Diversity Measures

N. Frerebeau

2020-03-19

# Load packages
library(tabula)
library(magrittr)

Thereafter, we denote by:

Diversity in ecology describes complex interspecific interactions between and within communities under a variety of environmental conditions (Bobrowsky and Ball 1989). This concept covers different components, allowing different aspects of interspecific interactions to be measured.

1 \(\alpha\)-diversity

1.1 Richness and Rarefaction

The number of different taxa, provides an instantly comprehensible expression of diversity. While the number of taxa within a sample is easy to ascertain, as a term, it makes little sense: some taxa may not have been seen, or there may not be a fixed number of taxa (e.g. in an open system; Peet 1974). As an alternative, richness (\(R\)) can be used for the concept of taxa number (McIntosh 1967). Richness refers to the variety of taxa/species/types present in an assemblage or community (Bobrowsky and Ball 1989) as “the number of species present in a collection containing a specified number of individuals” (Hurlbert 1971).

It is not always possible to ensure that all sample sizes are equal and the number of different taxa increases with sample size and sampling effort (Magurran 1988). Then, rarefaction (\(\hat{S}\)) is the number of taxa expected if all samples were of a standard size \(n\) (i.e. taxa per fixed number of individuals). Rarefaction assumes that imbalances between taxa are due to sampling and not to differences in actual abundances.

1.1.1 Definitions

Richness measures (Bobrowsky and Ball 1989). *: implemented.
Measure Reference
* \[ R_{1} = \frac{S - 1}{\ln N} \] Margalef (1958)
\[ R_{2} = \frac{S}{\log N} \] Odum, Cantlon, and Kornicker (1960)
* \[ R_{3} = \frac{S}{\sqrt{N}} \] Menhinick (1964)
\[ R_{4} = \frac{S}{\log A} \] Gleason (1922)
Rarefaction measures (Bobrowsky and Ball 1989). *: implemented.
Measure Reference
\[ \hat{S}_{1} = \alpha - \left[ \ln \left( 1 - x \right) \right] \] Fisher, Corbet, and Williams (1943)
\[ \hat{S}_{2} = y_{0} \hat{\sigma} \sqrt{2 \pi} \] Preston (1948)
\[ \hat{S}_{3} = 2.07 \left( \frac{N}{m} \right)^{0.262} \] Preston (1962a), Preston (1962b)
\[ \hat{S}_{4} = 2.07 \left( \frac{N}{m} \right)^{0.262} A^{0.262} \] Macarthur (1965)
\[ \hat{S}_{5} = k A^{d} \] Kilburn (1966)
\[ \hat{S}_{6} = \frac{a N}{1 + b N} \] family=Caprariis, Lindemann, and Collins (1976)
* \[ \hat{S}_{7} = \sum_{i = 1}^{S} 1 - \frac{{N - N_i} \choose n}{N \choose n} \] Hurlbert (1971), Sander (1968)

Where:

  • \(S\) is the number of observed species/types,
  • \(N_i\) is the number of individuals in the \(i\)-th species/type,
  • \(N = \sum_{i = 1}^{S} N_i\) is the total number of individuals,
  • \(A\) is the area of the isolate or collection.
  • \(m\) is the number of individuals in the rarest species/type.
  • \(\alpha\) is the Fisher’s slope constant,
  • \(y_{0}\) is the number of species/types in the modal class interval,
  • \(\hat{\sigma}\) is the estimate of the standard deviation,
  • \(n\) is the sub-sample size,
  • \(R\) is the constant of rate increment,
  • \(\hat{S}\) is the number of expected or predicted species/types,
  • \(k\), \(d\), \(a\) and \(b\) are empirically derived coefficients of regression.

1.1.1.1 Chao1 estimator (Chao 1984)

\[ \hat{S}_{Chao1} = \begin{cases} S + \frac{N - 1}{N} \frac{s_1^2}{2 s_2} & s_2 > 0 \\ S + \frac{N - 1}{N} \frac{s_1 (s_1 - 1)}{2} & s_2 = 0 \end{cases} \]

In the special case of homogeneous case, a bias-corrected estimator is:

\[ \hat{S}_{bcChao1} = S + \frac{N - 1}{N} \frac{s_1 (s_1 - 1)}{2 s_2 + 1}\]

