# beta.div.comp: Decompose D in replacement and richness difference components In adespatial: Multivariate Multiscale Spatial Analysis

## Description

Podani-family and Baselga-family decompositions of the Jaccard and S<c3><b8>rensen dissimilarity coefficients and their quantitative forms (Ruzicka and percentage difference) into replacement and richness difference components, for species presence-absence or abundance data, as described in Legendre (2014).

## Usage

 `1` ```beta.div.comp(mat, coef = "J", quant = FALSE, save.abc = FALSE) ```

## Arguments

 `mat` Community composition data (`data.frame` or `matrix`). `coef` Family of coefficients to be computed. "S" or "Sorensen": Podani family, Srensen-based indices. "J" or "Jaccard": Podani family, Jaccard-based indices. "BS" <80><93> Baselga family, Srensen-based indices. "BJ": Baselga family, Jaccard-based indices. "N": Podani & Schmera (2011) relativized nestedness index. The quantitative form of the Srensen dissimilarity is the percentage difference index. The quantitative form of the Jaccard dissimilarity is the Ruzicka index. `quant` If `TRUE`, compute the quantitative forms of replacement, nestedness and D. If `FALSE`, compute the presence-absence forms of the coefficients. `save.abc` If `TRUE`, save the matrices of parameters a, b and c used in presence-absence calculations.

## Details

For species presence-absence data, the dissimilarity coefficients are Jaccard = (b+c)/(a+b+c) and S<c3><b8>rensen = (b+c)/(2*a+b+c) with the usual a,b,c notation. For species abundance data, the dissimilarity coefficients are the Ruzicka index = (B+C)/(A+B+C) and Odum<e2><80><99>s percentage difference = (B+C)/(2A+B+C) (aka Bray-Curtis in some packages), where

• A = sum of the intersections (or minima) of species abundances at two sites,

• B = sum of abundances at site 1 minus A,

• C = sum of abundances at site 2 minus A.

The binary (`quant=FALSE`) and quantitative (`quant=TRUE`) forms of the S and J indices return the same values when computed for presence-absence data.

## Value

A list containing the following results:

• `repl`: Replacement matrix, class = dist.

• `rich`: Richness/abundance difference or nestedness matrix (class `dist`). With options "BJ", "BS" and "N", `rich` contains nestedness indices. With option "N", the repl[i,j] and rich[i,j] values do not add up to D[i,j].

• `D`: Dissimilarity matrix (class`dist`).

• `part`: Beta diversity partitioning vector:

1. BDtotal (total beta diversity) = sum(D.ij)/(n*(n-1)) (Legendre & De C<c3><a1>ceres 2013). This is equal to sum(d.ij^2)/(n*(n-1)) where d.ij = sqrt(D.ij). The dissimilarities are square-rooted because the Jaccard, S<c3><b8>rensen, Ruzicka and percentage difference indices are not Euclidean.

2. Repl = Total replacement diversity.

3. RichDiff|Nes = Total richness difference diversity (or nestedness).

4. Repl/BDtotal = Total replacement diversity/Total beta diversity.

5. RichDiff/BDtotal = Total richness difference diversity (or nestedness)/Total beta diversity.

• `note`: Name of the dissimilarity coefficient.

The Jaccard and S<c3><b8>rensen dissimilarity coefficients and their quantitative forms, the Ruzicka and percentage difference indices, all have upper bounds (Dmax) of 1. Hence, when all sites contain a different set of species with no species in common, the maximum value that BDtotal can take is 0.5. See Legendre & De Caceres (2013, p. 958), section Maximum value of BD. This differs form the values produced by function beta.div(): with methods "hellinger", "chord" and "profiles", which have maximum values of sqrt(2), BDtotal has a maximum value of 1 for these dissimilarities.

## Author(s)

Pierre Legendre [email protected]

## References

Baselga, A. (2010) Partitioning the turnover and nestedness components of beta diversity. Global Ecology and Biogeography, 19, 134<e2><80><93>143.

Baselga, A. (2012) The relationship between species replacement, dissimilarity derived from nestedness, and nestedness. Global Ecology and Biogeography, 21, 1223<e2><80><93>1232.

Baselga, A. (2013) Separating the two components of abundance-based dissimilarity: balanced changes in abundance vs. abundance gradients. Methods in Ecology and Evolution, 4, 552<e2><80><93>557.

Carvalho, J.C., Cardoso, P., Borges, P.A.V., Schmera, D. & Podani, J. (2013) Measuring fractions of beta diversity and their relationships to nestedness: a theoretical and empirical comparison of novel approaches. Oikos, 122, 825<e2><80><93>834.

Legendre, P. 2014. Interpreting the replacement and richness difference components of beta diversity. Global Ecology and Biogeography, 23, 1324-1334.

Legendre, P. and M. De C<c3><a1>ceres. 2013. Beta diversity as the variance of community data: dissimilarity coefficients and partitioning. Ecology Letters 16: 951-963.

Podani, J., Ricotta, C. & Schmera, D. (2013) A general framework for analyzing beta diversity, nestedness and related community-level phenomena based on abundance data. Ecological Complexity, 15, 52-61.

Podani, J. & Schmera, D. 2011. A new conceptual and methodological framework for exploring and explaining pattern in presence-absence data. Oikos, 120, 1625<e2><80><93>1638.

`LCBD.comp`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```if(require(ade4, quietly = TRUE)){ data(doubs) fish.sp = doubs\$fish[-8,] # Fish data; site 8 is removed because no fish were caught # Compute and partition a matrix of Jaccard indices (presence-absence data) out1 = beta.div.comp(fish.sp, coef="J", quant=FALSE) out1\$part # Compute and partition a matrix of percentage difference indices # (quantitative form of Sorensen index) out2 = beta.div.comp(fish.sp, coef="S", quant=TRUE) out2\$part # In paragraph Value, see the description of the 5 elements of vector part. # Is the fish beta diversity dominated by replacement or richness/abundance difference? } ```