is automatically (re)opened. If you want to save this file with the results of the current run and open a new, empty Statistics file (instead of deleting the results next time you run a different input file or the same input file with different parameters), follow these steps:
a. Quit EstimateS from the File menu.
b. Rename the Statistics.data (Macintosh) or Statistics.4DD and Statistics.4DR files (Windows) file with a name that is meaningful to you. (Any name is allowed. In Windows, both the .4DD and .4DR files must have the same name.)
c. Relaunch EstimateS, while holding down the Alt key (Windows) or the Option key (Macintosh) . (In Windows, you have to select the EstimateS icon or application name, then choose open from the Windows File menu, while holding down the Alt key.) An Open Data File window will appear.
d. Click the New button in the Open Data File window and give the new file an appropriate name (any name is allowed), with the extension .data (Macintosh) or .4DD (Windows).
Things You Should Know Before You Begin
Caveat Receptor
I have done my best to check all features of EstimateS 8 for usability and all computations and algorithms for accuracy, but the final responsibility for ensuring that your results are correct must rest with you.
In general, you should have little trouble understanding the output, by referring to Colwell & Coddington (1994), Chazdon et al. (1998), Gotelli & Colwell (2001), Colwell et. al. (2004), Chao et. al. (2005), or if necessary the references in Table 1 and 2.
Citing EstimateS
If you appreciate the effort that has gone into EstimateS, please credit the application and its author in any published work that makes use of results from EstimateS, citing EstimateS as an electronic publication and giving the EstimateS persistent URL (PURL) website address (http://purl.oclc.org/estimates) if the journal permits it. (This "permanent" address automatically transfers the visitor to http://viceroy.eeb.uconn.edu/EstimateS.) Here is one possible form for a References Cited entry:
Colwell, R. K. 2005. EstimateS: Statistical estimation of species richness and shared species from samples. Version 7.5. User's Guide and application published at: http://purl.oclc.org/estimates.
If the journal or book editor will not permit an entry in the References Cited section, you might try this text citation: "...computed using EstimateS (Version 7.5, R. K. Colwell, http://purl.oclc.org/estimates)...."
Failing that, you may be reduced to: "...computed using EstimateS (Version 7.5, R. K. Colwell, unpublished)...," perhaps slipping in the EstimateS website address (http://purl.oclc.org/estimates) in the Acknowledgment section.
I would be most grateful if you would kindly send a reprint of any paper based on your use of the program. Send a pdf to colwell@uconn.edu, or by post to: Robert K. Colwell, Department of Ecology and Evolutionary Biology, U43, University of Connecticut, Storrs, CT 062693043, USA.
What You Must Agree To: Copyright and Fair Use
EstimateS is a freeware application. By downloading and using EstimateS, you must agree not to distribute EstimateS in any commercial form.
You are most welcome to use EstimateS in any way you like for your own research, as long as such use is acknowledged as outlined above.
Sharing EstimateS With Others
To keep track of EstimateS users and to make sure that the latest version is in use, it is preferable that each new user downloads and registers his or her own copy of EstimateS from http://viceroy.eeb.uconn.edu/estimates or http://purl.oclc.org/estimates, rather than sharing someone else's (e.g. your) copy.
If you do share the program with a colleague, please be sure to make clear that the User's Guide is available online at http://viceroy.eeb.uconn.edu/estimates or http://purl.oclc.org/estimates, to save needless email support questions.
References Cited
References marked "Download pdf" are available here for downloading.
Brewer, A., & M. Williamson. 1994. A new relationship for rarefaction. Biodiversity and Conservation 3:373379.Bunge, J., & M. Fitzpatrick. 1993. Estimating the number of species: A review. Journal of the American Statistical Association 88, 364373.
Burnham, K.P. & W.S. Overton. 1978. Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika 65, 623633.
Burnham, K.P. & W.S. Overton. 1979. Robust estimation of population size when capture probabilities vary among animals. Ecology 60, 927936.
Butler, B. J., & R. L. Chazdon. 1998. Species richness, spatial variation, and abundance of the soil seed bank of a secondary tropical rain forest. Biotropica 30:214222. Download pdf.
Chao, A. 1984. Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics 11, 265270. Download pdf.
Chao, A. 1987. Estimating the population size for capturerecapture data with unequal catchability. Biometrics 43, 783791. Download pdf.
Chao, A. 2005. Species richness estimation, Pages 79097916 in N. Balakrishnan, C. B. Read, and B. Vidakovic, eds. Encyclopedia of Statistical Sciences. New York, Wiley. Download pdf.
Chao, A., R. L. Chazdon, R. K. Colwell, and T.J. Shen. 2005. A new statistical approach for assessing compositional similarity based on incidence and abundance data. Ecology Letters 8:148159. Download pdf. Spanish Version: Download pdf.
Chao, A., R. L. Chazdon, R. K. Colwell, and T.J. Shen. 2006. Abundancebased similarity indices and their estimation when there are unseen species in samples. BiometricsBiometrics 62, 361371. Download pdf.
Chao, A., W.H. Hwang, Y.C. Chen, and C.Y. Kuo. 2000. Estimating the number of shared species in two communities. Statistica Sinica 10:227246. Download pdf.
Chao, A. & S.M Lee. 1992 Estimating the number of classes via sample coverage. Journal of the American Statistical Association 87, 210217. Download pdf.
Chao, A., M.C. Ma, & M. C. K. Yang. 1993. Stopping rules and estimation for recapture debugging with unequal failure rates. Biometrika 80, 193201. Download pdf.
Chazdon, R. L., R. K. Colwell, J. S. Denslow, & M. R. Guariguata. 1998. Statistical methods for estimating species richness of woody regeneration in primary and secondary rain forests of NE Costa Rica. Pp. 285309 in F. Dallmeier and J. A. Comiskey, eds. Forest biodiversity research, monitoring and modeling: Conceptual background and Old World case studies. Parthenon Publishing, Paris. Download pdf.
