The function computes dissimilarity indices that are useful for or
popular with community ecologists. All indices use quantitative data,
although they would be named by the corresponding binary index, but
you can calculate the binary index using an appropriate argument. 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, Cao 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. Most of
these indices are discussed by Krebs (1999) and Legendre & Legendre
(2012), and their properties further compared by Wolda (1981) and
Legendre & De Cáceres (2012). Aitchison (1986) distance
is equivalent to Euclidean distance between CLR-transformed samples
("clr") and deals with positive compositional data.
Robust Aitchison distance by Martino et al. (2019) uses robust
CLR ("rlcr"), making it applicable to non-negative data
including zeroes (unlike the standard Aitchison).
vegdist(x, method="bray", binary=FALSE, diag=FALSE, upper=FALSE,
na.rm = FALSE, ...)Should provide a drop-in replacement for dist and
return a distance object of the same type.
Community data matrix.
Dissimilarity index, partial match to
"manhattan", "euclidean", "canberra",
"clark", "bray", "kulczynski",
"jaccard", "gower", "altGower",
"morisita", "horn", "mountford", "raup",
"binomial", "chao", "cao", "mahalanobis",
"chisq", "chord", "aitchison", or
"robust.aitchison".
Perform presence/absence standardization before analysis
using decostand.
Compute diagonals.
Return only the upper diagonal.
Pairwise deletion of missing observations when computing dissimilarities.
Other parameters. These are ignored, except in
method ="gower" which accepts range.global parameter of
decostand, and in method="aitchison", which
accepts pseudocount parameter of decostand used
in the clr transformation.
Jari Oksanen, with contributions from Tyler Smith (Gower index), Michael Bedward (Raup--Crick index), and Leo Lahti (Aitchison and robust Aitchison distance).
Jaccard ("jaccard"), Mountford ("mountford"),
Raup--Crick ("raup"), Binomial and Chao indices are discussed
later in this section. The function also finds indices for presence/
absence data by setting binary = TRUE. The following overview
gives first the quantitative version, where \(x_{ij}\)
\(x_{ik}\) refer to the quantity on species (column) \(i\)
and sites (rows) \(j\) and \(k\). In binary versions \(A\) and
\(B\) are the numbers of species on compared sites, and \(J\) is
the number of species that occur on both compared sites similarly as
in designdist (many indices produce identical binary
versions):
euclidean | \(d_{jk} = \sqrt{\sum_i (x_{ij}-x_{ik})^2}\) |
| binary: \(\sqrt{A+B-2J}\) | |
manhattan | \(d_{jk}=\sum_i |x_{ij}-x_{ik}|\) |
| binary: \(A+B-2J\) | |
gower | \(d_{jk} = (1/M) \sum_i \frac{|x_{ij}-x_{ik}|}{\max x_i-\min x_i}\) |
| binary: \((A+B-2J)/M\) | |
| where \(M\) is the number of columns (excluding missing values) | |
altGower | \(d_{jk} = (1/NZ) \sum_i |x_{ij} - x_{ik}|\) |
| where \(NZ\) is the number of non-zero columns excluding double-zeros (Anderson et al. 2006). | |
| binary: \(\frac{A+B-2J}{A+B-J}\) | |
canberra | \(d_{jk}=\frac{1}{NZ} \sum_i \frac{|x_{ij}-x_{ik}|}{|x_{ij}|+|x_{ik}|}\) |
| where \(NZ\) is the number of non-zero entries. | |
| binary: \(\frac{A+B-2J}{A+B-J}\) | |
clark | \(d_{jk}=\sqrt{\frac{1}{NZ} \sum_i (\frac{x_{ij}-x_{ik}}{x_{ij}+x_{ik}})^2}\) |
| where \(NZ\) is the number of non-zero entries. | |
| binary: \(\frac{A+B-2J}{A+B-J}\) | |
bray | \(d_{jk} = \frac{\sum_i |x_{ij}-x_{ik}|}{\sum_i (x_{ij}+x_{ik})}\) |
| binary: \(\frac{A+B-2J}{A+B}\) | |
kulczynski | \(d_{jk} = 1-0.5(\frac{\sum_i \min(x_{ij},x_{ik})}{\sum_i x_{ij}} + \frac{\sum_i \min(x_{ij},x_{ik})}{\sum_i x_{ik}} )\) |
| binary: \(1-(J/A + J/B)/2\) | |
morisita | \(d_{jk} = 1 - \frac{2 \sum_i x_{ij} x_{ik}}{(\lambda_j + \lambda_k) \sum_i x_{ij} \sum_i x_{ik}}\), where |
| \(\lambda_j = \frac{\sum_i x_{ij} (x_{ij} - 1)}{\sum_i x_{ij} \sum_i (x_{ij} - 1)}\) | |
| binary: cannot be calculated | |
