vegdist {vegan} | R Documentation

## Dissimilarity Indices for Community Ecologists

### Description

The function computes dissimilarity indices that are useful for or popular with community ecologists. If you do not find your favourite index here, you can see if it can be implemented using `designdist`. Gower, Bray–Curtis, Jaccard and Kulczynski indices are good in detecting underlying ecological gradients (Faith et al. 1987). Morisita, Horn–Morisita, Binomial and Chao indices should be able to handle different sample sizes (Wolda 1981, Krebs 1999, Anderson & Millar 2004), and Mountford (1962) and Raup-Crick indices for presence–absence data should be able to handle unknown (and variable) sample sizes.

### Usage

    vegdist(x, method="bray", binary=FALSE, diag=FALSE, upper=FALSE,
            na.rm = FALSE, ...)

### Arguments

`x` | Community data matrix.
`method` | Dissimilarity index, partial match to `"manhattan"`, `"euclidean"`, `"canberra"`, `"bray"`, `"kulczynski"`, `"jaccard"`, `"gower"`, `"morisita"`, `"horn"`, `"mountford"`, `"raup"` , `"binomial"` or `"chao"`.
`binary` | Perform presence/absence standardization before analysis using `decostand`.
`diag` | Compute diagonals.
`upper` | Return only the upper diagonal.
`na.rm` | Pairwise deletion of missing observations when computing dissimilarities.
`...` | Other parameters. These are ignored, except in `method ="gower"` which accepts `range.global` parameter of `decostand`. .

### Details

Jaccard (`"jaccard"`), Mountford (`"mountford"`), Raup–Crick (`"raup"`), Binomial and Chao indices are discussed below. The other indices are defined as:

`euclidean` | d[jk] = sqrt(sum (x[ij]-x[ik])^2)
`manhattan` | d[jk] = sum(abs(x[ij] - x[ik]))
`gower` | d[jk] = sum (abs(x[ij]-x[ik])/(max(x[i])-min(x[i]))
`canberra` | d[jk] = (1/NZ) sum ((x[ij]-x[ik])/(x[ij]+x[ik]))
| where NZ is the number of non-zero entries.
`bray` | d[jk] = (sum abs(x[ij]-x[ik])/(sum (x[ij]+x[ik]))
`kulczynski` | d[jk] 1 - 0.5*((sum min(x[ij],x[ik])/(sum x[ij]) + (sum min(x[ij],x[ik])/(sum x[ik]))
`morisita` | {d[jk] = 2*sum(x[ij]*x[ik])/((lambda[j]+lambda[k]) * sum(x[ij])*sum(x[ik])) }
| where lambda[j] = sum(x[ij]*(x[ij]-1))/sum(x[ij])*sum(x[ij]-1)
`horn` | Like `morisita`, but lambda[j] = sum(x[ij]^2)/(sum(x[ij])^2)
`binomial` | d[jk] = sum(x[ij]*log(x[ij]/n[i]) + x[ik]*log(x[ik]/n[i]) - n[i]*log(1/2))/n[i]
| where n[i] = x[ij] + x[ik]

Jaccard index is computed as 2B/(1+B), where B is Bray–Curtis dissimilarity.

Binomial index is derived from Binomial deviance under null hypothesis that the two compared communities are equal. It should be able to handle variable sample sizes. The index does not have a fixed upper limit, but can vary among sites with no shared species. For further discussion, see Anderson & Millar (2004).

Mountford index is defined as M = 1/α where α is the parameter of Fisher's logseries assuming that the compared communities are samples from the same community (cf. `fisherfit`, `fisher.alpha`). The index M is found as the positive root of equation exp(a*M) + exp(b*M) = 1 + exp((a+b-j)*M), where j is the number of species occurring in both communities, and a and b are the number of species in each separate community (so the index uses presence–absence information). Mountford index is usually misrepresented in the literature: indeed Mountford (1962) suggested an approximation to be used as starting value in iterations, but the proper index is defined as the root of the equation above. The function `vegdist` solves M with the Newton method. Please note that if either a or b are equal to j, one of the communities could be a subset of other, and the dissimilarity is 0 meaning that non-identical objects may be regarded as similar and the index is non-metric. The Mountford index is in the range 0 ... log(2), but the dissimilarities are divided by log(2) so that the results will be in the conventional range 0 ... 1.