The improved Chao1 estimator makes use of the additional information of tripletons and quadrupletons (Chiu et al. 2014):

\[ \hat{S}_{iChao1} = \hat{S}_{Chao1} + \frac{N - 3}{4 N} \frac{s_3}{s_4} \times \max\left(s_1 - \frac{N - 3}{N - 1} \frac{s_2 s_3}{2 s_4} , 0\right)\]

1.1.1.2 Abundance-based Coverage Estimator (Chao and Lee 1992)

\[ \hat{S}_{ACE} = \hat{S}_{abun} + \frac{\hat{S}_{rare}}{\hat{C}_{rare}} + \frac{s_1}{\hat{C}_{rare}} \times \hat{\gamma}^2_{rare} \]

Where \(\hat{S}_{rare} = \sum_{i = 1}^{k} s_i\) is the number of rare taxa, \(\hat{S}_{abun} = \sum_{i > k}^{N} s_i\) is the number of abundant taxa (for a given cut-off value \(k\)), \(\hat{C}_{rare} = 1 - \frac{s_1}{N_{rare}}\) is the Turing’s coverage estimate and:

\[ \hat{\gamma}^2_{rare} = \max\left[\frac{\hat{S}_{rare}}{\hat{C}_{rare}} \frac{\sum_{i = 1}^{k} i(i - 1)s_i}{\left(\sum_{i = 1}^{k} is_i\right)\left(\sum_{i = 1}^{k} is_i - 1\right)} - 1, 0\right] \]

1.1.1.3 Chao2 estimator (Chao 1987)

For replicated incidence data (i.e. a \(m \times p\) logical matrix), the Chao2 estimator is:

\[ \hat{S}_{Chao2} = \begin{cases} S + \frac{m - 1}{m} \frac{q_1^2}{2 q_2} & q_2 > 0 \\ S + \frac{m - 1}{m} \frac{q_1 (q_1 - 1)}{2} & q_2 = 0 \end{cases} \]

Improved Chao2 estimator (Chiu et al. 2014):

\[ \hat{S}_{iChao2} = \hat{S}_{Chao2} + \frac{m - 3}{4 m} \frac{q_3}{q_4} \times \max\left(q_1 - \frac{m - 3}{m - 1} \frac{q_2 q_3}{2 q_4} , 0\right)\]

1.1.1.4 Incidence-based Coverage Estimator (Chao and Chiu 2016)

\[ \hat{S}_{ICE} = \hat{S}_{freq} + \frac{\hat{S}_{infreq}}{\hat{C}_{infreq}} + \frac{q_1}{\hat{C}_{infreq}} \times \hat{\gamma}^2_{infreq} \]

Where \(\hat{S}_{infreq} = \sum_{i = 1}^{k} q_i\) is the number of infrequent taxa, \(\hat{S}_{freq} = \sum_{i > k}^{N} q_i\) is the number of frequent taxa (for a given cut-off value \(k\)), \(\hat{C}_{infreq} = 1 - \frac{Q_1}{\sum_{i = 1}^{k} iq_i}\) is the Turing’s coverage estimate and:

\[ \hat{\gamma}^2_{infreq} = \max\left[\frac{\hat{S}_{infreq}}{\hat{C}_{infreq}} \frac{m_{infreq}}{m_{infreq} - 1} \frac{\sum_{i = 1}^{k} i(i - 1)q_i}{\left(\sum_{i = 1}^{k} iq_i\right)\left(\sum_{i = 1}^{k} iq_i - 1\right)} - 1, 0\right] \]

Where \(m_{infreq}\) is the number of sampling units that include at least one infrequent species.

1.1.2 Usage

1.2 Heterogeneity and Evenness

1.2.1 Definitions

Diversity measurement assumes that all individuals in a specific taxa are equivalent and that all types are equally different from each other (Peet 1974). A measure of diversity can be achieved by using indices built on the relative abundance of taxa. These indices (sometimes referred to as non-parametric indices) benefit from not making assumptions about the underlying distribution of taxa abundance: they only take relative abundances of the species that are present and species richness into account. Peet (1974) refers to them as indices of heterogeneity (\(H\)).

Diversity indices focus on one aspect of the taxa abundance and emphasize either richness (weighting towards uncommon taxa) or dominance (weighting towards abundant taxa; Magurran 1988).

Evenness (\(E\)) is a measure of how evenly individuals are distributed across the sample.

