Received Date : 02-Sep-2015

Revised Date : 18-Apr-2016

Accepted Date : 16-Jun-2016
Accepted Article
Article type : Application
Editor : Greg McInerny

iNEXT: An R package for rarefaction and extrapolation of species

diversity (Hill numbers)

(Running Head: iNEXT package)

T. C. Hsieh, K. H. Ma, and Anne Chao*

Institute of Statistics, National Tsing Hua University, Hsin-Chu, Taiwan 30043

* Correspondence author E-mail: chao@stat.nthu.edu.tw

Summary
1. Hill numbers (or the effective number of species) have been increasingly used to quantify

the species/taxonomic diversity of an assemblage. The sample-size- and coverage-based

integration of rarefaction (interpolation) and extrapolation (prediction) of Hill numbers

represent a unified standardization method for quantifying and comparing species

This article has been accepted for publication and undergone full peer review but has not
been through the copyediting, typesetting, pagination and proofreading process, which may
lead to differences between this version and the Version of Record. Please cite this article as
doi: 10.1111/2041-210X.12613
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 diversity across multiple assemblages.

2. We briefly review the conceptual background of Hill numbers along with two approaches
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to standardization. We present an R package iNEXT (iNterpolation/EXTrapolation)

which provides simple functions to compute and plot the seamless rarefaction and

extrapolation sampling curves for the three most widely used members of the Hill number

family (species richness, Shannon diversity and Simpson diversity). Two types of

biodiversity data are allowed: individual-based abundance data and sampling-unit-based

incidence data.

3. Several applications of the iNEXT packages are reviewed: (1) Non-asymptotic analysis:

comparison of diversity estimates for equally-large or equally-complete samples. (2)

Asymptotic analysis: comparison of estimated asymptotic or true diversities. (3)

Assessment of sample completeness (sample coverage) across multiple samples. (4)

Comparison of estimated point diversities for a specified sample size or a specified level

of sample coverage.

4. Two examples are demonstrated, using the data (one for abundance data and the other for

incidence data) included in the package, to illustrate all R functions and graphical

displays.

Key words: abundance data, incidence data, sample coverage, Shannon diversity, Simpson

diversity, species richness.

Introduction

Hill numbers (or the effective number of species) have been increasingly used to

quantify the species/taxonomic diversity of an assemblage because they represent an intuitive

and statistically rigorous alternative to other diversity indices; see Chao et al. (2014) for a

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 recent review. Hill numbers are parameterized by a diversity order q, which determines the

measures’ sensitivity to species relative abundances. Hill numbers include the three most
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widely used species diversity measures as special cases: species richness (q=0), Shannon

diversity (q=1) and Simpson diversity (q=2).

For a given diversity measure, the goal in many diversity analyses is to make fair

comparison and assessment of diversities across multiple assemblages that may vary in the

size of their species pools or in the way in which they are sampled. For species richness, it is

well known that the empirical species richness in a sample is highly dependent on sample

size or sampling efforts. The traditional approach to compare species richnesses of different

assemblages is to use rarefaction to down-sample the larger samples until they contain the

same number of observed individuals or observations as the smallest sample. Ecologists then

compare the richnesses of these equally-large samples, but this necessitates that some data in

larger samples are thrown away.

To avoid discarding data, Colwell et al. (2012) proposed using a sample-size-based

rarefaction and extrapolation (R/E) sampling curve for species richness that can be rarefied to

smaller sample sizes or extrapolated to a larger sample size, guided by an estimated

asymptotic species richness. Chao & Jost (2012) showed that R/E to a given degree of sample

completeness was better able to judge the magnitude of the differences in richness among

communities, and ranked communities more efficiently, compared to traditional R/E to equal

sample sizes. The sample completeness is measured by sample coverage (the proportion of

the total number of individuals that belong to the species detected in the sample), a concept

originally developed by Alan Turing and I.J. Good in their cryptographic analysis during

World War II.

Hill number of any order (including species richness) is also dependent on sample size

and inventory completeness. Chao et al. (2014) extended Colwell et al. (2012) and Chao &

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 Jost (2012) to Hill numbers and proposed the sample-size- and coverage-based R/E of Hill

numbers as a unified framework for estimating species diversity, and for making statistical
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comparisons based on these estimates.

