varpart: Partition the Variation of Community Matrix by 2, 3, or 4...

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


The function partitions the variation in community data or community dissimilarities with respect to two, three, or four explanatory tables, using adjusted R-squared in redundancy analysis ordination (RDA) or distance-based redundancy analysis. If response is a single vector, partitioning is by partial regression. Collinear variables in the explanatory tables do NOT have to be removed prior to partitioning.


varpart(Y, X, ..., data, chisquare = FALSE, transfo, scale = FALSE,
    add = FALSE, sqrt.dist = FALSE, permutations)
showvarparts(parts, labels, bg = NULL, alpha = 63, Xnames,
    id.size = 1.2,  ...)
## S3 method for class 'varpart234'
plot(x, cutoff = 0, digits = 1, ...)



Data frame or matrix containing the response data table or dissimilarity structure inheriting from dist. In community ecology, that table is often a site-by-species table or a dissimilarity object.


Two to four explanatory models, variables or tables. These can be defined in three alternative ways: (1) one-sided model formulae beginning with ~ and then defining the model, (2) name of a single numeric or factor variable, or (3) name of matrix with numeric or data frame with numeric and factor variables. The model formulae can have factors, interaction terms and transformations of variables. The names of the variables in the model formula are found in data frame given in data argument, and if not found there, in the user environment. Single variables, data frames or matrices are found in the user environment. All entries till the next argument (data or transfo) are interpreted as explanatory models, and the names of these extra arguments cannot be abbreviated nor omitted.


Other parameters passed to functions. NB, arguments after dots cannot be abbreviated but they must be spelt out completely.


The data frame with the variables used in the formulae in X.


Partition Chi-square or the inertia of Correspondence Analysis (cca).


Transformation for Y (community data) using decostand. All alternatives in decostand can be used, and those preserving Euclidean metric include "hellinger", "chi.square", "total", "norm". Ignored if Y are dissimilarities.


Should the columns of Y be standardized to unit variance. Ignored if Y are dissimilarities.


Add a constant to the non-diagonal values to euclidify dissimilarities (see wcmdscale for details). Choice "lingoes" (or TRUE) use the recommended method of Legendre & Anderson (1999: “method 1”) and "cailliez" uses their “method 2”. The argument has an effect only when Y are dissimilarities.


Take square root of dissimilarities. This often euclidifies dissimilarities. NB., the argument name cannot be abbreviated. The argument has an effect only when Y are dissimilarities.


If chisquare = TRUE, the adjusted R-squared is estimated by permutations, and this paramater can be a list of control values for the permutations as returned by the function how, or the number of permutations required, or a permutation matrix where each row gives the permuted indices.


Number of explanatory tables (circles) displayed.


Labels used for displayed fractions. Default is to use the same letters as in the printed output.


Fill colours of circles or ellipses.


Transparency of the fill colour. The argument takes precedence over possible transparency definitions of the colour. The value must be in range 0...255, and low values are more transparent. Transparency is not available in all graphics devices or file formats.


Names for sources of variation. Default names are X1, X2, X3 and X4. Xnames=NA, Xnames=NULL and Xnames="" produce no names. The names can be changed to other names. It is often best to use short names.


A numerical value giving the character expansion factor for the names of circles or ellipses.


The varpart result.


The values below cutoff will not be displayed.


The number of significant digits; the number of decimal places is at least one higher.


The functions partition the variation in Y into components accounted for by two to four explanatory tables and their combined effects. If Y is a multicolumn data frame or matrix, the partitioning is based on redundancy analysis (RDA, see rda) or on constrained correspondence analysis if chisquare = TRUE (CCA, see cca). If Y is a single variable, the partitioning is based on linear regression. If Y are dissimilarities, the decomposition is based on distance-based redundancy analysis (db-RDA, see capscale) following McArdle & Anderson (2001). The input dissimilarities must be compatible to the results of dist. Vegan functions vegdist, designdist, raupcrick and betadiver produce such objects, as do many other dissimilarity functions in R packages. However, symmetric square matrices are not recognized as dissimilarities but must be transformed with as.dist. Partitioning will be made to squared dissimilarities analogously to using variance with rectangular data – unless sqrt.dist = TRUE was specified.

The function primarily uses adjusted R-squared to assess the partitions explained by the explanatory tables and their combinations (see RsquareAdj), because this is the only unbiased method (Peres-Neto et al., 2006). The raw R-squared for basic fractions are also displayed, but these are biased estimates of variation explained by the explanatory table. In correspondence analysis (chisquare = TRUE), the adjusted R-squared are found by permutation and they vary in repeated analyses.

The identifiable fractions are designated by lower case alphabets. The meaning of the symbols can be found in the separate document (use browseVignettes("vegan")), or can be displayed graphically using function showvarparts.

A fraction is testable if it can be directly expressed as an RDA or db-RDA model. In these cases the printed output also displays the corresponding RDA model using notation where explanatory tables after | are conditions (partialled out; see rda for details). Although single fractions can be testable, this does not mean that all fractions simultaneously can be tested, since the number of testable fractions is higher than the number of estimated models. The non-testable components are found as differences of testable components. The testable components have permutation variance in correspondence analysis (chisquare = TRUE), and the non-testable components have even higher variance.

An abridged explanation of the alphabetic symbols for the individual fractions follows, but computational details should be checked in the vignette (readable with browseVignettes("vegan")) or in the source code.

With two explanatory tables, the fractions explained uniquely by each of the two tables are [a] and [c], and their joint effect is [b] following Borcard et al. (1992).

With three explanatory tables, the fractions explained uniquely by each of the three tables are [a] to [c], joint fractions between two tables are [d] to [f], and the joint fraction between all three tables is [g].

