Partition the Variation of Community Matrix by 2, 3, or 4 Explanatory Matrices
The function partitions the variation of response table Y with respect to two, three, or four explanatory tables, using adjusted R-squared 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.
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Data frame or matrix containing the response data table. In community ecology, that table is often a site-by-species table.
Two to four explanatory models, variables or tables. These can
be defined in three alternative ways: (1) one-sided model formulae
The data frame with the variables used in the formulae in
Should the columns of
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
A numerical value giving the character expansion factor for the names of circles or ellipses.
The values below
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
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 R-squared to assess the partitions explained by the explanatory tables and their combinations, 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.
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
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
| 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
With two explanatory tables, the fractions explained
uniquely by each of the two tables are
[c], and their joint effect
[b] following Borcard et al. (1992).
With three explanatory tables, the fractions explained uniquely
by each of the three tables are
[c], joint fractions between two tables are
[f], and the joint fraction between all three
With four explanatory tables, the fractions explained uniquely by each
of the four tables are
[d], joint fractions between two tables are
[j], joint fractions between three variables are
[n], and the joint fraction between all four tables is
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
character size should allow users to place names of reasonably short
lengths on the
varpart returns an
object of class
"varpart" with items
transfo (can be missing) which hold information on
tables which contains names of explanatory
call with the function
varpart calls function
varpart4 which return an object of class
"varpart234" and saves its result in the item
The items in this object are:
Sum of squares of matrix
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
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
contr2 are all data frames with
DfDegrees of freedom of numerator of the F-statistic for the fraction.
R.squareRaw R-squared. This is calculated only for
fractand this is
NAin other items.
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
The functions frequently give negative estimates of variation. Adjusted R-squared can be negative for any fraction; unadjusted R-squared of testable fractions always 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.
The current function will only use RDA in multivariate partitioning. It is much more complicated to estimate the adjusted R-squares 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
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
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.
For analysing testable fractions, see
anova.cca. For data transformation, see
(unadjusted) components of variation for each species or site
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.
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data(mite) data(mite.env) data(mite.pcnm) # Two explanatory matrices -- Hellinger-transform 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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