Description Usage Arguments Details Value Author(s) References See Also Examples
Function cca
performs correspondence analysis, or optionally
constrained correspondence analysis (a.k.a. canonical correspondence
analysis), or optionally partial constrained correspondence
analysis. Function rda
performs redundancy analysis, or
optionally principal components analysis.
These are all very popular ordination techniques in community ecology.
1 2 3 4 5 6 7 8 9 10 
formula 
Model formula, where the left hand side gives the
community data matrix, right hand side gives the constraining variables,
and conditioning variables can be given within a special function

data 
Data frame containing the variables on the right hand side of the model formula. 
X 
Community data matrix. 
Y 
Constraining matrix, typically of environmental variables.
Can be missing. It is better to use 
Z 
Conditioning matrix, the effect of which is removed (‘partialled out’) before next step. Can be missing. 
scale 
Scale species to unit variance (like correlations). 
na.action 
Handling of missing values in constraints or
conditions. The default ( 
subset 
Subset of data rows. This can be a logical vector which
is 
... 
Other arguments for 
Since their introduction (ter Braak 1986), constrained, or canonical,
correspondence analysis and its spinoff, redundancy analysis, have
been the most popular ordination methods in community ecology.
Functions cca
and rda
are similar to popular
proprietary software Canoco
, although the implementation is
completely different. The functions are based on Legendre &
Legendre's (2012) algorithm: in cca
Chisquare transformed data matrix is subjected to weighted linear
regression on constraining variables, and the fitted values are
submitted to correspondence analysis performed via singular value
decomposition (svd
). Function rda
is similar, but uses
ordinary, unweighted linear regression and unweighted SVD. Legendre &
Legendre (2012), Table 11.5 (p. 650) give a skeleton of the RDA
algorithm of vegan. The algorithm of CCA is similar, but
involves standardization by row and column weights.
The functions can be called either with matrixlike entries for
community data and constraints, or with formula interface. In
general, the formula interface is preferred, because it allows a
better control of the model and allows factor constraints. Some
analyses of ordination results are only possible if model was fitted
with formula (e.g., most cases of anova.cca
, automatic
model building).
In the following sections, X
, Y
and Z
, although
referred to as matrices, are more commonly data frames.
In the matrix interface, the
community data matrix X
must be given, but the other data
matrices may be omitted, and the corresponding stage of analysis is
skipped. If matrix Z
is supplied, its effects are removed from
the community matrix, and the residual matrix is submitted to the next
stage. This is called ‘partial’ correspondence or redundancy
analysis. If matrix
Y
is supplied, it is used to constrain the ordination,
resulting in constrained or canonical correspondence analysis, or
redundancy analysis.
Finally, the residual is submitted to ordinary correspondence
analysis (or principal components analysis). If both matrices
Z
and Y
are missing, the
data matrix is analysed by ordinary correspondence analysis (or
principal components analysis).
Instead of separate matrices, the model can be defined using a model
formula
. The left hand side must be the
community data matrix (X
). The right hand side defines the
constraining model.
The constraints can contain ordered or unordered factors,
interactions among variables and functions of variables. The defined
contrasts
are honoured in factor
variables. The constraints can also be matrices (but not data
frames).
The formula can include a special term Condition
for conditioning variables (“covariables”) “partialled out” before
analysis. So the following commands are equivalent:
cca(X, Y, Z)
, cca(X ~ Y + Condition(Z))
, where Y
and Z
refer to constraints and conditions matrices respectively.
Constrained correspondence analysis is indeed a constrained method:
CCA does not try to display all variation in the
data, but only the part that can be explained by the used constraints.
Consequently, the results are strongly dependent on the set of
constraints and their transformations or interactions among the
constraints. The shotgun method is to use all environmental variables
as constraints. However, such exploratory problems are better
analysed with
unconstrained methods such as correspondence analysis
(decorana
, corresp
) or nonmetric
multidimensional scaling (metaMDS
) and
environmental interpretation after analysis
(envfit
, ordisurf
).
