dbrda.RdDistance-based redundancy analysis (dbRDA) is an ordination method
similar to Redundancy Analysis (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 dbrda is 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. Function
capscale is a simplified version based on Euclidean
approximation of dissimilarities. The functions can also perform
unconstrained principal coordinates analysis (PCO), optionally using
extended dissimilarities. pco() is a wrapper to dbrda(),
which performs PCO.
dbrda(formula, data, distance = "euclidean", sqrt.dist = FALSE,
add = FALSE, dfun = vegdist, metaMDSdist = FALSE,
na.action = na.fail, subset = NULL, ...)
capscale(formula, data, distance = "euclidean", sqrt.dist = FALSE,
comm = NULL, add = FALSE, dfun = vegdist, metaMDSdist = FALSE,
na.action = na.fail, subset = NULL, ...)
pco(X, ...)Model formula. The function can be called only with the
formula interface. Most usual features of 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 dbRDA.
Community data matrix.
Data frame containing the variables on the right hand side of the model formula.
The name of the dissimilarity (or distance) index if
the LHS of the formula is a data frame instead of
dissimilarity matrix.
Take square roots of dissimilarities. See section
Details below.
Community data frame which will be used for finding
species scores when the LHS of the formula was a
dissimilarity matrix. This is not used if the LHS is a data
frame. If this is not supplied, the “species scores” are
unavailable when dissimilarities were supplied. N.B., this is
only available in capscale: dbrda does not return
species scores. Function sppscores can be used to add
species scores if they are missing.
Add a constant to the non-diagonal dissimilarities such
that all eigenvalues are non-negative in the underlying Principal
Co-ordinates Analysis (see wcmdscale for
details). "lingoes" (or TRUE) uses the
recommended method of Legendre & Anderson (1999: “method
1”) and "cailliez" uses their “method 2”. The
latter is the only one in cmdscale.
Distance or dissimilarity function used. Any function
returning standard "dist" and taking the index name as the
first argument can be used.
Use 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.
Handling of missing values in constraints or
conditions. The default (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.
Subset of data rows. This can be a logical vector
which is 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).
Other parameters passed to underlying functions (e.g.,
metaMDSdist). For pco() argument are passed to
dbrda().
Functions dbrda and capscale provide two alternative
implementations of dbRDA. Function dbrda is based on McArdle
& Anderson (2001) and directly decomposes dissimilarities. With
Euclidean distances results are identical to rda.
Non-Euclidean dissimilarities may give negative eigenvalues
associated with imaginary axes. Function capscale is based on
Legendre & Anderson (1999): the dissimilarity data are first
ordinated using metric scaling, and the ordination results are
analysed as rda. capscale ignores the imaginary
component and will not give negative eigenvalues (but will report
the magnitude on imaginary component).
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 compatible 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.
Function dbrda does not return species scores, and they can
also be missing in capscale, but they can be added after the
analysis using function sppscores.
Non-Euclidean dissimilarities can produce negative eigenvalues
(Legendre & Anderson 1999, 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 positive eigenvalues. If negative
eigenvalues are disturbing, functions let you distort the
dissimilarities so that only non-negative eigenvalues will be
produced with argument add = TRUE. Alternatively, with
sqrt.dist = TRUE, square roots of dissimilarities can 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 right hand side, or comm ~ 1. The new function
pco() implements principal coordinates analysis via
dbrda() directly, using this formula. 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.
The functions return an object of class dbrda or
capscale which inherit from rda. See
cca.object for description of the result object. Function
pco() returns an object of class "vegan_pco" (which
inherits from class "dbrda") to avoid clashes with other packages.
Anderson, M.J. & Willis, T.J. (2003). Canonical analysis of principal coordinates: a useful method of constrained ordination for ecology. Ecology 84, 511–525.
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.
Function dbrda implements real distance-based RDA and is
preferred over capscale. capscale was originally
developed as a variant of constrained analysis of proximities
(Anderson & Willis 2003), but these developments made it more
similar to dbRDA. However, it discards the imaginary dimensions with
negative eigenvalues and ordination and significance tests area only
based on real dimensions and positive eigenvalues. capscale
may be removed from vegan in the future. It has been in
vegan since 2003 (CRAN release 1.6-0) while dbrda was
first released in 2016 (version 2.4-0), and removal of
capscale may be disruptive to historical examples and
scripts, but in modern times dbrda should be used.
