varpart {vegan}

Partition the Variation of Community Matrix by 2, 3, or 4 Explanatory Matrices

Description

The function partitions the variation in community data or community dissimilarities with respect to two, three, or four explanatory tables, using adjusted R^2 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.

Usage

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

Arguments

Y

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.

X

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.

data

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

chisquare

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

transfo

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.

scale

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

add

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.

sqrt.dist

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.

permutations

If chisquare = TRUE, the adjusted R^2 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.

parts

Number of explanatory tables (circles) displayed.

labels

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

bg

Fill colours of circles or ellipses.

alpha

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.

Xnames

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.

id.size

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

x, object

The varpart result.

cutoff

The values below cutoff will not be displayed.

digits

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

Details

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 dbrda) 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. 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^2 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^2 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^2 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 [b], and their joint effect is [c].

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].

summary will give an overview of unique and and overall contribution of each group of variables. The overall contribution (labelled as “Contributed”) consists of the unique contribution of the variable and equal shares of each fraction where the variable contributes. The summary tabulates how each fraction is divided between the variables, and the contributed component is the sum of all these divided fractions. The summary is based on the idea of Lai et al. (2022), and is similar to the output of their rdacca.hp package.

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.

Value

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:

SS.Y

Sum of squares of matrix Y.

n

Number of observations (rows).

nsets

Number of explanatory tables

bigwarning

Warnings on collinearity.

fract

Basic fractions from all estimated constrained models.

indfract

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

contr1

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

contr2

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:

Df:

Degrees of freedom of numerator of the F-statistic for the fraction.

R.square:

Raw R^2. This is calculated only for fract and this is NA in other items.

Adj.R.square:

Adjusted R^2.

Testable:

If the fraction can be expressed as a (partial) RDA model, it is directly Testable, and this field is TRUE. In that case the fraction label also gives the specification of the testable RDA model.

Note

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^2 can be negative for any fraction; unadjusted R^2 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 dbrda). These negative eigenvalues in the underlying analysis can be avoided with arguments sqrt.dist and add which have a similar effect as in dbrda: 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.

Author(s)

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

References

(a) References on variation partitioning

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

Lai J., Y. Zou, J. Zhang & P. Peres-Neto. 2022. Generalizing hierarchical and variation partitioning in multiple regression and canonical analysis using the rdacca.hp R package. Methods in Ecology and Evolution, 13: 782–788.

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^2 that are similar to the current function also for partial models.

Examples

data(mite)
data(mite.env)
data(mite.pcnm)

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

## 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)
anova(aFrac)
## RsquareAdj gives the same result as component [a] of varpart
RsquareAdj(aFrac)

## 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")
mod
summary(mod)
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)
anova(rda.result)

## Four explanatory tables
mod <- varpart(mite, ~ SubsDens + WatrCont, ~Substrate + Shrub + Topo,
  mite.pcnm[,1:11], mite.pcnm[,12:22], data=mite.env, transfo="hel")
mod
summary(mod)
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)