Here we first introduce two types of biodiversity data, briefly review the conceptual

background of Hill numbers and present two approaches to standardization. We introduce

iNEXT (iNterpolation/EXTrapolation), an R package that provides simple functions to

compute and plot sample-size and coverage-based R/E sampling curves, along with

confidence bands. We focus on the three most widely used members of the family of Hill

numbers (q = 0, 1 and 2) based on two types of biodiversity data: abundance data and

incidence data. The estimated asymptote along with a confidence interval for each diversity

measure is also computed. iNEXT offers three graphic displays. In addition to plots of the

sample-size- and coverage-based R/E sampling curves, iNEXT also plots the sample

completeness curve: this curve plots the sample completeness (as measured by sample

coverage) with respect to sample size. Several applications of iNEXT packages are reviewed.

Two examples (one for abundance data and the other one for incidence data) are used to

illustrate the use of iNEXT. A quick introduction to iNEXT via examples is provided in the

Appendix which is now included as a new vignette document for iNEXT in R and can be

viewed using the command vignette("Introduction", package="iNEXT"). This document and

detailed information about iNEXT functions are also available at

http://chao.stat.nthu.edu.tw/wordpress/software_download/. An online version of iNEXT

(https://chao.shinyapps.io/iNEXT/) is also available for users without an R background.

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 Two Types of Data
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Assume that there are S species in the focal assemblage, where S is unknown. Let

{ p1 , p2 ,..., pS } denote the true, unknown species relative abundances. In most biological

surveys, data can be generally classified into two types: individual-based abundance data and

sampling-unit-based incidence data, as described below.

ABUNDANCE DATA

For abundance data, the sampling unit is an individual. We assume a reference sample of n

individuals is selected from the assemblage. Let Xi denote the sample abundance or frequency

of the i-th species in the reference sample, i = 1, 2,..., S . Only those species with abundance

X ≥1 are detected in the sample. The input data for iNEXT for a single assemblage is the

sample abundance vector ( X 1 , X 2 ,..., X S ) . When there are N assemblages, input data consist

of an S by N abundance matrix or N lists of species abundances. In iNEXT, this type of

abundance data (from 1 to N assemblages) is specified by an argument

datatype="abundance".

INCIDENCE DATA

For incidence data, the sampling unit is usually a trap, net, quadrat, plot, or timed survey. A

reference sample consists of a species-by-sampling-unit incidence matrix with S rows and T

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 columns, where T denotes the number of sampling units; the (i, j) element is 1 if species i is
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detected in sampling unit j, and 0 otherwise. Let Yi be the row sum, the number of sampling

units in which species i is detected; here Yi is referred to as the sample species incidence

frequency and is analogous to Xi in the abundance data. There are two kinds of incidence

input data for iNEXT: (1) Incidence-raw data: for each assemblage, input data consist of a

species-by-sampling-unit matrix; when there are N assemblages, input data consist of N

matrices via a list object, with each matrix being a species-by-sampling-unit matrix. In

iNEXT, this type of data is specified by datatype="incidence_raw". (2)

Incidence-frequency data: input data for each assemblage consist of the number of sampling

units (T) followed by the observed incidence frequencies (Y1 , Y2 ,..., YS ) . When there are N

assemblages, input data consist of an S+1 by N matrix or N lists of species incidence

frequencies. The first entry of each column/list must be the total number of sampling units,

followed by the species incidence frequencies. In iNEXT, this type of data is specified by

datatype="incidence_freq".

Conceptual Background

HILL NUMBERS

Here, we briefly review the concept of Hill numbers for abundance data; see Chao et al.

(2014) for a similar conceptual background based on incidence data. Complete agreement

was reached in an Ecology forum (Ellison 2010) that Hill numbers should be the diversity

measure of choice. Hill (1973) integrated species richness and species relative abundances

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 into a class of diversity measures later called Hill numbers, or effective numbers of species,

defined as
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1 / (1− q )
 S 
q
D =   piq  . eqn 1
 i =1 

The parameter q determines the sensitivity of the measure to the relative frequencies.

