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.

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

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.

- 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.

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.

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

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. 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.

(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.

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.

```
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)
```

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