vegan (version 2.0-4)

capscale: [Partial] Constrained Analysis of Principal Coordinates or distance-based RDA

Description

Constrained Analysis of Principal Coordinates (CAP) 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 capscale will be much more inefficient. Function capscale is a constrained version of metric scaling, a.k.a. principal coordinates analysis, which is based on the Euclidean distance but can be used, and is more useful, with other dissimilarity measures. The function can also perform unconstrained principal coordinates analysis, optionally using extended dissimilarities.

Usage

capscale(formula, data, distance = "euclidean", sqrt.dist = FALSE,
    comm = NULL, add = FALSE,  dfun = vegdist, metaMDSdist = FALSE,
    na.action = na.fail, subset = NULL, ...)

Arguments

formula

Model formula. The function can be called only with the formula interface. Most usual features of formula hold, especially as defined in cca and

data

Data frame containing the variables on the right hand side of the model formula.

distance

The name of the dissimilarity (or distance) index if the LHS of the formula is a data frame instead of dissimilarity matrix.

sqrt.dist

Take square roots of dissimilarities. See section Notes below.

comm

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 the axes

add

Logical indicating if an additive constant should be computed, and added to the non-diagonal dissimilarities such that all eigenvalues are non-negative in the underlying Principal Co-ordinates Analysis (see

dfun

Distance or dissimilarity function used. Any function returning standard "dist" and taking the index name as the first argument can be used.

metaMDSdist

Use metaMDSdist similarly as in metaMDS. This means automatic data transformation and using extended flexible shortest path dissimilarities (fu

na.action

Handling of missing values in constraints or conditions. The default (na.fail) is to stop with missing values. Choices na.omit and

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 parameters passed to rda or to metaMDSdist.

Value

• The function returns an object of class capscale which is identical to the result of rda. At the moment, capscale does not have specific methods, but it uses cca and rda methods plot.cca, scores.rda etc. Moreover, you can use anova.cca for permutation tests of ``significance'' of the results.

Details

Canonical Analysis of Principal Coordinates (CAP) is simply a Redundancy Analysis of results of Metric (Classical) Multidimensional Scaling (Anderson & Willis 2003). Function capscale uses two steps: (1) it ordinates the dissimilarity matrix using cmdscale and (2) analyses these results using rda. If the user supplied a community data frame instead of dissimilarities, the function will find the needed dissimilarity matrix using vegdist with specified distance. However, the method will accept dissimilarity matrices from vegdist, dist, or any other method producing similar matrices. 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'' CAP can be performed.

The current implementation differs from the method suggested by Anderson & Willis (2003) in three major points which actually make it similar to distance-based redundancy analysis (Legendre & Anderson 1999):

1. Anderson & Willis used the orthonormal solution ofcmdscale, whereascapscaleuses axes weighted by corresponding eigenvalues, so that the ordination distances are the best approximations of original dissimilarities. In the original method, later ``noise'' axes are just as important as first major axes.
2. Anderson & Willis take only a subset of axes, whereascapscaleuses all axes with positive eigenvalues. The use of subset is necessary with orthonormal axes to chop off some ``noise'', but the use of all axes guarantees that the results are the best approximation of original dissimilarities.
3. Functioncapscaleadds species scores as weighted sums of (residual) community matrix (if the matrix is available), whereas Anderson & Willis have no fixed method for adding species scores.

With these definitions, function capscale with Euclidean distances will be identical to rda in eigenvalues and in site, species and biplot scores (except for possible sign reversal). However, it makes no sense to use capscale with Euclidean distances, since direct use of rda is much more efficient. Even with non-Euclidean dissimilarities, the rest of the analysis will be metric and linear.

The function 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.

References

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. (1998). Numerical Ecology. 2nd English Edition. Elsevier

See Also

rda, cca, plot.cca, anova.cca, vegdist, dist, cmdscale.

Examples

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)