Raup–Crick dissimilarity (`method = "raup"`) is a probabilistic index based on presensec/absence data. It is defined as 1 - prob(j), or based on the probability of observing at least j species in shared in compared communities. Legendre & Legendre (1998) suggest using simulations to assess the probability, but the current function uses analytic result from hypergeometric distribution (`phyper`) instead. This probability (and the index) is dependent on the number of species missing in both sites, and adding all-zero species to the data or removing missing species from the data will influence the index. The probability (and the index) may be almost zero or almost one for a wide range of parameter values. The index is nonmetric: two communities with no shared species may have a dissimilarity slightly below one, and two identical communities may have dissimilarity slightly above zero.

Chao index tries to take into account the number of unseen species pairs, similarly as Chao's method in `specpool`. Function `vegdist` implements a Jaccard type index defined as d_{jk} = U_j U_k/(U_j + U_k - U_j U_k), where U_j = C_j/N_j + (N_k - 1)/N_k times a_1/(2 a_2) times S_j/N_j. Here C_j is the total number of individuals in species shared with site k, N is the total number of individuals, a_1 and a_2 are number of species occurring only with one or two individuals in another site, and S_j is the number of individuals in species that occur only with one individual in another site (Chao et al. 2005).

Morisita index can be used with genuine count data (integers) only. Its Horn–Morisita variant is able to handle any abundance data.

Euclidean and Manhattan dissimilarities are not good in gradient separation without proper standardization but are still included for comparison and special needs.

Bray–Curtis and Jaccard indices are rank-order similar, and some other indices become identical or rank-order similar after some standardizations, especially with presence/absence transformation of equalizing site totals with `decostand`. Jaccard index is metric, and probably should be preferred instead of the default Bray-Curtis which is semimetric.

The naming conventions vary. The one adopted here is traditional rather than truthful to priority. The function finds either quantitative or binary variants of the indices under the same name, which correctly may refer only to one of these alternatives For instance, the Bray index is known also as Steinhaus, Czekanowski and Sørensen index. The quantitive version of Jaccard should probably called Ružička index. The abbreviation `"horn"` for the Horn–Morisita index is misleading, since there is a separate Horn index. The abbreviation will be changed if that index is implemented in `vegan`.

### Value

Should provide a drop-in replacement for `dist` and return a distance object of the same type.

### Note

The function is an alternative to `dist` adding some ecologically meaningful indices. Both methods should produce similar types of objects which can be interchanged in any method accepting either. Manhattan and Euclidean dissimilarities should be identical in both methods. Canberra index is divided by the number of variables in `vegdist`, but not in `dist`. So these differ by a constant multiplier, and the alternative in `vegdist` is in range (0,1). Function `daisy` (package cluster) provides alternative implentation of Gower index for mixed data of numeric and class variables (but it works for mixed variables only).

Most dissimilarity indices in `vegdist` are designed for community data, and they will give misleading values if there are negative data entries. The results may also be misleading or `NA` or `NaN` if there are empty sites. In principle, you cannot study species compostion without species and you should remove empty sites from community data.

### Author(s)

Jari Oksanen, with contributions from Tyler Smith (Gower index) and Michael Bedward (Raup–Crick index).

### References

Anderson, M.J. and Millar, R.B. (2004). Spatial variation and effects of habitat on temperate reef fish assemblages in northeastern New Zealand. Journal of Experimental Marine Biology and Ecology 305, 191–221.

Chao, A., Chazdon, R. L., Colwell, R. K. and Shen, T. (2005). A new statistical approach for assessing similarity of species composition with incidence and abundance data. Ecology Letters 8, 148–159.

Faith, D. P, Minchin, P. R. and Belbin, L. (1987). Compositional dissimilarity as a robust measure of ecological distance. Vegetatio 69, 57–68.

Krebs, C. J. (1999). Ecological Methodology. Addison Wesley Longman.

Legendre, P, & Legendre, L. (1998) Numerical Ecology. 2nd English Edition. Elsevier.

Mountford, M. D. (1962). An index of similarity and its application to classification problems. In: P.W.Murphy (ed.), Progress in Soil Zoology, 43–50. Butterworths.

Wolda, H. (1981). Similarity indices, sample size and diversity. Oecologia 50, 296–302.

### See Also

Function `designdist` can be used for defining your own dissimilarity index. Alternative dissimilarity functions include `dist` in base R, `daisy` (package cluster), and `dsvdis` (package labdsv).

### Examples

    data(varespec)
    vare.dist <- vegdist(varespec)
    # Orlóci's Chord distance: range 0 .. sqrt(2)
    vare.dist <- vegdist(decostand(varespec, "norm"), "euclidean")

[Package vegan version 1.8-8 Index]