capscale {vegan}R Documentation

[Partial] Constrained Analysis of Principal Coordinates

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 may be useful with other dissimilarity measures, since Euclidean distances inherent in rda are generally poor with community data

Usage

capscale(formula, data, distance = "euclidean", comm = 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 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 will be used to find the dissimilarities. RHS defines the constraints. The constraints can be continuous or factors, they can be transformed within the formula, and they can have interactions as in typical formula. The RHS can have a special term Condition that defines variables ``partialled out'' before constraints, just like in rda or cca. This allows the use of partial CAP.
data Data frame containing the variables on the right hand side of the model formula.
distance Dissimilarity (or distance) index in vegdist used if the LHS of the formula is a data frame instead of dissimilarity matrix.
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 of initial metric scaling (cmdscale) and may be confusing.
... Other parameters passed to rda.

Details

The 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:

  1. Anderson & Willis used orthonormal solution of cmdscale, whereas capscale uses axes weighted by corresponding eigenvalues, so that the ordination distances are 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, whereas capscale uses 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. Function capscale adds 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.

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, summary.rda etc. Moreover, you can use anova.cca for permutation tests of ``significance'' of the results.

Note

Function rda usually divides the ordination results by number of sites minus one. In this way, the inertia is variance instead of sum of squares, and the eigenvalues sum up to variance. Many dissimilarity measures are in the range 0 to 1, so they have already made a similar division. If the largest original dissimilarity is less or equal to 4 (allowing for stepacross), this division is undone in capscale and original dissimilarities are used. The inertia is called as squared dissimilarity (as defined in the dissimilarity matrix), but keyword mean is added to the inertia in cases where division was made, e.g. in Euclidean and Manhattan distances.

Author(s)

Jari Oksanen

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.

See Also

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

Examples

data(varespec)
data(varechem)
vare.cap <- capscale(varespec ~ N + P + K + Condition(Al), varechem, dist="bray")
vare.cap
plot(vare.cap)
anova(vare.cap)

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