When q=0, 0D is simply species richness, which counts species equally without regard to

their relative abundances. For q=1, eqn 1 is undefined, but its limit as q tends to 1 is the

exponential of the familiar Shannon index, referred to as Shannon diversity (Chao et al.

2014):

 S 
1
D = lim q D = exp −  pi log pi  .
q→1
 i =1 

The measure for q=1 counts individuals equally and thus counts species in proportion to their

abundances; the measure 1D can be interpreted as the effective number of common species in

the assemblage. The measure for q=2, referred to as Simpson diversity, discounts all but the

dominant species and can be interpreted as the effective number of dominant species in the

assemblage.

STANDARDIZATION: SAMPLE-SIZE-BASED R/E

Like species richness, the expected diversity in a sample of size m, denoted as qD(m), m

= 1, 2,…, is a non-decreasing function of size m; see Chao et al. (2014) for the theoretical

formula and its derivation. A diversity accumulation curve depicts qD(m) as a function of m.

In the special case of q=0, the curve reduces to the familiar species accumulation curve.

Based on a reference sample of size n, Chao et al. (2014) derived an interpolated diversity

estimator q
Dˆ (m ) for any rarefied sample of size m<n and an extrapolated diversity estimator

q
Dˆ ( n + m* ) for any enlarged sample of size n+m*; see Tables 1 and 2 of their paper. The

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 sample-size-based R/E curve includes a rarefaction part (which plots q
Dˆ (m ) as a function of

m<n), and an extrapolation part (which plots q

Dˆ ( n + m* ) as a function of n+m*); both join
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smoothly at the reference point (n, Sobs), where Sobs denotes the observed species richness in

the reference sample. The confidence intervals based on the bootstrap method developed by

Chao et al. (2014) also join smoothly.

For species richness, the size in the R/E curve can be extrapolated to at most double or

triple the minimum observed sample size, guided by an estimated asymptote. For Shannon

diversity and Simpson diversity, if the data are not too sparse, the extrapolation can be

reliably extended to infinity to attain the estimated asymptote provided in Chao et al. (2014).

STANDARDIZATION: COVERAGE-BASED R/E

The expected sample coverage for a hypothetical sample of size m, C(m), is also a

function of m. Chao & Jost (2012) derived an interpolated coverage estimator Cˆ ( m) for any

rarefied sample of size m<n and an extrapolated coverage estimator Cˆ ( n + m* ) for any

enlarged sample of size n+m*. The coverage-based sampling curve includes a rarefaction part

(which plots q
Dˆ (m ) as a function of Cˆ ( m) ), and an extrapolation part (which plots

q
Dˆ ( n + m* ) as a function of Cˆ ( n + m* ) ); both join smoothly at the reference sample point

( Cˆ ( n ) , Sobs), where Cˆ ( n ) denotes the estimated sample coverage for the reference sample.

The confidence intervals based on the bootstrap method (Chao & Jost, 2012) also join

smoothly. The curve can be extended to the coverage of the maximum size used in the

sample-size-based sampling curve.

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 Package Description
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The iNEXT package provides simple functions for computing and plotting seamless R/E

sampling curves for Hill numbers. The iNEXT package is available on CRAN

(https://cran.r-project.org/web/packages/iNEXT/index.html) and can be downloaded with a

standard installation procedure. For first-time installations, an additional visualization

extension package (ggplot2) must be loaded. The list of the functions in iNEXT and their

description are shown in Table 1; we demonstrate the use of these functions in Examples.

<Table 1 near here>

Applications

The functions in the iNEXT package have been applied to various types of data and have

potential to be useful in many research fields. These applications can be classified into four

categories as summarized below. In each category, we only provide one representative

reference due to space restriction.

(1) A non-asymptotic analysis: comparison of estimated diversities of standardized samples

with a common sample size or sample completeness. This analysis aims to compare

diversity estimates for equally-large or equally-complete samples; it is based on the

seamless rarefaction and extrapolation sampling curves of Hill numbers via our main

iNEXT() function (see Eren et al. 2016 for archaeological data).

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 (2) An asymptotic analysis: comparison of the estimated asymptotic diversities. It is based on
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statistical estimation of Hill numbers via three functions: ChaoRichness(),

ChaoShannon()and ChaoSimpson(); these functions return respectively the

estimated asymptote for Hill numbers of order q = 0, 1 and 2 (see Kendrick et al. 2015 for

ant data).