1.2.1.1 Information theory index

1.2.1.1.1 Shannon-Wiener diversity index

The Shannon-Wiener index (Shannon 1948) assumes that individuals are randomly sampled from an infinite population and that all taxa are represented in the sample (it does not reflect the sample size). The main source of error arises from the failure to include all taxa in the sample: this error increases as the proportion of species discovered in the sample declines (Peet 1974; Magurran 1988). The maximum likelihood estimator (MLE) is used for the relative abundance, this is known to be negatively biased by sample size.

Heterogeneity for an infinite sample:

\[ H' = - \sum_{i = 1}^{S} p_i \ln p_i \]

Heterogeneity for a finite sample:

\[ H' = - \sum_{i = 1}^{S} \frac{n_i}{N} \ln \frac{n_i}{N} \]

Evenness:

\[ E = \frac{H}{H_{max}} = \frac{H'}{\ln S} = - \sum_{i = 1}^{S} p_i \log_S p_i \]

When \(p_i\) is unknown in the population, an estimate is given by \(\hat{p}_i =\frac{n_i}{N}\) (maximum likelihood estimator - MLE). As the use of \(\hat{p}_i\) results in a biased estimate, Hutcheson (1970) and Bowman et al. (1971) suggest the use of:

\[ \hat{H}' = - \sum_{i = 1}^{S} \hat{p}_i \ln \hat{p}_i - \frac{S - 1}{N} + \frac{1 - \sum_{i = 1}^{S} \hat{p}_i^{-1}}{12N^2} + \frac{\sum_{i = 1}^{S} (\hat{p}_i^{-1} - \hat{p}_i^{-2})}{12N^3} + \cdots \]

This error is rarely significant (Peet 1974), so the unbiased form is not implemented here (for now).

1.2.1.1.2 Brillouin diversity index

The Brillouin index (Brillouin 1956) describes a known collection: it does not assume random sampling in an infinite population. Pielou (1975) and Laxton (1978) argues for the use of the Brillouin index in all circumstances, especially in preference to the Shannon index.

Diversity:

\[ H' = \frac{\ln (N!) - \sum_{i = 1}^{S} \ln (n_i!)}{N} \]

Evenness:

\[ E = \frac{H'}{H'_{max}} \]

with:

\[ H'_{max} = \frac{1}{N} \ln \frac{N!}{\left( \lfloor \frac{N}{S} \rfloor! \right)^{S - r} \left[ \left( \lfloor \frac{N}{S} \rfloor + 1 \right)! \right]^{r}} \]

where: \(r = N - S \lfloor \frac{N}{S} \rfloor\).

1.2.1.2 Dominance index

The following methods return a dominance index, not the reciprocal or inverse form usually adopted, so that an increase in the value of the index accompanies a decrease in diversity.

1.2.1.2.1 Simpson index

The Simpson index (Simpson 1949) expresses the probability that two individuals randomly picked from a finite sample belong to two different types. It can be interpreted as the weighted mean of the proportional abundances. This metric is a true probability value, it ranges from \(0\) (perfectly uneven) to \(1\) (perfectly even).

Dominance for an infinite sample:

\[ D = \sum_{i = 1}^{S} p_i^2 \]

Dominance for a finite sample:

\[ D = \sum_{i = 1}^{S} \frac{n_i \left( n_i - 1 \right)}{N \left( N - 1 \right)} \]

1.2.1.2.2 McIntosh index

The McIntosh index (McIntosh 1967) expresses the heterogeneity of a sample in geometric terms. It describes the sample as a point of a \(S\)-dimensional hypervolume and uses the Euclidean distance of this point from the origin.

Dominance:

\[ D = \frac{N - U}{N - \sqrt{N}} \]

Evenness:

\[ E = \frac{N - U}{N - \frac{N}{\sqrt{S}}} \]

where \(U\) is the distance of the sample from the origin in an \(S\) dimensional hypervolume:

\[U = \sqrt{\sum_{i = 1}^{S} n_i^2}\]

1.2.1.2.3 Berger-Parker index

The Berger-Parker index (Berger and Parker 1970) expresses the proportional importance of the most abundant type. This metric is highly biased by sample size and richness, moreover it does not make use of all the information available from sample.

Dominance:

\[ D = \frac{n_{max}}{N} \]

1.2.2 Usage

Note that berger, mcintosh and simpson methods return a dominance index, not the reciprocal form usually adopted, so that an increase in the value of the index accompanies a decrease in diversity.

