Description

Distance-based redundancy analysis (dbRDA) 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 dbRDA will be less efficient. Functions capscale and dbrda are constrained versions of metric scaling, a.k.a. principal coordinates analysis, which are based on the Euclidean distance but can be used, and are more useful, with other dissimilarity measures. The functions can also perform unconstrained principal coordinates analysis, optionally using extended dissimilarities.

Usage

Arguments

Details

Functions capscale and dbrda provide two alternative implementations of dbRDA. Function capscale is based on Legendre & Anderson (1999): the dissimilarity data are first ordinated using metric scaling, and the ordination results are analysed as rda. Function dbrda is based on McArdle & Anderson (2001) and directly decomposes dissimilarities. It does not use rda but a parallel implementation adapted for analysing dissimilarities and returns a subset of rda items. With Euclidean distances both results are identical to rda. Other dissimilarities may give negative eigenvalues associated with imaginary axes. Negative eigenvalues are handled differently: capscale ignores imaginary axes and analyses only real axes with positive eigenvalues, and dbrda directly analyses dissimilarities and can give negative eigenvalues in any component.

If the user supplied a community data frame instead of dissimilarities, the functions will find dissimilarities using vegdist or distance function given in dfun with specified distance. The functions will accept distance objects from vegdist, dist, or any other method producing compatible objects. 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” analysis can be performed.

Function dbrda does not return species scores, and they can also be missing in capscale, but they can be added after the analysis using function sppscores.

Non-Euclidean dissimilarities can produce negative eigenvalues (Legendre & Anderson 1999, McArdle & Anderson 2001). If there are negative eigenvalues, the printed output of capscale will add a column with sums of positive eigenvalues and an item of sum of negative eigenvalues, and dbrda will add a column giving the number of real dimensions with positive eigenvalues. If negative eigenvalues are disturbing, functions let you to distort the dissimilarities so that only non-negative eigenvalues will be produced with argument add = TRUE. Alternatively, with sqrt.dist = TRUE, square roots of dissimilarities will be used which may help in avoiding negative eigenvalues (Legendre & Anderson 1999).

The functions 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.

Value

The functions return an object of class capscale or dbrda which inherits from rda. See cca.object for description of the result object.

Note

The function capscale was originally developed as a variant of constrained analysis of proximities (Anderson & Willis 2003), but these developments made it similar to dbRDA. However, it discards the imaginary dimensions with negative eigenvalues and ordination and significance tests area only based on real dimensions and positive eigenvalues.

The inertia is named after the dissimilarity index as defined in the dissimilarity data, or as unknown distance if such information is missing. If the largest original dissimilarity was larger than 4, capscale handles input similarly as rda and bases its analysis on variance instead of sum of squares. Keyword mean is added to the inertia in these cases, e.g. with Euclidean and Manhattan distances. Inertia is based on squared index, and keyword squared is added to the name of distance, unless data were square root transformed (argument sqrt.dist=TRUE). If an additive constant was used with argument add, Lingoes or Cailliez adjusted is added to the the name of inertia, and the value of the constant is printed.

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.

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. (2012). Numerical Ecology. 3rd English Edition. Elsevier.

McArdle, B.H. & Anderson, M.J. (2001). Fitting multivariate models to community data: a comment on distance-based redundancy analysis. Ecology 82, 290–297.

See Also

rda, cca, plot.cca, anova.cca, vegdist, dist, cmdscale, wcmdscale for underlying and related functions. Function sppscores can add species scores or replace existing species scores.

The function returns similar result object as rda (see cca.object). This section for rda gives a more complete list of functions that can be used to access and analyse dbRDA results.

Examples