(3) Assessment of sample completeness of multiple samples via the iNEXT() function (see

Uchida & Ushimaru 2015 for grassland plants and herbivorous insects data).

(4) Comparison of point diversities for a specified sample size or a specified level of sample

coverage via the estimateD()function (see Mateo‐Tomás et al. 2015 for vertebrate

scavenger data).

Examples

Several data sets are included in the package for demonstration. Here we illustrate all

graphical displays using two data sets (spider for abundance data and ant for incidence

data). These data are presented in the supporting information; see Chao et al. (2014) for

analysis details and data sources/interpretations. Here we first use the spider data to explain

the arguments of the main function iNEXT(); the data consist of abundance data from two

canopy manipulation treatments (“Girdled” and “Logged”) of hemlock trees.

MAIN FUNCTION: iNEXT()

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 The main function iNEXT()returns the "iNEXT" object, including three data frames:
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$DataInfo, $iNextEst and $AsyEst, as explained for the spider example below.

For the spider data, the following code can be run to obtain output after the iNEXT

package is installed and the package ggplot2 is loaded (see the Appendix for more details):

> iNEXT(spider, q=c(0,1,2), datatype="abundance").

The first list, $DataInfo, summarizes the dataset: in the Girdled site, there were 26 species

among 168 individuals; in the Logged site, there were 37 species among 252 individuals. In

the Girdled treatment site, by default, 40 equally spaced knots (samples sizes) between 1 and

336 (=2x168, double the reference sample size) are selected. Diversity estimates and related

statistics are computed for these 40 knots (corresponding to sample sizes m = 1, 10, 19, …,

336), which locates the reference sample at the mid-point of the selected knots. The list

$iNextEst (as shown in the Appendix) includes diversity estimates along with related

statistics for rarefied samples of sizes m = 1, 10, …, 159, and also for extrapolated samples of

sizes m=177, 186, …, 336. In the list, the sample coverage estimate along with the 95% lower

and upper confidence limits are also shown; these estimates and confidence limits are used

for plotting the sample completeness curve and coverage-based R/E curves. The list

$AsyEst shows the asymptotic diversity estimates along with related statistics for Hill

numbers of order q=0, 1 and 2. These estimated asymptotes are calculated via the functions

ChaoRichness(), ChaoShannon(), and ChaoSimpson().

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 GRAPHIC DISPLAYS: ggiNEXT()
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The ggiNEXT() function, a wrapper around the ggplot2 package, serves to create a R/E

curve. The resulting object can be manipulated using the ggplot2 tools. The following

command returns the sample-size-based R/E curve:

ggiNEXT(x, type=1, se=TRUE, grey=FALSE, …)

Here x is an "iNEXT" object. Three types of curves are allowed: (1) Sample-size-based R/E

curve (type=1) with confidence intervals (if se=TRUE); see Figs. 1a (for q = 0, 1 and 2)

and 2a (for q = 0 only). (2) Sample completeness curve (type=2) with confidence intervals

(if se=TRUE); see Figs. 1b and 2b. This curve plots the sample coverage with respect to

sample size. (3) Coverage-based R/E curve (type=3) with confidence intervals (if

se=TRUE); see Figs. 1c (for q = 0, 1 and 2) and 2c (for q = 0 only). The user may also use

the argument grey=TRUE to plot black/white figures. Note that ggiNEXT allows

ggplot2 functions such as xlim(), ylim(), theme(), theme_bw(), etc., to be used

to modify the display settings; see the Appendix for examples.

INCIDENCE DATA

We use the tropical ant data (in the file ant) at five elevations (50m, 500m, 1070m, 1500m,

and 2000m) collected by Longino & Colwell (2011) in Costa Rica for illustration. The first

entry of each list must be the total number of sampling units. Figure 2 shows the three types

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 of sampling curves for species richness without grey backgrounds. Details are omitted here
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due to space restrictions.

POINT ESTIMATION FUNCTION: estimateD()

We also supply the function estimateD() to compute diversity estimates with q = 0, 1, 2

(all three levels of q are reported) for any particular level of sample size or any specified level

of sample coverage for either abundance data or incidence data. For example, the following

command returns the species diversity with a specified level of sample coverage of 98.5% for

the ant data.