Corresponding evenness can also be computed :

2 \(\beta\)-diversity

2.1 Turnover

2.1.1 Definitions

The following methods can be used to ascertain the degree of turnover in taxa composition along a gradient on qualitative (presence/absence) data. This assumes that the order of the matrix rows (from 1 to \(m\)) follows the progression along the gradient/transect.

We denote the \(m \times p\) incidence matrix by \(X = \left[ x_{ij} \right] ~\forall i \in \left[ 1,m \right], j \in \left[ 1,p \right]\) and the \(p \times p\) corresponding co-occurrence matrix by \(Y = \left[ y_{ij} \right] ~\forall i,j \in \left[ 1,p \right]\), with row and column sums:

\[\begin{align} x_{i \cdot} = \sum_{j = 1}^{p} x_{ij} && x_{\cdot j} = \sum_{i = 1}^{m} x_{ij} && x_{\cdot \cdot} = \sum_{j = 1}^{p} \sum_{i = 1}^{m} x_{ij} && \forall x_{ij} \in \lbrace 0,1 \rbrace \\ y_{i \cdot} = \sum_{j \geqslant i}^{p} y_{ij} && y_{\cdot j} = \sum_{i \leqslant j}^{p} y_{ij} && y_{\cdot \cdot} = \sum_{i = 1}^{p} \sum_{j \geqslant i}^{p} y_{ij} && \forall y_{ij} \in \lbrace 0,1 \rbrace \end{align}\]

Turnover measures. *: implemented.
Measure Reference
* \[ \beta_W = \frac{S}{\alpha} - 1 \] Whittaker (1960)
* \[ \beta_C = \frac{g(H) + l(H)}{2} - 1 \] Cody (1975)
* \[ \beta_R = \frac{S^2}{2 y_{\cdot \cdot} + S} - 1 \] Routledge (1977)
* \[ \beta_I = \log x_{\cdot \cdot} - \frac{\sum_{j = 1}^{p} x_{\cdot j} \log x_{\cdot j}}{x_{\cdot \cdot}} - \frac{\sum_{i = 1}^{m} x_{i \cdot} \log x_{i \cdot}}{x_{\cdot \cdot}} \] Routledge (1977)
* \[ \beta_E = \exp(\beta_I) - 1 \] Routledge (1977)
* \[ \beta_T = \frac{g(H) + l(H)}{2\alpha} \] Wilson and Shmida (1984)

Where:

  • \(\alpha\) is the mean sample diversity: \(\alpha = \frac{x_{\cdot \cdot}}{m}\),
  • \(g(H)\) is the number of taxa gained along the transect,
  • \(l(H)\) is the number of taxa lost along the transect.

2.1.2 Usage

2.2 Similarity

2.2.1 Definitions

Similarity between two samples \(a\) and \(b\) or between two types \(x\) and \(y\) can be measured as follow.

These indices provide a scale of similarity from \(0\)-\(1\) where \(1\) is perfect similarity and \(0\) is no similarity, with the exception of the Brainerd-Robinson index which is scaled between \(0\) and \(200\).

Qualitative similarity measures (between samples). *: implemented.
Measure Reference
* \[ C_J = \frac{o_j}{S_a + S_b - o_j} \] Jaccard
* \[ C_S = \frac{2 \times o_j}{S_a + S_b} \] Sorenson
Quantitative similarity measures (between samples). *: implemented.
Measure Reference
* \[ C_{BR} = 200 - \sum_{j = 1}^{S} \left| \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right|\] Brainerd (1951), Robinson (1951)
* \[ C_N = \frac{2 \sum_{j = 1}^{S} \min(a_j, b_j)}{N_a + N_b} \] Bray and Curtis (1957), Sorenson
* \[ C_{MH} = \frac{2 \sum_{j = 1}^{S} a_j \times b_j}{(\frac{\sum_{j = 1}^{S} a_j^2}{N_a^2} + \frac{\sum_{j = 1}^{S} b_j^2}{N_b^2}) \times N_a \times N_b} \] Morisita-Horn
Quantitative similarity measures (between types). *: implemented.
Measure Reference
* \[ C_{Bi} = \frac{o_i - N \times p}{\sqrt{N \times p \times (1 - p)}} \] Kintigh (2006)

Where:

  • \(S_a\) and \(S_b\) denote the total number of taxa observed in samples \(a\) and \(b\), respectively,
  • \(N_a\) and \(N_b\) denote the total number of individuals in samples \(a\) and \(b\), respectively,
  • \(a_j\) and \(b_j\) denote the number of individuals in the \(j\)-th type/taxon, \(j \in \left[ 1,S \right]\),
  • \(x_i\) and \(y_i\) denote the number of individuals in the \(i\)-th sample/case, \(i \in \left[ 1,m \right]\),
  • \(o_i\) denotes the number of sample/case common to both type/taxon: \(o_i = \sum_{k = 1}^{m} x_k \cap y_k\),
  • \(o_j\) denotes the number of type/taxon common to both sample/case: \(o_j = \sum_{k = 1}^{S} a_k \cap b_k\).