Chen, Y.C., W.H. Hwang, A. Chao, & C.Y. Kuo. 1995. Estimating the number of common species. Analysis of the number of common bird species in KeYar Stream and ChungKang Stream. (In Chinese with English abstract.) Journal of the Chinese Statistical Association 33, 373393.
Coleman, B.D. 1981. On random placement and speciesarea relations. Mathematical Biosciences 54, 191215.
Coleman, B.D., Mares, M.A., Willig, M.R. & Hsieh, Y.H. 1982. Randomness, area, and species richness. Ecology 63, 11211133.
Colwell, R. K. 2006. Biota: The biodiversity database manager, Version 2. Sinauer Associates, Sunderland, MA.
Colwell, R. K., & J. A. Coddington. 1994. Estimating terrestrial biodiversity through extrapolation. Philosophical Transactions of the Royal Society (Series B) 345, 101118. Download low resolution pdf. or download high resolution pdf.
Colwell, R. K., C. X. Mao, & J. Chang. 2004. Interpolating, extrapolating, and comparing incidencebased species accumulation curves. Ecology 85, 27172727. Download pdf. Spanish Version: Download pdf.
Gotelli, N., & R. K. Colwell. 2001. Quantifying biodiversity: Procedures and pitfalls in the measurement and comparison of species richness. Ecology Letters 4 , 379391. Download pdf.
Hayek, L. C., & M. A. Buzas. 1996. Surveying natural populations. Columbia University Press, NY.
Heck, K.L., Jr., van Belle, G. & Simberloff, D. 1975. Explicit calculation of the rarefaction diversity measurement and the determination of sufficient sample size. Ecology 56, 14591461.
Heltshe, J. & Forrester, N.E. 1983 . Estimating species richness using the jackknife procedure. Biometrics 39, 111.
Lee, S.M., and A. Chao. 1994. Estimating population size via sample coverage for closed capturerecapture models. Biometrics 50, 8897. Download pdf.
Mao, C. X., R. K. Colwell, and J. Chang. 2005. Estimating species accumulation curves using mixtures. Biometrics 61:433–441. Download pdf.
Magurran, A. E. 1988. Ecological diversity and its measurement. Princeton University Press, Princeton, N. J.
Magurran, A. E. 2004. Measuring biological diversity. Blackwell.
Palmer, M.W. 1991. Estimating species richness: The secondorder jackknife reconsidered. Ecology 72, 15121513.
Raaijmakers, J. G. W. 1987. Statistical analysis of the MichaelisMenten equation. Biometrics 43, 793803.
Savitch, Walter J. 1992. Turbo Pascal : an introduction to the art and science of programming. 3rd ed. Benjamin/Cummings, Redwood City, Calif.
Smith, E.P. & van Belle, G. 1984. Nonparametric estimation of species richness. Biometrics 40, 119129.
Soberón, J., & J. Llorente. 1993. The use of species accumulation functions for the prediction of species richness. Conservation Biology 7 , 480488.
Ugland, K. I., J. S. Gray, & K. E. Ellingsen. 2003. The speciesaccumulation curve and estimation of species richness. Journal of Animal Ecology 72 , 888897.
Walther, B. A., and J. L. Moore. 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, 815829.
Appendix A: Samplebased Rarefaction Curve (Expected Species Accumulation Curve)
The resamplingbased S_{obs} curve computed by previous versions of EstimateS was an approximation to the curve computed by this analytical method. For details and interpretation, see Colwell et al. (2004).
Let s_{j}
stand for the number of species found in exactly j samples of the empirical sample set,
which has a total of H
samples. Thus s_{1} is the number of species found in
precisely 1 sample, s_{2} is the number of species found in
precisely 2 samples, and so on. The observed richness in the empirical
sample set is therefore
.
For samplebased rarefaction (species
accumulation based on samples),
is an unbiased estimator of the species richness expected in h samples pooled, where
This estimator is based on the s_{j}, appropriately weighted by combinatorial
coefficients:
where the combinatorial coefficients are defined by
.
Note
that
.
Because the coefficient alpha in equation (1) is 0 for h = H, estimated richness for the full empirical sample set is
We consider the observed
richness S_{obs}
to be measured with error. This approach is critical to the derivation
of an unconditioned variance estimator for t(h) at h<H.
One can
construct approximate 95% confidence intervals
using variance estimator
where Stilde is an estimator for the unknown total species
richness S. A form of the Chao2 richness estimator is a simple
option for Stilde:
EstimateS 7+ uses equation (3)
to compute Stilde when s_{2} > 0. When s_{2} = 0,
Chao's full, biascorrected formula is used
instead.
Appendix B: Nonparametric Estimators of Species Richness
Please note that nonparametic estimators of species richness are minimum estimators: their computed values should be viewed as lower bounds of total species numbers, given the information in a sample or sample set.
Definition of variables
S_{est} 
Estimated species richness, where est is replaced in the formula by the name of the estimator 
S_{obs} 
Total number of species observed in all samples pooled 
S_{rare} 
Number of rare species (each with 10 or fewer individuals) when all samples are pooled 
S_{abund} 
Number of abundant species (each with more than 10 individuals) when all samples are pooled 
S_{infr} 
Number of infrequent species (each found in 10 or fewer samples) 
S_{freq} 
Number of frequent species (each found in more than 10 samples) 
m 
Total number of samples 
m_{infr} 
Number of samples that have at least one infrequent species 
F_{i} 
Number of species that have exactly i individuals when all samples are pooled (F_{1} is the frequency of singletons, F<sub>2</sub>_{ }the frequency of doubletons) 
Q_{j} 
Number of species that occur in exactly j samples (Q_{1} is the frequency of uniques, Q_{2} the frequency of duplicates) 
p_{k} 
Proportion of samples that contain species k 
N_{rare} 
Total number of individuals in rare species 
N_{infr} 
Total number of incidences (occurrences) of infrequent species 
C_{ace} 
Sample abundance coverage estimator 
C_{ice} 
Sample incidence coverage estimator 
Estimated coefficient of variation of the F_{i} for rare species 

