vegan (version 2.4-2)

# cca: [Partial] [Constrained] Correspondence Analysis and Redundancy Analysis

## Description

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.

## Usage

```"cca"(formula, data, na.action = na.fail, subset = NULL, ...)
"cca"(X, Y, Z, ...)
"rda"(formula, data, scale=FALSE, na.action = na.fail, subset = NULL, ...)
"rda"(X, Y, Z, scale=FALSE, ...)```

## Arguments

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 `Condition`.
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 `formula` instead of this argument, and some further analyses only work when `formula` was used.
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 (`na.fail`) is to stop with missing value. Choice `na.omit` removes all rows with missing values. Choice `na.exclude` keeps all observations but gives `NA` for results that cannot be calculated. The WA scores of rows may be found also for missing values in constraints. Missing values are never allowed in dependent community data.
subset
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.
...
Other arguments for `print` or `plot` functions (ignored in other functions).

## Value

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.

## Details

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. 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` 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 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 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 `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 non-metric 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 Chi-squared 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 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 `summary` and `plot` methods which are documented separately (see `plot.cca`, `summary.cca`).

## References

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, `cca` and `rda`. A related method, distance-based 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) \$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 `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`.

## Examples

Run this code
``````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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