horn | Like morisita, but \(\lambda_j = \sum_i
x_{ij}^2/(\sum_i x_{ij})^2\) |
| binary: \(\frac{A+B-2J}{A+B}\) | |
binomial | \(d_{jk} = \sum_i [x_{ij} \log (\frac{x_{ij}}{n_i}) + x_{ik} \log (\frac{x_{ik}}{n_i}) - n_i \log(\frac{1}{2})]/n_i\), |
| where \(n_i = x_{ij} + x_{ik}\) | |
| binary: \(\log(2) \times (A+B-2J)\) | |
cao | \(d_{jk} = \frac{1}{S} \sum_i \log \left(\frac{n_i}{2}\right) - (x_{ij} \log(x_{ik}) + x_{ik} \log(x_{ij}))/n_i\), |
| where \(S\) is the number of species in compared sites and \(n_i = x_{ij}+x_{ik}\) |
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).
Cao index or CYd index (Cao et al. 1997) was suggested as a minimally biased index for high beta diversity and variable sampling intensity. Cao index does not have a fixed upper limit, but can vary among sites with no shared species. The index is intended for count (integer) data, and it is undefined for zero abundances; these are replaced with arbitrary value \(0.1\) following Cao et al. (1997). Cao et al. (1997) used \(\log_{10}\), but the current function uses natural logarithms so that the values are approximately \(2.30\) times higher than with 10-based logarithms. Anderson & Thompson (2004) give an alternative formulation of Cao index to highlight its relationship with Binomial index (above).
Mountford index is defined as \(M = 1/\alpha\) where \(\alpha\)
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 \(\exp(aM) +
\exp(bM) = 1 + \exp[(a+b-j)M]\), 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 \(0 \dots \log(2)\), but the dissimilarities
are divided by \(\log(2)\) so that the results will be in
the conventional range \(0 \dots 1\).
Raup--Crick dissimilarity (method = "raup") is a probabilistic
index based on presence/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. The current function uses
analytic result from hypergeometric distribution
(phyper) to find the probabilities. 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. The
index uses equal occurrence probabilities for all species, but Raup
and Crick originally suggested that sampling probabilities should be
proportional to species frequencies (Chase et al. 2011). A simulation
approach with unequal species sampling probabilities is implemented in
raupcrick function following Chase et al. (2011). The
index can be also used for transposed data to give a probabilistic
dissimilarity index of species co-occurrence (identical to Veech
2013).
Chao index tries to take into account the number of unseen species
pairs, similarly as in method = "chao" in
specpool. Function vegdist implements a
Jaccard, index defined as
\(1-\frac{U \times V}{U + V - U \times V}\);
other types can be defined with function chaodist. In Chao
equation, \(U = C_j/N_j + (N_k - 1)/N_k \times a_1/(2 a_2) \times
S_j/N_j\),
and \(V\) is similar except for site index
\(k\). \(C_j\) is the total number of individuals in the
species of site \(j\) that are shared with site \(k\),
\(N_j\) is the total number of individuals at site \(j\),
\(a_1\) (and \(a_2\)) are the number of species
occurring in site \(j\) that have only one (or two) individuals in
site \(k\), and \(S_j\) is the total number of individuals
in the species present at site \(j\) that occur with only one
individual in site \(k\) (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.
Mahalanobis distances are Euclidean distances of a matrix where columns are centred, have unit variance, and are uncorrelated. The index is not commonly used for community data, but it is sometimes used for environmental variables. The calculation is based on transforming data matrix and then using Euclidean distances following Mardia et al. (1979). The Mahalanobis transformation usually fails when the number of columns is larger than the number of rows (sampling units). When the transformation fails, the distances are nearly constant except for small numeric noise. Users must check that the returned Mahalanobis distances are meaningful.