Estimated coefficient of variation of the Q_{i} for infrequent species 
The estimators
Chao 1 and Chao2: Different equations are used to compute the Chao1 and Chao2 richness estimators, their estimated variance, and the corresponding loglinear 95% confidence intervals, depending on (1) the number of singletons and doubletons (in abundancebased data) or uniques and duplicates (for incidencebased data), and (2) the settings you select "Chao 1 and Chao 2 bias correction" panel in the Estimators tab of the Diversity Settings screen (Diversity menu). The table below specifies the equations used in each case. The equations referred to appear below the table. This section was developed in personal communication with Anne Chao, Institute of Statistics, National Tsing Hua University, Taiwan, to whom I am most grateful.
Estimator 
Singletons (F_{1} ) or Uniques 
Doubletons (F_{2}) or Duplicates 
Setting 
Estimate 
Variance 
95% CI 
Chao1 
F_{1} > 0 
F_{2} > 0 
Classic 
Eq. 1 
Eq. 5 
Eq. 13 



Biascorrected 
Eq. 2 
Eq. 6 
Eq. 13 

F_{1} > 0 
F_{2} = 0 
Either 
Eq. 2 
Eq. 7 
Eq. 13 

F_{1} = 0 
F_{2} > or = 0 
Either 
Eq. 2 
Eq. 8 
Eq. 14 
Chao2 
Q_{1} > 0 
Q_{2} > 0 
Classic 
Eq. 3 
Eq. 9 
Eq. 13 