The inertia is named after the dissimilarity index as defined in the
dissimilarity data, or as unknown distance if such
information is missing. If the largest original dissimilarity was
larger than 4, capscale handles input similarly as rda
and bases its analysis on variance instead of sum of
squares. Keyword mean is added to the inertia in these cases,
e.g. with Euclidean and Manhattan distances. Inertia is based on
squared index, and keyword squared is added to the name of
distance, unless data were square root transformed (argument
sqrt.dist=TRUE). If an additive constant was used with
argument add, Lingoes or Cailliez adjusted is
added to the the name of inertia, and the value of the constant is
printed.
rda, cca, plot.cca,
anova.cca, vegdist,
dist, cmdscale, wcmdscale
for underlying and related functions. Function sppscores
can add species scores or replace existing species scores.
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, varechem)
## dbrda
dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="bray")
#>
#> Call: dbrda(formula = varespec ~ N + P + K + Condition(Al), data =
#> varechem, distance = "bray")
#>
#> Inertia Proportion Rank RealDims
#> Total 4.5444 1.0000
#> Conditional 0.9726 0.2140 1
#> Constrained 0.9731 0.2141 3 3
#> Unconstrained 2.5987 0.5718 19 13
#>
#> Inertia is squared Bray distance
#>
#> Eigenvalues for constrained axes:
#> dbRDA1 dbRDA2 dbRDA3
#> 0.5362 0.3198 0.1171
#>
#> Eigenvalues for unconstrained axes:
#> MDS1 MDS2 MDS3 MDS4 MDS5 MDS6 MDS7 MDS8
#> 0.9054 0.5070 0.3336 0.2581 0.2027 0.1605 0.1221 0.0825
#> (Showing 8 of 19 unconstrained eigenvalues)
#>
## avoid negative eigenvalues with sqrt distances
dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="bray",
sqrt.dist = TRUE)
#>
#> Call: dbrda(formula = varespec ~ N + P + K + Condition(Al), data =
#> varechem, distance = "bray", sqrt.dist = TRUE)
#>
#> Inertia Proportion Rank
#> Total 6.9500 1.0000
#> Conditional 0.9535 0.1372 1
#> Constrained 1.2267 0.1765 3
#> Unconstrained 4.7698 0.6863 19
#>
#> Inertia is Bray distance
#>
#> Eigenvalues for constrained axes:
#> dbRDA1 dbRDA2 dbRDA3
#> 0.5817 0.4086 0.2365
#>
#> Eigenvalues for unconstrained axes:
#> MDS1 MDS2 MDS3 MDS4 MDS5 MDS6 MDS7 MDS8
#> 0.9680 0.6100 0.4469 0.3837 0.3371 0.3012 0.2558 0.2010
#> (Showing 8 of 19 unconstrained eigenvalues)
#>
## avoid negative eigenvalues also with Jaccard distances
(m <- dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="jaccard"))
#>
#> Call: dbrda(formula = varespec ~ N + P + K + Condition(Al), data =
#> varechem, distance = "jaccard")
#>
#> Inertia Proportion Rank
#> Total 6.5044 1.0000
#> Conditional 1.0330 0.1588 1
#> Constrained 1.2068 0.1855 3
#> Unconstrained 4.2646 0.6557 19
#>
#> Inertia is squared Jaccard distance
#>
#> Eigenvalues for constrained axes:
#> dbRDA1 dbRDA2 dbRDA3
#> 0.5992 0.3994 0.2082
#>
#> Eigenvalues for unconstrained axes:
#> MDS1 MDS2 MDS3 MDS4 MDS5 MDS6 MDS7 MDS8
#> 1.0388 0.6441 0.4518 0.3759 0.3239 0.2785 0.2279 0.1644
#> (Showing 8 of 19 unconstrained eigenvalues)
#>
## add species scores
sppscores(m) <- wisconsin(varespec)
## pco
pco(varespec, dist = "bray", sqrt.dist = TRUE)
#>
#> Call: pco(X = varespec, dist = "bray", sqrt.dist = TRUE)
#>
#> Inertia Rank
#> Total 6.95
#> Unconstrained 6.95 23
#>
#> Inertia is Bray distance
#>
#> Eigenvalues for unconstrained axes:
#> MDS1 MDS2 MDS3 MDS4 MDS5 MDS6 MDS7 MDS8
#> 1.6348 1.1428 0.5658 0.4780 0.3737 0.3716 0.3074 0.2665
#> (Showing 8 of 23 unconstrained eigenvalues)
#>