# Beta Diversity

## N. Frerebeau

### 2025-09-04

\(\beta\)-diversity measures how different local systems are from one another (Moreno and Rodríguez 2010).

tabula allows to calculate several turnover and similarity measures from a count table (absolute frequencies giving the number of individuals for each category, i.e. a contingency table). It assumes that you keep your data tidy: each variable (type/taxa) must be saved in its own column and each observation (sample/case) must be saved in its own row.

    ## Install extra packages (if needed)
    # install.packages("folio") # Datasets

    ## Load packages
    library(tabula)

    ## Ceramic data from Lipo et al. 2015
    data("mississippi", package = "folio")

    ## Turnover
    turnover(mississippi, method = "whittaker")
    #> [1] 0.4925373

    ## Similarity
    BR <- similarity(mississippi, method = "brainerd")

    ## Plot
    plot_spot(BR, col = color("YlOrBr")(12))

plot of chunk intro

Under the hood, the `index_*()` functions are called (see details below).

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}

    ## Data from Magurran 1988, p. 162
    woodland <- matrix(
      data = c(TRUE, TRUE, TRUE, FALSE, FALSE, FALSE,
               TRUE, TRUE, TRUE, TRUE, TRUE, TRUE,
               FALSE, FALSE, TRUE, FALSE, TRUE, FALSE,
               FALSE, FALSE, FALSE, TRUE, TRUE, TRUE,
               FALSE, FALSE, FALSE, FALSE, TRUE, TRUE,
               FALSE, FALSE, FALSE, TRUE, FALSE, TRUE),
      nrow = 6, ncol = 6
    )
    colnames(woodland) <- c("Birch", "Oak", "Rowan", "Beech", "Hazel", "Holly")

# 1 Turnover

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.

Data are standardized on a presence/absence scale (\(0\)/$1$) beforehand.

## 1.1 Whittaker (1960)

$$ \beta_W = \frac{S}{\alpha} - 1 $$

    index_whittaker(woodland)
    #> [1] 1

Where \(\alpha\) is the mean sample diversity: \(\alpha = \frac{x_{\cdot \cdot}}{m}\)

## 1.2 Cody (1975)

$$ \beta_C = \frac{g(H) + l(H)}{2} - 1 $$

Where:

- \(g(H)\) is the number of taxa gained along the transect,
- \(l(H)\) is the number of taxa lost along the transect.

    index_cody(woodland)
    #> [1] 3

## 1.3 Routledge (1977)

## 1.4 Routledge 1

$$ \beta_R = \frac{S^2}{2 y_{\cdot \cdot} + S} - 1 $$

    index_routledge1(woodland)
    #> [1] 0.2857143

## 1.5 Routledge 2

$$ \beta_I = \log x_{\cdot \cdot} - \frac{\sum_{j = 1}^{p} x_{\cdot j} \log x_{\cdot j}}{x_{\cdot \cdot}} - \frac{\sum_{i = 1}^{m} x_{i \cdot} \log x_{i \cdot}}{x_{\cdot \cdot}} $$

    index_routledge2(woodland)
    #> [1] 0.5594978

## 1.6 Routledge 3

$$ \beta_E = \exp(\beta_I) - 1 $$

    index_routledge3(woodland)
    #> [1] 1.749794

## 1.7 Wilson & Shmida (1984)

$$ \beta_T = \frac{g(H) + l(H)}{2\alpha} $$

    index_wilson(woodland)
    #> [1] 1

# 2 Similarity

Similarity between two samples \(a\) and \(b\) 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 \(200\).

Thereafter, we denote by:

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

## 2.1 Qualitative similarity measures

Data are standardized on a presence/absence scale (\(0\)/$1$) beforehand.

### 2.1.1 Jaccard

$$ C_J = \frac{o_j}{S_a + S_b - o_j} $$

### 2.1.2 Dice (1945) - Sorensen (1948)

$$ C_S = \frac{2 \times o_j}{S_a + S_b} $$

## 2.2 Quantitative similarity measures

### 2.2.1 Brainerd (1951) - Robinson (1951)

$$ C_{BR} = 200 - \sum_{j = 1}^{S} \left| \frac{a_j \times 100}{\sum_{j = 1}^{S} a_j} - \frac{b_j \times 100}{\sum_{j = 1}^{S} b_j} \right|$$

### 2.2.2 Bray-Curtis

Bray and Curtis (1957) modified version of the Dice-Sorensen index.

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

### 2.2.3 Morisita-Horn

Horn (1966) modified version of the Morisita (1959) overlap index.

$$ C_{MH} = \frac{2 \sum_{j = 1}^{S} a_j \times b_j}{(\frac{\sum_{j = 1}^{S} a_j^2}{N_a^2} + \frac{\sum_{j = 1}^{S} b_j^2}{N_b^2}) \times N_a \times N_b} $$

# 3 References

Brainerd, G. W. 1951. The Place of Chronological Ordering in Archaeological Analysis. American Antiquity, 16(4), 301-313. DOI: 10.2307/276979.

Bray, J. R. & Curtis, J. T. (1957). An Ordination of the Upland Forest Communities of Southern Wisconsin. Ecological Monographs, 27(4), 325-349. DOI: 10.2307/1942268.

Cody, M. L. (1975). Towards a Theory of Continental Species Diversity: Bird Distributions Over Mediterranean Habitat Gradients. In M. L. Cody & J. M. Diamond (Eds.), Ecology and Evolution of Communities, 214-257. Cambridge, MA: Harvard University Press.

Dice, L. R. (1945). Measures of the Amount of Ecologic Association Between Species. Ecology, 26(3): 297-302. DOI: 10.2307/1932409.

Horn, H. S. (1966). Measurement of “Overlap” in Comparative Ecological Studies. The American Naturalist, 100(914): 419-424. DOI: 10.1086/282436.

Moreno, C. E. & Rodríguez, P. (2010). A Consistent Terminology for Quantifying Species Diversity? Oecologia, 163(2), 279-782. DOI: 10.1007/s00442-010-1591-7.

Mosrisita, M. (1959). Measuring of interspecific association and similarity between communities. Memoirs of the Faculty of Science, Kyushu University, Series E, 3:65-80.

Robinson, W. S. (1951). A Method for Chronologically Ordering Archaeological Deposits. American Antiquity, 16(4), 293-301. DOI: 10.2307/276978.

Routledge, R. D. (1977). On Whittaker’s Components of Diversity. Ecology, 58(5), 1120-1127. DOI: 10.2307/1936932.

Sorensen, T. (1948). A Method of Establishing Groups of Equal Amplitude in Plant Sociology Based on Similarity of Species Content and Its Application to Analyses of the Vegetation on Danish Commons. Kongelige Danske Videnskabernes Selskab, 5(4): 1-34.

Whittaker, R. H. (1960). Vegetation of the Siskiyou Mountains, Oregon and California. Ecological Monographs, 30(3), 279-338. DOI: 10.2307/1943563..

Wilson, M. V. & Shmida, A. (1984). Measuring Beta Diversity with Presence-Absence Data. The Journal of Ecology, 72(3), 1055-1064. DOI: 10.2307/2259551.