cca performs correspondence analysis, or optionally
constrained correspondence analysis (a.k.a. canonical correspondence
analysis), or optionally partial constrained correspondence
rda performs redundancy analysis, or
optionally principal components analysis.
These are all very popular ordination techniques in community ecology.
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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 frame containing the variables on the right hand side of the model formula.
Community data matrix.
Constraining matrix, typically of environmental variables.
Can be missing. It is better to use
Conditioning matrix, the effect of which is removed (‘partialled out’) before next step. Can be missing.
Scale species to unit variance (like correlations).
Handling of missing values in constraints or
conditions. The default (
Subset of data rows. This can be a logical vector which
Other arguments for
Since their introduction (ter Braak 1986), constrained, or canonical,
correspondence analysis and its spin-off, redundancy analysis, have
been the most popular ordination methods in community ecology.
rda are similar to popular
Canoco, although the implementation is
completely different. The functions are based on Legendre &
Legendre's (2012) algorithm: in
Chi-square 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
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 matrix-like 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
In the following sections,
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
Finally, the residual is submitted to ordinary correspondence
analysis (or principal components analysis). If both matrices
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
The constraints can contain ordered or unordered factors,
interactions among variables and functions of variables. The defined
contrasts are honoured in
variables. The constraints can also be matrices (but not data
The formula can include a special term
for conditioning variables (“covariables”) “partialled out” before
analysis. So the following commands are equivalent:
cca(X, Y, Z),
cca(X ~ Y + Condition(Z)), where
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
unconstrained methods such as correspondence analysis
corresp) or non-metric
multidimensional scaling (
environmental interpretation after analysis
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 Chi-squared metric. However, standardized
Euclidean distance may be an appropriate measures (see Hellinger
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 Chi-square 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
plot methods which are
documented separately (see
cca returns a huge object of class
is described separately in
rda returns an object of class
inherits from class
cca and is described in
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, 2617-2623.
Palmer, M. W. (1993) Putting things in even better order: The advantages of canonical correspondence analysis. Ecology 74,2215-2230.
Ter Braak, C. J. F. (1986) Canonical Correspondence Analysis: a new eigenvector technique for multivariate direct gradient analysis. Ecology 67, 1167-1179.
This help page describes two constrained ordination functions,
rda. A related method, distance-based
redundancy analysis (dbRDA) is described separately
capscale). All these functions return similar objects
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
The main plotting functions are
plot.cca for all
biplot.rda for RDA and dbRDA. However,
generic vegan plotting functions can also handle the results.
The scores can be accessed and scaled with
and summarized with
summary.cca. The eigenvalues can
be accessed with
eigenvals.cca and the regression
coefficients for constraints with
eigenvalues can be plotted with
screeplot.cca, and the
(adjusted) R-squared 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
Diagnostic statistics can be found with
as.mlm.cca refits the
result object as a multiple
lm object, and this allows
finding influence statistics (
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
ordiR2step (for RDA) are special functions for
constrained ordination. Randomized data sets can be generated with
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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)
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