> estimateD(ant, datatype="incidence_freq", base="coverage",

level=0.985). See the Appendix for details.

Alternative software
There is alternative software and R functions that provide similar tools for

rarefaction and extrapolation curves.

(1) The freeware EstimateS (Colwell 2013) with a full graphical user interface

obtains R/E sampling curves with confidence intervals for both abundance and

incidence data. All these tools in EstimateS are designed for species richness.

iNEXT is more comprehensive because iNEXT also provides the corresponding

output for Shannon diversity and Simpson diversity. EstimateS is a GUI interface,

which makes it hard to do reproducible science with it, whereas iNEXT R

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 package does do that. An on-line version of iNEXT is also available for users

without an R background; see the Introduction.

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(2) The function "rarefy", available in the R package vegan (Oksanen et al.

2015), provides rarefaction curves for species richness, but this function does not

include extrapolation.

Conclusion
We have reviewed the standardization methods for Hill numbers, presented the iNEXT

package, and illustrated the use of iNEXT in constructing two types (sample-size- and

coverage-based) of rarefaction and extrapolation curves with Hill numbers, along with a

sample completeness curve that links the two types of curves. For each type of curve, the

sampling curves with confidence intervals for species richness, Shannon diversity, and

Simpson diversity are suggested to quantify and compare species diversities in a unified

framework. Figures 1 and 2, respectively, show the sampling curves for abundance data and

incidence data. The package iNEXT provides an easy-to-use interface and efficiently uses all

data to make more robust and detailed inferences about the sampled assemblages, and also to

make objective comparisons of multiple assemblages. iNEXT will be soon extended to its

phylogenetic generalization, iNextPD (https://github.com/JohnsonHsieh/iNextPD), for

analyzing phylogenetic data.

Acknowledgements

The authors thank the Editor (Jana Vamosi), an Associate Editor, Robert Colwell, Jonathan

Lefcheck, Scott Chamberlain and an anonymous reviewer for very helpful and thoughtful

comments and suggestions. This research is supported by Taiwan Ministry of Science and

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 Technology under Contract 103-2628-M007-007. TCH was supported by a post-doctoral

fellowship, Taiwan Ministry of Science and Technology, Taiwan.

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Data accessibility

All data used in this paper are presented in the supporting information.

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 & O’Brien, M.J. (2016) Statistical Analysis of paradigmatic class richness supports greater
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paleoindian projectile-point diversity in the Southeast. American Antiquity, 81, 174–192.

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 Supporting Information
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Additional Supporting Information may be found in the online version of this article.

Appendix S1. A quick introduction to iNEXT via examples.

Table 1. List of the functions in the iNEXT package and their description

Type Function Description

Main function iNEXT() Interpolation and extrapolation of Hill
numbers

Asymptotic ChaoRichness() Estimation of species richness

diversity ChaoShannon() Estimation of Shannon diversity
estimation ChaoSimpson() Estimation of Simpson diversity
function

Point estimation estimateD() Estimation of species diversity with a

function particular sample size/coverage

Graphic displays ggiNEXT() ggplot2 extension for iNEXT object to

function plot rarefaction/extrapolation curves

Others DataInfo() Summary of basic data information

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 Figure Legends:
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Fig. 1. (a) Sample-size-based and (c) coverage-based rarefaction (solid line segment) and

extrapolation (dotted line segments) sampling curves with 95% confidence intervals (shaded

areas) for the spider data of two treatments, separately by diversity order: q=0 (species

richness, left panel), q=1 (Shannon diversity, middle panel) and q=2 (Simpson diversity, right

panel). The solid dots/triangles represent the reference samples. (b) Sample completeness

curves linking curves in (a) and (c).

Fig. 2. (a) Sample-size-based and (c) coverage-based rarefaction (solid line segment) and

extrapolation (dotted line segments) sampling curves for species richness (q=0) with 95%

confidence intervals (shaded areas) for the tropical ant data at five elevations. The solid dots

and the other four symbols represent the reference samples. (b) Sample completeness curves

linking curves in (a) and (c). iNEXT offers a customized graphic theme to change grey

background to black-and-white; see the Appendix for details.

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