rda), but it allows
non-Euclidean dissimilarity indices, such as Manhattan or
Bray--Curtis distance. Despite this non-Euclidean feature, the analysis
is strictly linear and metric. If called with Euclidean distance,
the results are identical to rda, but dbRDA
will be less efficient. Functions capscale and dbrda are
constrained versions of metric scaling, a.k.a. principal coordinates
analysis, which are based on the Euclidean distance but can be used,
and are more useful, with other dissimilarity measures. The functions
can also perform unconstrained principal coordinates analysis,
optionally using extended dissimilarities.
capscale(formula, data, distance = "euclidean", sqrt.dist = FALSE, comm = NULL, add = FALSE, dfun = vegdist, metaMDSdist = FALSE, na.action = na.fail, subset = NULL, ...)
dbrda(formula, data, distance = "euclidean", sqrt.dist = FALSE, add = FALSE, dfun = vegdist, metaMDSdist = FALSE, na.action = na.fail, subset = NULL, ...)formula hold,
especially as defined in cca and rda. The
LHS must be either a community data matrix or a dissimilarity matrix,
e.g., from
vegdist or dist.
If the LHS is a data matrix, function vegdist or
function given in dfun
will be used to find the dissimilarities. The RHS defines the
constraints. The constraints can be continuous variables or factors,
they can be transformed within the formula, and they can have
interactions as in a typical formula. The RHS can have a
special term Condition that defines variables to be
``partialled out'' before constraints, just like in rda
or cca. This allows the use of partial CAP.formula is a data frame instead of
dissimilarity matrix.Details below.formula was a
dissimilarity matrix. This is not used if the LHS is a data
frame. If this is not supplied, the ``species scores'' unavailable.wcmdscale for
details). Choice "lingoes" (or TRUE) use the
recommended method of Legendre & Anderson (1999: “method
1”) and "cailliez" uses their “method 2”. The
latter is the only one in cmdscale."dist" and taking the index name as the
first argument can be used. metaMDSdist similarly as in
metaMDS. This means automatic data transformation and
using extended flexible shortest path dissimilarities (function
stepacross) when there are many dissimilarities based on
no shared species.na.fail) is to stop
with missing values. Choices na.omit and
na.exclude delete rows with missing values, but
differ in representation of results. With na.omit only
non-missing site scores are shown, but na.exclude gives
NA for scores of missing observations. Unlike in
rda, no WA scores are available for missing
constraints or conditions.TRUE for kept observations, or a logical
expression which can contain variables in the working
environment, data or species names of the community data
(if given in the formula or as comm argument).rda or to
metaMDSdist. capscale or
dbrda which inherits from rda. See
cca.object for description of the result object. Functions capscale and dbrda provide two alternative
implementations of dbRDA. Function capscale is based on
Legendre & Anderson (1999): the dissimilarity data are first
ordinated using metric scaling, and the ordination results are
analysed with rda. Function dbrda is based on
McArdle & Anderson (2001) and directly decomposes
dissimilarities. It does not use rda but a parallel
implementation adapted for analysing dissimilarities and returns a
subset of rda items. With Euclidean distances both
results are identical to rda. Other dissimmilarities
may give negative eigenvalues associated with imaginary
axes. Negative eigenvalues are handled differently: capscale
ignores imaginary axes and analyses only real axes with positive
eigenvalues, and dbrda directly analyses dissimilarities and
can give negative eigenvalues in any component. Both methods define
total inertia of conditions, constraints and residuals identically.
If the user supplied a community data frame instead of
dissimilarities, the functions will find dissimilarities using
vegdist or distance function given in dfun with
specified distance. The functions will accept distance
objects from vegdist, dist, or any other
method producing similar objects. The constraining variables can be
continuous or factors or both, they can have interaction terms, or
they can be transformed in the call. Moreover, there can be a
special term Condition just like in rda and
cca so that ``partial'' analysis can be performed.
Non-Euclidean dissimilarities can produce negative eigenvalues
(Legendre & Anderson 1999, McArdle & Anderson 2001). The total
inertia and anova.cca tests for constraints will also
include the effects of imaginary axes with negative eigenvalues
following McArdle & Anderson (2001). If there are negative
eigenvalues, the printed output of capscale will add a column
with sums of positive eigenvalues and an item of sum of negative
eigenvalues, and dbrda will add a column giving the number of
real dimensions with postive eigenvalues. If negative eigenvalues
are disturbing, capscale lets you to distort the
dissimilarities so that only non-negative eigenvalues will be
produced using argument add = TRUE (this argument is passed
to cmdscale). Alternatively, with
sqrt.dist = TRUE, square roots of dissimilarities will be used
which may help in avoiding negative eigenvalues (Legendre & Anderson
1999).
The functions can be also used to perform ordinary metric scaling
a.k.a. principal coordinates analysis by using a formula with only a
constant on the left hand side, or comm ~ 1. With
metaMDSdist = TRUE, the function can do automatic data
standardization and use extended dissimilarities using function
stepacross similarly as in non-metric multidimensional
scaling with metaMDS.
Gower, J.C. (1985). Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra and its Applications 67, 81--97.
Legendre, P. & Anderson, M. J. (1999). Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecological Monographs 69, 1--24.
Legendre, P. & Legendre, L. (2012). Numerical Ecology. 3rd English Edition. Elsevier.
McArdle, B.H. & Anderson, M.J. (2001). Fitting multivariate models to community data: a comment on distance-based redundancy analysis. Ecology 82, 290--297.
rda, cca, plot.cca,
anova.cca, vegdist,
dist, cmdscale, wcmdscale. The function returns similar result object as rda (see
cca.object). This section for rda gives a
more complete list of functions that can be used to access and
analyse dbRDA results.
data(varespec)
data(varechem)
## Basic Analysis
vare.cap <- capscale(varespec ~ N + P + K + Condition(Al), varechem,
dist="bray")
vare.cap
plot(vare.cap)
anova(vare.cap)
## Avoid negative eigenvalues with additive constant
capscale(varespec ~ N + P + K + Condition(Al), varechem,
dist="bray", add =TRUE)
## Avoid negative eigenvalues by taking square roots of dissimilarities
capscale(varespec ~ N + P + K + Condition(Al), varechem,
dist = "bray", sqrt.dist= TRUE)
## Principal coordinates analysis with extended dissimilarities
capscale(varespec ~ 1, dist="bray", metaMDS = TRUE)
## dbrda
dbrda(varespec ~ N + P + K + Condition(Al), varechem,
dist="bray")
## avoid negative eigenvalues also with Jaccard distances
dbrda(varespec ~ N + P + K + Condition(Al), varechem,
dist="jaccard")
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