CCA is a good choice if the user has
clear and strong a priori hypotheses on constraints and is not
interested in the major structure in the data set.
CCA is able to correct the curve artefact commonly found in correspondence analysis by forcing the configuration into linear constraints. However, the curve artefact can be avoided only with a low number of constraints that do not have a curvilinear relation with each other. The curve can reappear even with two badly chosen constraints or a single factor. Although the formula interface makes it easy to include polynomial or interaction terms, such terms often produce curved artefacts (that are difficult to interpret), these should probably be avoided.
According to folklore, rda
should be used with “short
gradients” rather than cca
. However, this is not based
on research which finds methods based on Euclidean metric as uniformly
weaker than those based on Chisquared metric. However, standardized
Euclidean distance may be an appropriate measures (see Hellinger
standardization in decostand
in particular).
Partial CCA (pCCA; or alternatively partial RDA) can be used to remove
the effect of some
conditioning or “background” or “random” variables or
“covariables” before CCA proper. In fact, pCCA compares models
cca(X ~ Z)
and cca(X ~ Y + Z)
and attributes their
difference to the effect of Y
cleansed of the effect of
Z
. Some people have used the method for extracting
“components of variance” in CCA. However, if the effect of
variables together is stronger than sum of both separately, this can
increase total Chisquare after “partialling out” some
variation, and give negative “components of variance”. In general,
such components of “variance” are not to be trusted due to
interactions between two sets of variables.
The functions have summary
and plot
methods which are
documented separately (see plot.cca
, summary.cca
).
Function cca
returns a huge object of class cca
, which
is described separately in cca.object
.
Function rda
returns an object of class rda
which
inherits from class cca
and is described in cca.object
.
The scaling used in rda
scores is described in a separate
vignette with this package.
The responsible author was Jari Oksanen, but the code borrows heavily from Dave Roberts (Montana State University, USA).
The original method was by ter Braak, but the current implementation follows Legendre and Legendre.
Legendre, P. and Legendre, L. (2012) Numerical Ecology. 3rd English ed. Elsevier.
McCune, B. (1997) Influence of noisy environmental data on canonical correspondence analysis. Ecology 78, 26172623.
Palmer, M. W. (1993) Putting things in even better order: The advantages of canonical correspondence analysis. Ecology 74,22152230.
Ter Braak, C. J. F. (1986) Canonical Correspondence Analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology 67, 11671179.
This help page describes two constrained ordination functions,
cca
and rda
. A related method, distancebased
redundancy analysis (dbRDA) is described separately
(capscale
). All these functions return similar objects
(described in cca.object
). There are numerous support
functions that can be used to access the result object. In the list
below, functions of type cca
will handle all three constrained
ordination objects, and functions of rda
only handle rda
and capscale
results.
The main plotting functions are plot.cca
for all
methods, and biplot.rda
for RDA and dbRDA. However,
generic vegan plotting functions can also handle the results.
The scores can be accessed and scaled with scores.cca
,
and summarized with summary.cca
. The eigenvalues can
be accessed with eigenvals.cca
and the regression
coefficients for constraints with coef.cca
. The
eigenvalues can be plotted with screeplot.cca
, and the
(adjusted) Rsquared can be found with
RsquareAdj.rda
. The scores can be also calculated for
new data sets with predict.cca
which allows adding
points to ordinations. The values of constraints can be inferred
from ordination and community composition with
calibrate.cca
.
Diagnostic statistics can be found with goodness.cca
,
inertcomp
, spenvcor
,
intersetcor
, tolerance.cca
, and
vif.cca
. Function as.mlm.cca
refits the
result object as a multiple lm
object, and this allows
finding influence statistics (lm.influence
,
cooks.distance
etc.).
