The function partitions the variation of response table Y with respect to two, three, or four explanatory tables, using adjusted Rsquared in redundancy analysis ordination (RDA). If Y contains a single vector, partitioning is by partial regression. Collinear variables in the explanatory tables do NOT have to be removed prior to partitioning.
1 2 3 4 5 
Y 
Data frame or matrix containing the response data table. In community ecology, that table is often a sitebyspecies table. 
X 
Two to four explanatory models, variables or tables. These can
be defined in three alternative ways: (1) onesided model formulae
beginning with 
data 
The data frame with the variables used in the formulae in

transfo 
Transformation for 
scale 
Should the columns of 
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 
id.size 
A numerical value giving the character expansion factor for the names of circles or ellipses. 
x 
The 
cutoff 
The values below 
digits 
The number of significant digits; the number of decimal places is at least one higher. 
... 
Other parameters passed to functions. 
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
), and if Y
is a single variable, the
partitioning is based on linear regression.
The function primarily uses adjusted Rsquared to assess the partitions explained by the explanatory tables and their combinations, because this is the only unbiased method (PeresNeto et al., 2006). The raw Rsquared for basic fractions are also displayed, but these are biased estimates of variation explained by the explanatory table.
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 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 there number of testable fractions is higher than
the number of estimated models.
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:
SS.Y 
Sum of squares of matrix 
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 
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. 
Items fract
,
indfract
, contr1
and contr2
are all data frames with
items:
DfDegrees of freedom of numerator of the Fstatistic for the fraction.
R.squareRaw Rsquared. This is calculated only for
fract
and this is NA
in other items.
Adj.R.squareAdjusted Rsquared.
TestableIf 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.
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 Rsquared can be negative for any fraction; unadjusted Rsquared of testable fractions always will be nonnegative. Nontestable 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 nontestable fractions.
The current function will only use RDA in multivariate partitioning. It is much more complicated to estimate the adjusted Rsquares for CCA, and unbiased analysis of CCA is not currently implemented.
A simplified, fast version of RDA is used (function
simpleRDA2
). 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. Adapted to vegan 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
PeresNeto, P., P. Legendre, S. Dray and D. Borcard. 2006. Variation partitioning of species data matrices: estimation and comparison of fractions. Ecology 87: 2614–2625.
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 Rsquared that are similar to the current
function also for partial models.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47  data(mite)
data(mite.env)
data(mite.pcnm)
# Two explanatory matrices  Hellingertransform Y
# Formula shortcut "~ ." means: use all variables in 'data'.
mod < varpart(mite, ~ ., mite.pcnm, data=mite.env, transfo="hel")
mod
## Use fill colours
showvarparts(2, bg = c("hotpink","skyblue"))
plot(mod, bg = c("hotpink","skyblue"))
# Alternative way of to conduct this partitioning
# Change the data frame with factors into numeric model matrix
mm < model.matrix(~ SubsDens + WatrCont + Substrate + Shrub + Topo, mite.env)[,1]
mod < varpart(decostand(mite, "hel"), mm, mite.pcnm)
# Test fraction [a] using partial RDA:
aFrac < rda(decostand(mite, "hel"), mm, mite.pcnm)
anova(aFrac, step=200, perm.max=200)
# RsquareAdj gives the same result as component [a] of varpart
RsquareAdj(aFrac)
# Three explanatory matrices
mod < varpart(mite, ~ SubsDens + WatrCont, ~ Substrate + Shrub + Topo,
mite.pcnm, data=mite.env, transfo="hel")
mod
showvarparts(3, bg=2:4)
plot(mod, bg=2:4)
# An alternative formulation of the previous model using
# matrices mm1 amd mm2 and Hellinger transformed species data
mm1 < model.matrix(~ SubsDens + WatrCont, mite.env)[,1]
mm2 < model.matrix(~ Substrate + Shrub + Topo, mite.env)[, 1]
mite.hel < decostand(mite, "hel")
mod < varpart(mite.hel, mm1, mm2, mite.pcnm)
# Use RDA to test fraction [a]
# Matrix can be an argument in formula
rda.result < rda(mite.hel ~ mm1 + Condition(mm2) +
Condition(as.matrix(mite.pcnm)))
anova(rda.result, step=200, perm.max=200)
# 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
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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