Principal Coordinates Analysis and [Partial] Distance-based Redundancy Analysis

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 dbrda is 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. Function capscale is a simplified version based on Euclidean approximation of dissimilarities. The functions can also perform unconstrained principal coordinates analysis (PCO), optionally using extended dissimilarities. pco() is a wrapper to dbrda(), which performs PCO.

Usage

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

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 or function given in dfun will be used to find the dissimilarities. The RHS defines the constraints. The constraints can be continuous variables or factors, they can be transformed within the formula, and they can have interactions as in a typical formula. The RHS can have a special term Condition that defines variables to be “partialled out” before constraints, just like in rda or cca. This allows the use of partial dbRDA.
X: Community data matrix.
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 Details 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 unavailable when dissimilarities were supplied. N.B., this is only available in capscale: dbrda does not return species scores. Function sppscores can be used to add species scores if they are missing.
add: Add a constant to the non-diagonal dissimilarities such that all eigenvalues are non-negative in the underlying Principal Co-ordinates Analysis (see wcmdscale for details). "lingoes" (or TRUE) uses the recommended method of Legendre & Anderson (1999: “method 1”) and "cailliez" uses their “method 2”. The latter is the only one in cmdscale.
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 (function stepacross) when there are many dissimilarities based on no shared species.
na.action: Handling of missing values in constraints or conditions. The default (na.fail) is to stop with missing values. Choices na.omit and na.exclude delete rows with missing values, but differ in representation of results. With na.omit only non-missing site scores are shown, but na.exclude gives NA for scores of missing observations. Unlike in rda, no WA scores are available for missing constraints or conditions.
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 (if given in the formula or as comm argument).
...: Other parameters passed to underlying functions (e.g., metaMDSdist). For pco() argument are passed to dbrda().

Details

Functions dbrda and capscale provide two alternative implementations of dbRDA. Function dbrda is based on McArdle & Anderson (2001) and directly decomposes dissimilarities. With Euclidean distances results are identical to rda. Non-Euclidean dissimilarities may give negative eigenvalues associated with imaginary axes. 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. capscale ignores the imaginary component and will not give negative eigenvalues (but will report the magnitude on imaginary 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 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 can 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 right hand side, or comm ~ 1. The new function pco() implements principal coordinates analysis via dbrda() directly, using this formula. 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 dbrda or capscale which inherit from rda. See cca.object for description of the result object. Function pco() returns an object of class "vegan_pco" (which inherits from class "dbrda") to avoid clashes with other packages.

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.

Author

Jari Oksanen

Note

Function dbrda implements real distance-based RDA and is preferred over capscale. capscale was originally developed as a variant of constrained analysis of proximities (Anderson & Willis 2003), but these developments made it more 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. capscale may be removed from vegan in the future. It has been in vegan since 2003 (CRAN release 1.6-0) while dbrda was first released in 2016 (version 2.4-0), and removal of capscale may be disruptive to historical examples and scripts, but in modern times dbrda should be used.

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.

Examples

data(varespec, varechem)
## dbrda
dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="bray")
#> Call: dbrda(formula = varespec ~ N + P + K + Condition(Al), data =
#> varechem, distance = "bray")
#> 
#> -- Model Summary --
#> 
#>               Inertia Proportion Rank RealDims
#> Total          4.5444     1.0000              
#> Conditional    0.9726     0.2140    1         
#> Constrained    0.9731     0.2141    3        3
#> Unconstrained  2.5987     0.5718   19       13
#> 
#> Inertia is squared Bray distance
#> 
#> -- Eigenvalues --
#> 
#> Eigenvalues for constrained axes:
#> dbRDA1 dbRDA2 dbRDA3 
#> 0.5362 0.3198 0.1171 
#> 
#> Eigenvalues for unconstrained axes:
#>   MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8 
#> 0.9054 0.5070 0.3336 0.2581 0.2027 0.1605 0.1221 0.0825 
#> (Showing 8 of 19 unconstrained eigenvalues)
#> 
## avoid negative eigenvalues with sqrt distances
dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="bray",
     sqrt.dist = TRUE)
#> Call: dbrda(formula = varespec ~ N + P + K + Condition(Al), data =
#> varechem, distance = "bray", sqrt.dist = TRUE)
#> 
#> -- Model Summary --
#> 
#>               Inertia Proportion Rank
#> Total          6.9500     1.0000     
#> Conditional    0.9535     0.1372    1
#> Constrained    1.2267     0.1765    3
#> Unconstrained  4.7698     0.6863   19
#> 
#> Inertia is Bray distance
#> 
#> -- Eigenvalues --
#> 
#> Eigenvalues for constrained axes:
#> dbRDA1 dbRDA2 dbRDA3 
#> 0.5817 0.4086 0.2365 
#> 
#> Eigenvalues for unconstrained axes:
#>   MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8 
#> 0.9680 0.6100 0.4469 0.3837 0.3371 0.3012 0.2558 0.2010 
#> (Showing 8 of 19 unconstrained eigenvalues)
#> 
## avoid negative eigenvalues also with Jaccard distances
(m <- dbrda(varespec ~ N + P + K + Condition(Al), varechem, dist="jaccard"))
#> Call: dbrda(formula = varespec ~ N + P + K + Condition(Al), data =
#> varechem, distance = "jaccard")
#> 
#> -- Model Summary --
#> 
#>               Inertia Proportion Rank
#> Total          6.5044     1.0000     
#> Conditional    1.0330     0.1588    1
#> Constrained    1.2068     0.1855    3
#> Unconstrained  4.2646     0.6557   19
#> 
#> Inertia is squared Jaccard distance
#> 
#> -- Eigenvalues --
#> 
#> Eigenvalues for constrained axes:
#> dbRDA1 dbRDA2 dbRDA3 
#> 0.5992 0.3994 0.2082 
#> 
#> Eigenvalues for unconstrained axes:
#>   MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8 
#> 1.0388 0.6441 0.4518 0.3759 0.3239 0.2785 0.2279 0.1644 
#> (Showing 8 of 19 unconstrained eigenvalues)
#> 
## add species scores
sppscores(m) <- wisconsin(varespec)
## pco
pco(varespec, dist = "bray", sqrt.dist = TRUE)
#> Call: pco(X = varespec, dist = "bray", sqrt.dist = TRUE)
#> 
#> -- Model Summary --
#> 
#>               Inertia Rank
#> Total            6.95     
#> Unconstrained    6.95   23
#> 
#> Inertia is Bray distance
#> 
#> -- Eigenvalues --
#> 
#> Eigenvalues for unconstrained axes:
#>   MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8 
#> 1.6348 1.1428 0.5658 0.4780 0.3737 0.3716 0.3074 0.2665 
#> (Showing 8 of 23 unconstrained eigenvalues)
#>