3 Abundance models

Ranks vs abundance plot can be used for abundance models (model fitting will be implemented in a future release):

mississippi %>%
  as_count() %>%
  plot_rank(log = "xy", facet = FALSE) +
  ggplot2::theme_bw() +
  khroma::scale_color_discreterainbow()

References

Berger, W. H., and F. L. Parker. 1970. “Diversity of Planktonic Foraminifera in Deep Sea Sediments.” Science 168 (3937): 1345–7. https://doi.org/10.1126/science.168.3937.1345.

Bobrowsky, Peter, T., and Bruce F. Ball. 1989. “The Theory and Mechanics of Ecological Diversity in Archaeology.” In Quantifying Diversity in Archaeology, edited by Robert D. Leonard and George T. Jones, 4–12. New Directions in Archaeology. Cambridge: Cambridge University Press.

Bowman, K. O., K. Hutcheson, E. P. Odum, and L. R. Shenton. 1971. “Comments on the Distribution of Indices of Diversity.” In Statistical Ecology, edited by E. C. Patil, E. C. Pielou, and W. E. Waters, 3:315–66. University Park, PA: Pennsylvania State University Press.

Brainerd, George W. 1951. “The Place of Chronological Ordering in Archaeological Analysis.” American Antiquity 16 (04): 301–13. https://doi.org/10.2307/276979.

Bray, J. Roger, and J. T. Curtis. 1957. “An Ordination of the Upland Forest Communities of Southern Wisconsin.” Ecological Monographs 27 (4): 325–49. https://doi.org/10.2307/1942268.

Brillouin, Leon. 1956. Science and Information Theory. New York: Academic Press.

Chao, Anne. 1984. “Nonparametric Estimation of the Number of Classes in a Population.” Scandinavian Journal of Statistics 11 (4): 265–70.

———. 1987. “Estimating the Population Size for Capture Recapture Data with Unequal Catchability.” Biometrics 43 (4): 783–91. https://doi.org/10.2307/2531532.

Chao, Anne, and Chun-Huo Chiu. 2016. “Species Richness: Estimation and Comparison.” In Wiley StatsRef: Statistics Reference Online, edited by N. Balakrishnan, Theodore Colton, Brian Everitt, Walter Piegorsch, Fabrizio Ruggeri, and Jozef L. Teugels, 1–26. Chichester, UK: John Wiley & Sons, Ltd. https://doi.org/10.1002/9781118445112.stat03432.pub2.

Chao, Anne, and Shen-Ming Lee. 1992. “Estimating the Number of Classes via Sample Coverage.” Journal of the American Statistical Association 87 (417): 210–17. https://doi.org/10.1080/01621459.1992.10475194.

Chiu, Chun-Huo, Yi-Ting Wang, Bruno A. Walther, and Anne Chao. 2014. “An Improved Nonparametric Lower Bound of Species Richness via a Modified Good-Turing Frequency Formula.” Biometrics 70 (3): 671–82. https://doi.org/10.1111/biom.12200.

Cody, M. L. 1975. “Towards a Theory of Continental Species Diversity: Bird Distributions over Mediterranean Habitat Gradients.” In Ecology and Evolution of Communities, edited by M. L. Cody and J. M. Diamond, 214–57. Cambridge, MA: Harvard University Press.

family=Caprariis, prefix=de, given=Pascal, Richard H. Lindemann, and Catharine M. Collins. 1976. “A Method for Determining Optimum Sample Size in Species Diversity Studies.” Journal of the International Association for Mathematical Geology 8 (5): 575–81. https://doi.org/10.1007/BF01042995.

Fisher, R. A., A. Steven Corbet, and C. B. Williams. 1943. “The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population.” The Journal of Animal Ecology 12 (1): 42. https://doi.org/10.2307/1411.

