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.vegdist(x, method="bray", binary=FALSE, diag=FALSE, upper=FALSE,
na.rm = FALSE, ...)"manhattan",
"euclidean", "canberra", "bray", "kulczynski",
"jaccard", "gower", "morisita", "horndecostand.method ="gower" which accepts range.global parameter of
decostand. .dist and
return a distance object of the same type."jaccard"), Mountford ("mountford"),
Raup--Crick ("raup"), Binomial and Chao indices are discussed below.
The other indices are defined as:
euclidean
$d_{jk} = \sqrt{\sum_i (x_{ij}-x_{ik})^2}$
manhattan
$d_{jk} = \sum_i |x_{ij} - x_{ik}|$
gower
$d_{jk} = (1/M) \sum_i \frac{|x_{ij}-x_{ik}|}{\max x_i-\min
x_i}$
where $M$ is the number of columns (excluding missing
values)
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.
bray
$d_{jk} = \frac{\sum_i |x_{ij}-x_{ik}|}{\sum_i (x_{ij}+x_{ik})}$
kulczynski
$d_{jk} = 1-0.5(\frac{\sum_i \min(x_{ij},x_{ik})}{\sum_i x_{ij}} +
\frac{\sum_i \min(x_{ij},x_{ik})}{\sum_i x_{ik}} )$
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 $\lambda_j = \frac{\sum_i x_{ij} (x_{ij} - 1)}{\sum_i
x_{ij} \sum_i (x_{ij} - 1)}$
horn
Like morisita, but $\lambda_j = \sum_i
x_{ij}^2/(\sum_i x_{ij})^2$
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 $n_i = x_{ij} + x_{ik}$
}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 $M = 1/\alpha$ 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 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 "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.
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.
designdist can be used for defining your own
dissimilarity index. Alternative dissimilarity functions include
dist in base R,
daisy (package dsvdis (package 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")Run the code above in your browser using DataLab