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Beta Diversity

N. Frerebeau

2025-05-01

Source: vignettes/beta.Rmd
beta.Rmd

β\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))

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

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

xi⋅=∑j=1pxijx⋅j=∑i=1mxijx⋅⋅=∑j=1p∑i=1mxij∀xij∈{0,1}yi⋅=∑j≥ipyijy⋅j=∑i≤jpyijy⋅⋅=∑i=1p∑j≥ipyij∀yij∈{0,1}\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")

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 mm) follows the progression along the gradient/transect.

Data are standardized on a presence/absence scale (00/11) beforehand.

Whittaker (1960)

βW=Sα−1 \beta_W = \frac{S}{\alpha} - 1 

index_whittaker(woodland)
#> [1] 1

Where α\alpha is the mean sample diversity: α=x⋅⋅m\alpha = \frac{x_{\cdot \cdot}}{m}

Cody (1975)

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

Where:

  • g(H)g(H) is the number of taxa gained along the transect,
  • l(H)l(H) is the number of taxa lost along the transect.
index_cody(woodland)
#> [1] 3

Routledge (1977)

Routledge 1

βR=S22y⋅⋅+S−1 \beta_R = \frac{S^2}{2 y_{\cdot \cdot} + S} - 1 

index_routledge1(woodland)
#> [1] 0.2857143

Routledge 2

βI=log⁡x⋅⋅−∑j=1px⋅jlog⁡x⋅jx⋅⋅−∑i=1mxi⋅log⁡xi⋅x⋅⋅ \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

Routledge 3

βE=exp⁡(βI)−1 \beta_E = \exp(\beta_I) - 1 

index_routledge3(woodland)
#> [1] 1.749794

Wilson & Shmida (1984)

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

index_wilson(woodland)
#> [1] 1

Similarity

Similarity between two samples aa and bb can be measured as follow.

These indices provide a scale of similarity from 00-11 where 11 is perfect similarity and 00 is no similarity, with the exception of the Brainerd-Robinson index which is scaled between 00 and 200200.

Thereafter, we denote by:

  • SaS_a and SbS_b the total number of taxa observed in samples aa and bb, respectively,
  • NaN_a and NbN_b the total number of individuals in samples aa and bb, respectively,
  • aja_j and bjb_j the number of individuals in the jj-th type/taxon, j∈[1,S]j \in \left[ 1,S \right],
  • ojo_j the number of type/taxon common to both sample/case: oj=∑k=1Sak∩bko_j = \sum_{k = 1}^{S} a_k \cap b_k.

Qualitative similarity measures

Data are standardized on a presence/absence scale (00/11) beforehand.

Jaccard

CJ=ojSa+Sb−oj C_J = \frac{o_j}{S_a + S_b - o_j}

Dice (1945) - Sorensen (1948)

CS=2×ojSa+Sb C_S = \frac{2 \times o_j}{S_a + S_b}

Quantitative similarity measures

Brainerd (1951) - Robinson (1951)

CBR=200−∑j=1S|aj×100∑j=1Saj−bj×100∑j=1Sbj| 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|

Bray-Curtis

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

CN=2∑j=1Smin⁡(aj,bj)Na+Nb C_N = \frac{2 \sum_{j = 1}^{S} \min(a_j, b_j)}{N_a + N_b}

Morisita-Horn

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

CMH=2∑j=1Saj×bj(∑j=1Saj2Na2+∑j=1Sbj2Nb2)×Na×Nb 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}

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.