```
# Load packages
library(tabula)
library(folio) # Datasets
```

Thereafter, we denote by:

- \(S\) the total number of taxa recorded,
- \(i\) the rank of the taxon
- \(N_i\) the number of individuals in the \(i\)-th taxon,
- \(N = \sum N_i\) the total number of individuals,
- \(p_i\) the relative proportion of the \(i\)-th taxon in the population,
- \(s_k\) the number of taxa with \(k\) individuals,
- \(q_i\) the incidence of the \(i\)-th taxon.

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.

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.

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) |

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) |

\[ \hat{S}_{6} = \frac{a N}{1 + b N} \] | de 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.

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

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):

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

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

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:

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

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):

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

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

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.

```
richness(mississippi, method = "margalef")
#> [1] 0.5963696 0.4524421 0.6971143 0.6193544 0.5599404 0.4577237 0.7292886
#> [8] 0.7779583 1.0304965 0.9224182 1.1892416 1.1412278 1.5518107 1.2645413
#> [15] 1.2090820 1.0903435 1.1570758 1.1892416 1.2552092 1.0158754
```

```
composition(mississippi, method = "chao1")
#> [1] 4.000000 4.000000 6.000000 5.000000 5.000000 3.000000 5.000000
#> [8] 5.000000 7.984375 6.000000 8.000000 7.000000 8.494505 10.000000
#> [15] 10.000000 8.998371 8.498821 8.249306 10.000000 8.499491
```

*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.

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).

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

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.

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:

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

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:

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

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:

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

```
heterogeneity(mississippi, method = "shannon")
#> [1] 1.2027955 0.7646565 0.9293974 0.8228576 0.7901428 0.9998430 1.2051989
#> [8] 1.1776226 1.1533432 1.2884172 1.1725355 1.5296294 1.7952443 1.1627477
#> [15] 1.0718463 0.9205717 1.1751002 0.7307620 1.1270126 1.0270291
```

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 :

```
evenness(mississippi, method = "shannon")
#> [1] 0.8676335 0.5515831 0.5187066 0.5112702 0.4909433 0.9100964 0.7488322
#> [8] 0.7316981 0.6436931 0.7190793 0.5638704 0.7860740 0.8633300 0.5049749
#> [15] 0.4654969 0.4427014 0.5651037 0.3514222 0.4894554 0.4938966
```

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:

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

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.

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 \(200\).

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 * |

Measure | Reference |
---|---|

\[ C_{BR} = 200 - \sum_{j = 1}^{S} \left| \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right|\] | (brainerd1951?), (robinson1951?) * |

\[ C_N = \frac{2 \sum_{j = 1}^{S} \min(a_j, b_j)}{N_a + N_b} \] | 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 * |

Measure | Reference |
---|---|

\[ C_{Bi} = \frac{o_i - N \times p}{\sqrt{N \times p \times (1 - p)}} \] | (kintigh2006?) * |

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: \(o_j = \sum_{k = 1}^{S} a_k \cap b_k\).

```
# Brainerd-Robinson (similarity between assemblages)
<- similarity(mississippi, method = "brainerd")
BR plot_spot(BR) +
::scale_colour_YlOrBr() khroma
```

```
# Binomial co-occurrence (similarity between types)
<- similarity(mississippi, method = "binomial")
BI plot_spot(BI) +
::scale_colour_PRGn() khroma
```

```
# Baxter rarefaction
<- rarefaction(mississippi, sample = 100, method = "baxter")
RA plot(RA)
```

Kintigh (1989)

```
## Data from Conkey 1980, Kintigh 1989, p. 28
<- heterogeneity(chevelon, method = "shannon")
HE <- simulate(HE)
HE_sim plot(HE_sim)
<- richness(chevelon, method = "count")
RI <- simulate(RI)
RI_sim plot(RI_sim)
```

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

```
plot_rank(mississippi, log = "xy") +
::theme_bw() +
ggplot2::scale_color_discreterainbow() khroma
```

Berger, W. H., and F. L. Parker. 1970. “Diversity of
Planktonic Foraminifera in Deep Sea
Sediments.” *Science* 168 (3937): 1345–47. 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.

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.

Caprariis, Pascal de, 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.

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.

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 W. 1989. “Sample Size,
Significance, and Measures of
Diversity.” In *Quantifying
Diversity in Archaeology*, edited by
Robert D. Leonard and George T. Jones, 25–36. New
Directions in Archaeology.
Cambridge: Cambridge University Press.

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.

Routledge, R. D. 1977. “On Whittaker’s
Components of Diversity.”
*Ecology* 58 (5): 1120–27. 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.