With four explanatory tables, the fractions explained uniquely by each of the four tables are [a] to [d], joint fractions between two tables are [e] to [j], joint fractions between three variables are [k] to [n], and the joint fraction between all four tables is [o].

There is a plot function that displays the Venn diagram and labels each intersection (individual fraction) with the adjusted R squared if this is higher than cutoff. A helper function showvarpart displays the fraction labels. The circles and ellipses are labelled by short default names or by names defined by the user in argument Xnames. Longer explanatory file names can be written on the varpart output plot as follows: use option Xnames=NA, then add new names using the text function. A bit of fiddling with coordinates (see locator) and character size should allow users to place names of reasonably short lengths on the varpart plot.


Function varpart returns an object of class "varpart" with items scale and transfo (can be missing) which hold information on standardizations, tables which contains names of explanatory tables, and call with the function call. The function varpart calls function varpart2, varpart3 or varpart4 which return an object of class "varpart234" and saves its result in the item part. The items in this object are:


Sum of squares of matrix Y.


Number of observations (rows).


Number of explanatory tables


Warnings on collinearity.


Basic fractions from all estimated constrained models.


Individual fractions or all possible subsections in the Venn diagram (see showvarparts).


Fractions that can be found after conditioning on single explanatory table in models with three or four explanatory tables.


Fractions that can be found after conditioning on two explanatory tables in models with four explanatory tables.

Fraction Data Frames

Items fract, indfract, contr1 and contr2 are all data frames with items:


You can use command browseVignettes("vegan") to display document which presents Venn diagrams showing the fraction names in partitioning the variation of Y with respect to 2, 3, and 4 tables of explanatory variables, as well as the equations used in variation partitioning.

The functions frequently give negative estimates of variation. Adjusted R-squared can be negative for any fraction; unadjusted R-squared of testable fractions of variances will be non-negative. Non-testable fractions cannot be found directly, but by subtracting different models, and these subtraction results can be negative. The fractions are orthogonal, or linearly independent, but more complicated or nonlinear dependencies can cause negative non-testable fractions. Any fraction can be negative for non-Euclidean dissimilarities because the underlying db-RDA model can yield negative eigenvalues (see capscale, dbrda). These negative eigenvalues in the underlying analysis can be avoided with arguments sqrt.dist and add which have a similar effect as in capscale: the square roots of several dissimilarities do not have negative eigenvalues, and no negative eigenvalues are produced after Lingoes or Cailliez adjustment, which in effect add random variation to the dissimilarities.

A simplified, fast version of RDA, CCA adn dbRDA are used (functions simpleRDA2, simpleCCA and simpleDBRDA). The actual calculations are done in functions varpart2 to varpart4, but these are not intended to be called directly by the user.


Pierre Legendre, Departement de Sciences Biologiques, Universite de Montreal, Canada. Further developed by Jari Oksanen.


(a) References on variation partitioning

Borcard, D., P. Legendre & P. Drapeau. 1992. Partialling out the spatial component of ecological variation. Ecology 73: 1045–1055.

Legendre, P. & L. Legendre. 2012. Numerical ecology, 3rd English edition. Elsevier Science BV, Amsterdam.

(b) Reference on transformations for species data

Legendre, P. and E. D. Gallagher. 2001. Ecologically meaningful transformations for ordination of species data. Oecologia 129: 271–280.

(c) Reference on adjustment of the bimultivariate redundancy statistic

Peres-Neto, P., P. Legendre, S. Dray and D. Borcard. 2006. Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology 87: 2614–2625.

(d) References on partitioning of dissimilarities

Legendre, P. & Anderson, M. J. (1999). Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecological Monographs 69, 1–24.

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

For analysing testable fractions, see rda and anova.cca. For data transformation, see decostand. Function inertcomp gives (unadjusted) components of variation for each species or site separately. Function rda displays unadjusted components in its output, but RsquareAdj will give adjusted R-squared that are similar to the current function also for partial models.



# Two explanatory data frames -- Hellinger-transform Y
mod <- varpart(mite, mite.env, mite.pcnm, transfo="hel")

## Use fill colours
showvarparts(2, bg = c("hotpink","skyblue"))
plot(mod, bg = c("hotpink","skyblue"))
## Test fraction [a] using partial RDA, '~ .' in formula tells to use
## all variables of data mite.env.
aFrac <- rda(decostand(mite, "hel"), mite.env, mite.pcnm)
## RsquareAdj gives the same result as component [a] of varpart

## Partition Bray-Curtis dissimilarities
varpart(vegdist(mite), mite.env, mite.pcnm)
## Three explanatory tables with formula interface
mod <- varpart(mite, ~ SubsDens + WatrCont, ~ Substrate + Shrub + Topo,
   mite.pcnm, data=mite.env, transfo="hel")
showvarparts(3, bg=2:4)
plot(mod, bg=2:4)

## Use RDA to test fraction [a]
## Matrix can be an argument in formula
rda.result <- rda(decostand(mite, "hell") ~ SubsDens + WatrCont +
   Condition(Substrate + Shrub + Topo) +
   Condition(as.matrix(mite.pcnm)), data = mite.env)

## Four explanatory tables
mod <- varpart(mite, ~ SubsDens + WatrCont, ~Substrate + Shrub + Topo,
  mite.pcnm[,1:11], mite.pcnm[,12:22], data=mite.env, transfo="hel")
plot(mod, bg=2:5)
## Show values for all partitions by putting 'cutoff' low enough:
plot(mod, cutoff = -Inf, cex = 0.7, bg=2:5)

vegan documentation built on May 17, 2018, 5:03 p.m.