Permutation based significance for the overall model, single
constraining variables or axes can be found with
anova.cca
. Automatic model building with R
step
function is possible with
deviance.cca
, add1.cca
and
drop1.cca
. Functions ordistep
and
ordiR2step
(for RDA) are special functions for
constrained ordination. Randomized data sets can be generated with
simulate.cca
.
Separate methods based on constrained ordination model are principal
response curves (prc
) and variance partitioning between
several components (varpart
).
Design decisions are explained in vignette
on “Design decisions” which can be accessed with
browseVignettes("vegan")
.
Package ade4 provides alternative constrained ordination
functions cca
and pcaiv
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  data(varespec)
data(varechem)
## Common but bad way: use all variables you happen to have in your
## environmental data matrix
vare.cca < cca(varespec, varechem)
vare.cca
plot(vare.cca)
## Formula interface and a better model
vare.cca < cca(varespec ~ Al + P*(K + Baresoil), data=varechem)
vare.cca
plot(vare.cca)
## `Partialling out' and `negative components of variance'
cca(varespec ~ Ca, varechem)
cca(varespec ~ Ca + Condition(pH), varechem)
## RDA
data(dune)
data(dune.env)
dune.Manure < rda(dune ~ Manure, dune.env)
plot(dune.Manure)

Loading required package: permute
Loading required package: lattice
This is vegan 2.44
Call: cca(X = varespec, Y = varechem)
Inertia Proportion Rank
Total 2.0832 1.0000
Constrained 1.4415 0.6920 14
Unconstrained 0.6417 0.3080 9
Inertia is mean squared contingency coefficient
Eigenvalues for constrained axes:
CCA1 CCA2 CCA3 CCA4 CCA5 CCA6 CCA7 CCA8 CCA9 CCA10 CCA11
0.4389 0.2918 0.1628 0.1421 0.1180 0.0890 0.0703 0.0584 0.0311 0.0133 0.0084
CCA12 CCA13 CCA14
0.0065 0.0062 0.0047
Eigenvalues for unconstrained axes:
CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8 CA9
0.19776 0.14193 0.10117 0.07079 0.05330 0.03330 0.01887 0.01510 0.00949
Call: cca(formula = varespec ~ Al + P * (K + Baresoil), data =
varechem)
Inertia Proportion Rank
Total 2.083 1.000
Constrained 1.046 0.502 6
Unconstrained 1.038 0.498 17
Inertia is mean squared contingency coefficient
Eigenvalues for constrained axes:
CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
0.3756 0.2342 0.1407 0.1323 0.1068 0.0561
Eigenvalues for unconstrained axes:
CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8
0.27577 0.15411 0.13536 0.11803 0.08887 0.05511 0.04919 0.03781
(Showed only 8 of all 17 unconstrained eigenvalues)
Call: cca(formula = varespec ~ Ca, data = varechem)
Inertia Proportion Rank
Total 2.08320 1.00000
Constrained 0.15722 0.07547 1
Unconstrained 1.92598 0.92453 22
Inertia is mean squared contingency coefficient
Eigenvalues for constrained axes:
CCA1
0.15722
Eigenvalues for unconstrained axes:
CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8
0.4745 0.2939 0.2140 0.1954 0.1748 0.1171 0.1121 0.0880
(Showed only 8 of all 22 unconstrained eigenvalues)
Call: cca(formula = varespec ~ Ca + Condition(pH), data = varechem)
Inertia Proportion Rank
Total 2.0832 1.0000
Conditional 0.1458 0.0700 1
Constrained 0.1827 0.0877 1
Unconstrained 1.7547 0.8423 21
Inertia is mean squared contingency coefficient
Eigenvalues for constrained axes:
CCA1
0.18269
Eigenvalues for unconstrained axes:
CA1 CA2 CA3 CA4 CA5 CA6 CA7 CA8
0.3834 0.2749 0.2123 0.1760 0.1701 0.1161 0.1089 0.0880
(Showed only 8 of all 21 unconstrained eigenvalues)
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