Euclidean and Manhattan dissimilarities are not good in gradient separation without proper standardization but are still included for comparison and special needs.
Chi-square distances ("chisq") are Euclidean distances of
Chi-square transformed data (see decostand). This is
the internal standardization used in correspondence analysis
(cca, decorana). Weighted principal
coordinates analysis of these distances with row sums as weights is
equal to correspondence analysis (see the Example in
wcmdscale). Chi-square distance is intended for
non-negative data, such as typical community data. However, it can
be calculated as long as all margin sums are positive, but warning
is issued on negative data entries.
Chord distances ("chord") are Euclidean distance of a matrix
where rows are standardized to unit norm (their sums of squares are
1) using decostand. Geometrically this standardization
moves row points to a surface of multidimensional unit sphere, and
distances are the chords across the hypersphere. Community data are
non-negative, and all points will be on the positive quadrant of the
hypersphere, and the distances have upper limit
\(\sqrt{2}\). If data can have negative values, the upper
limit is 2.
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.
Aitchison distance (1986) and robust Aitchison distance (Martino et al. 2019) are metrics that deal with compositional data. Aitchison distance has been said to outperform Jensen-Shannon divergence and Bray-Curtis dissimilarity, due to a better stability to subsetting and aggregation, and it being a proper distance (Aitchison et al., 2000).
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 quantitative 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.
Aitchison, J. The Statistical Analysis of Compositional Data (1986). London, UK: Chapman & Hall.
Aitchison, J., Barceló-Vidal, C., Martín-Fernández, J.A., Pawlowsky-Glahn, V. (2000). Logratio analysis and compositional distance. Math. Geol. 32, 271–275.
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.
Anderson, M.J., Ellingsen, K.E. & McArdle, B.H. (2006). Multivariate dispersion as a measure of beta diversity. Ecology Letters 9, 683--693.
Anderson, M.J & Thompson, A.A. (2004). Multivariate control charts for ecological and environmental monitoring. Ecological Applications 14, 1921--1935.
Cao, Y., Williams, W.P. & Bark, A.W. (1997). Similarity measure bias in river benthic Auswuchs community analysis. Water Environment Research 69, 95--106.
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.
Chase, J.M., Kraft, N.J.B., Smith, K.G., Vellend, M. and Inouye, B.D. (2011). Using null models to disentangle variation in community dissimilarity from variation in \(\alpha\)-diversity. Ecosphere 2:art24 tools:::Rd_expr_doi("10.1890/ES10-00117.1")
Faith, D. P, Minchin, P. R. and Belbin, L. (1987). Compositional dissimilarity as a robust measure of ecological distance. Vegetatio 69, 57--68.
Gower, J. C. (1971). A general coefficient of similarity and some of its properties. Biometrics 27, 623--637.
Krebs, C. J. (1999). Ecological Methodology. Addison Wesley Longman.
Legendre, P. & De Cáceres, M. (2012). Beta diversity as the variance of community data: dissimilarity coefficients and partitioning. Ecology Letters 16, 951--963. tools:::Rd_expr_doi("10.1111/ele.12141")
Legendre, P. and Legendre, L. (2012) Numerical Ecology. 3rd English ed. Elsevier.
Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979). Multivariate analysis. Academic Press.
Martino, C., Morton, J.T., Marotz, C.A., Thompson, L.R., Tripathi, A., Knight, R. & Zengler, K. (2019) A novel sparse compositional technique reveals microbial perturbations. mSystems 4, 1.
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.
Veech, J. A. (2013). A probabilistic model for analysing species co-occurrence. Global Ecology and Biogeography 22, 252--260.
Wolda, H. (1981). Similarity indices, sample size and diversity. Oecologia 50, 296--302.
Function designdist can be used for defining
your own dissimilarity index. Function betadiver
provides indices intended for the analysis of beta diversity.
data(varespec)
vare.dist <- vegdist(varespec)
# Orlóci's Chord distance: range 0 .. sqrt(2)
vare.dist <- vegdist(decostand(varespec, "norm"), "euclidean")
# Anderson et al. (2006) version of Gower
vare.dist <- vegdist(decostand(varespec, "log"), "altGower")
# Range standardization with "altGower" (that excludes double-zeros)
vare.dist <- vegdist(decostand(varespec, "range"), "altGower")
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