Biascorrected 
Eq. 4 
Eq. 10 
Eq. 13 

Q_{1} > 0 
Q_{2} = 0 
Either 
Eq. 4 
Eq. 11 
Eq. 13 

Q_{1} = 0 
Q_{2} > or = 0 
Either 
Eq. 4 
Eq. 12 
Eq. 14 
Equations referenced in the table above:
Jackknife 1: Firstorder
jackknife estimator of species richness (incidencebased) (Burnham and Overton 1978,1979; Heltshe and Forrester 1983)
.
Jackknife 2: Secondorder
jackknife estimator of species richness (incidencebased) (Smith and van Belle 1984)
Bootstrap: Bootstrap estimator of
species richness (incidencebased) (Smith
and van Belle 1984)
.
ACE: Abundancebased Coverage Estimator of species richness (Chao and Lee 1992, Chao, Ma, and Yang 1993)
First note that
.
The sample coverage estimate based on abundance data is
,
where.
Thus, this sample coverage estimate is the proportion of all individuals in rare species that are not singletons. Then the ACE estimator of species richness is
where the estimate the coefficient of variation of the F_{i}'s, is.
Note: The formula for ACE is undefined when all Rare species are Singletons (F_{1} = N_{rare}, yielding C = 0). In this case, EstimateS computes the biascorrected form of Chao1 instead (on Anne Chao's advice).
ICE: Incidencebased Coverage Estimator of species richness (Lee and Chao 1994)
First note that
.
The sample coverage estimate based on incidence data is
,
where.
Thus, the sample coverage estimate is the proportion of all individuals in infrequent species that are not uniques. Then the ICE estimator of species richness is.
where the estimate the coefficient of variation estimates the coefficient of variation of the Q_{j}'s, is.
Note: The formula for ICE is undefined when all Infrequent species are Uniques (Q_{1} = N_{infr}, yielding C = 0). In this case, EstimateS computes the biascorrected form of Chao2 instead (on Anne Chao's advice).
Appendix C: Coveragebased Estimator of Shared Species
This appendix and its implementation in EstimateS is based on Chao et al. (2000) and on personal communication with Anne Chao, Institute of Statistics, National Tsing Hua University, Taiwan.
Definition of variables
Estimated number of species shared by samples j and k
Observed number of species shared by samples j and k
Observed number of shared, abundant species (>10 individuals in sample j, in sample k, or in both)
Observed number of shared, rare species (< or = 10 individuals in sample j AND < or = 10 individuals in sample k)
Number of individuals of rare, shared species i in sample j
Number of individuals of rare, shared species i in sample k
Total number of singletons (X_{i }= 1) among rare, shared species in sample j
Total number of singletons (Y_{i }= 1) among rare, shared species in sample k
Number of rare, shared species that are singletons in sample j but have Y_{i} > 1 in sample k
Number of rare, shared species that are singletons in sample k but have X_{i} > 1 in sample j
Number of rare, shared species that are singletons in both samples j and k
Number of individuals in sample k for rare, shared species that are singletons in sample j
Number of individuals in sample j for rare, shared species that are singletons in sample k
Sample coverage for rare, shared species
The estimator
Sample coverage for rare, shared species is estimated by
,
where the summation is taken over all rare, shared species. An estimate
of the true number of rare, shared species for samples j and k,
uncorrected for variation (among species) and covariation (among
species between samples) in abundance is
.
With variation and covariation in abundance taken into account,
estimated true number of shared species for samples j and k (the result
that EstimateS produces) is then
,
where the gamma terms are estimates of the coefficients of variation
and covariation in abundance among rare, shared species. The gamma
terms are computed as
,
where, taking all summations over
we have
.
Note: Sample size terms in the
numerator and denominator of the gammas, of the form n/(n1),
appear in Chao et al. (2000). Since these
ratios are effectively unity, they have been omitted above and for
computational purposes in EstimateS.
Appendix D: Chao's Abundancebased Jaccard and Sorensen Similarity Indexes and Their Estimators
This appendix and its implementation in EstimateS is based on Chao et al. (2005) and on personal communication with Anne Chao, Institute of Statistics, National Tsing Hua University, Taiwan.
Appendix D is a pdf document. Click here to display Appendix D in Acrobat or Acrobat Reader. For full details, download Chao et al. (2005).