Gleason, Henry Allan. 1922. “On the Relation Between Species and Area.” Ecology 3 (2): 158–62. https://doi.org/10.2307/1929150.

Hurlbert, Stuart H. 1971. “The Nonconcept of Species Diversity: A Critique and Alternative Parameters.” Ecology 52 (4): 577–86. https://doi.org/10.2307/1934145.

Hutcheson, K. 1970. “A Test for Comparing Diversity Based on the Shannon Formula.” Journal of Theoretical Biology 29 (1): 151–54. https://doi.org/10.1016/0022-5193(70)90124-4.

Kilburn, Paul D. 1966. “Analysis of the Species-Area Relation.” Ecology 47 (5): 831–43. https://doi.org/10.2307/1934269.

Kintigh, Keith. 2006. “Ceramic Dating and Type Associations.” In Managing Archaeological Data: Essays in Honor of Sylvia W. Gaines, edited by Jeffrey Hantman and Rachel Most, 17–26. Anthropological Research Paper 57. Tempe, AZ: Arizona State University. https://doi.org/10.6067/XCV8J38QSS.

Laxton, R.R. 1978. “The Measure of Diversity.” Journal of Theoretical Biology 70 (1): 51–67. https://doi.org/10.1016/0022-5193(78)90302-8.

Macarthur, Robert H. 1965. “Patterns of Species Diversity.” Biological Reviews 40 (4): 510–33. https://doi.org/10.1111/j.1469-185X.1965.tb00815.x.

Magurran, Anne E. 1988. Ecological Diversity and Its Measurement. Princeton, NJ: Princeton University Press.

Margalef, R. 1958. “Information Theory in Ecology.” General Systems 3: 36–71.

McIntosh, Robert P. 1967. “An Index of Diversity and the Relation of Certain Concepts to Diversity.” Ecology 48 (3): 392–404. https://doi.org/10.2307/1932674.

Menhinick, Edward F. 1964. “A Comparison of Some Species-Individuals Diversity Indices Applied to Samples of Field Insects.” Ecology 45 (4): 859–61. https://doi.org/10.2307/1934933.

Odum, H. T., J. E. Cantlon, and L. S. Kornicker. 1960. “An Organizational Hierarchy Postulate for the Interpretation of Species-Individual Distributions, Species Entropy, Ecosystem Evolution, and the Meaning of a Species-Variety Index.” Ecology 41 (2): 395–95. https://doi.org/10.2307/1930248.

Peet, R. K. 1974. “The Measurement of Species Diversity.” Annual Review of Ecology and Systematics 5 (1): 285–307. https://doi.org/10.1146/annurev.es.05.110174.001441.

Pielou, E. C. 1975. Ecological Diversity. New York: Wiley.

Preston, F. W. 1948. “The Commonness, and Rarity, of Species.” Ecology 29 (3): 254–83. https://doi.org/10.2307/1930989.

———. 1962a. “The Canonical Distribution of Commonness and Rarity: Part I.” Ecology 43 (2): 185. https://doi.org/10.2307/1931976.

———. 1962b. “The Canonical Distribution of Commonness and Rarity: Part Ii.” Ecology 43 (3): 410–32. https://doi.org/10.2307/1933371.

Robinson, W. S. 1951. “A Method for Chronologically Ordering Archaeological Deposits.” American Antiquity 16 (04): 293–301. https://doi.org/10.2307/276978.

Routledge, R. D. 1977. “On Whittaker’s Components of Diversity.” Ecology 58 (5): 1120–7. https://doi.org/10.2307/1936932.

Sander, Howard L. 1968. “Marine Benthic Diversity: A Comparative Study.” The American Naturalist 102 (925): 243–82. https://www.jstor.org/stable/2459027.

Shannon, C. E. 1948. “A Mathematical Theory of Communication.” The Bell System Technical Journal 27: 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.

Simpson, E. H. 1949. “Measurement of Diversity.” Nature 163 (4148): 688–88. https://doi.org/10.1038/163688a0.

Whittaker, R. H. 1960. “Vegetation of the Siskiyou Mountains, Oregon and California.” Ecological Monographs 30 (3): 279–338. https://doi.org/10.2307/1943563.

Wilson, M. V., and A. Shmida. 1984. “Measuring Beta Diversity with Presence-Absence Data.” The Journal of Ecology 72 (3): 1055–64. https://doi.org/10.2307/2259551.