capscale: [Partial] Constrained Analysis of Principal Coordinates or...

Description Usage Arguments Details Value Note Author(s) References See Also Examples


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.


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



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. 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 CAP.


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


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


Take square roots of dissimilarities. See section Notes below.


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.


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 cmdscale for details). This implements “correction method 2” of Legendre & Legendre (2012, p. 503). The negative eigenvalues are caused by using semi-metric or non-metric dissimilarities with basically metric cmdscale. They are harmless and ignored in capscale, but you also can avoid warnings with this option.


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


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.


Handling of missing values in constraints or conditions. The default ( 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 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 rda or to metaMDSdist.


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 of cmdscale, whereas capscale uses 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, 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.

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.


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.


The function produces negative eigenvalues with non-Euclidean dissimilarity indices. The non-Euclidean component of inertia is given under the title Imaginary in the printed output. The Total inertia is the sum of all eigenvalues, but the sum of all non-negative eigenvalues is given as Real Total (which is higher than the Total). The ordination is based only on the real dimensions with positive eigenvalues, and therefore the proportions of inertia components only apply to the Real Total and ignore the Imaginary component. Permutation tests with anova.cca use only the real solution of positive eigenvalues. Function adonis gives similar significance tests, but it also handles the imaginary dimensions (negative eigenvalues) and therefore its results may differ from permutation test results of capscale.

If the negative eigenvalues are disturbing, you can use argument add = TRUE passed to cmdscale, or, preferably, a distance measure that does not cause these warnings. Alternatively, after square root transformation of distances (argument sqrt.dist = TRUE) many indices do not produce negative eigenvalues.

The inertia is named after the dissimilarity index as defined in the dissimilarity data, or as unknown distance if such an information is missing. Function rda usually divides the ordination scores 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 than or equal to 4 (allowing for stepacross), this division is undone in capscale and original dissimilarities are used. Keyword mean is added to the inertia in cases where division was made, e.g. in 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, keyword euclidified is added to the the name of inertia, and the value of the constant is printed (argument add = TRUE).


Jari Oksanen


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

See Also

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

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 capscale results.


## Basic Analysis
vare.cap <- capscale(varespec ~ N + P + K + Condition(Al), varechem,
## 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)

Example output

Loading required package: permute
Loading required package: lattice
This is vegan 2.4-3
Call: capscale(formula = varespec ~ N + P + K + Condition(Al), data =
varechem, distance = "bray")

              Inertia Proportion Eigenvals Rank
Total          4.5444     1.0000    4.8034     
Conditional    0.9726     0.2140    0.9772    1
Constrained    0.9731     0.2141    0.9972    3
Unconstrained  2.5987     0.5718    2.8290   15
Imaginary                          -0.2590    8
Inertia is squared Bray distance 

Eigenvalues for constrained axes:
  CAP1   CAP2   CAP3 
0.5413 0.3265 0.1293 

Eigenvalues for unconstrained axes:
  MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8   MDS9  MDS10  MDS11 
0.9065 0.5127 0.3379 0.2626 0.2032 0.1618 0.1242 0.0856 0.0689 0.0583 0.0501 
 MDS12  MDS13  MDS14  MDS15 
0.0277 0.0208 0.0073 0.0013 

Permutation test for capscale under reduced model
Permutation: free
Number of permutations: 999

Model: capscale(formula = varespec ~ N + P + K + Condition(Al), data = varechem, distance = "bray")
         Df SumOfSqs      F Pr(>F)   
Model     3  0.97314 2.3717  0.005 **
Residual 19  2.59866                 
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Call: capscale(formula = varespec ~ N + P + K + Condition(Al), data =
varechem, distance = "bray", add = TRUE)

              Inertia Proportion Rank
Total          6.2496     1.0000     
Conditional    1.0468     0.1675    1
Constrained    1.1956     0.1913    3
Unconstrained  4.0073     0.6412   19
Inertia is Lingoes adjusted squared Bray distance 

Eigenvalues for constrained axes:
  CAP1   CAP2   CAP3 
0.6103 0.3940 0.1913 

Eigenvalues for unconstrained axes:
  MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8 
0.9796 0.5811 0.4077 0.3322 0.2769 0.2346 0.1962 0.1566 
(Showed only 8 of all 19 unconstrained eigenvalues)

Constant added to distances: 0.07413903 

Call: capscale(formula = varespec ~ N + P + K + Condition(Al), data =
varechem, distance = "bray", sqrt.dist = TRUE)

              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 for constrained axes:
  CAP1   CAP2   CAP3 
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 
(Showed only 8 of all 19 unconstrained eigenvalues)

Square root transformation
Wisconsin double standardization
Call: capscale(formula = varespec ~ 1, distance = "bray", metaMDSdist =

               Inertia Eigenvals Rank
Total          2.54753   2.59500     
Unconstrained  2.54753   2.59500   19
Imaginary               -0.04747    4
Inertia is squared Bray distance 

Eigenvalues for unconstrained axes:
  MDS1   MDS2   MDS3   MDS4   MDS5   MDS6   MDS7   MDS8 
0.6075 0.3820 0.3335 0.2046 0.1731 0.1684 0.1505 0.1163 
(Showed only 8 of all 19 unconstrained eigenvalues)

metaMDSdist transformed data: wisconsin(sqrt(varespec)) 

vegan documentation built on May 2, 2019, 5:51 p.m.