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

Description

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.

Usage

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varpart(Y, X, ..., data, transfo, scale = FALSE)
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. In community ecology, that table is often a site-by-species table.

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 variable, or (3) name of data frame or matrix with numeric 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 numeric 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 arguments cannot be abbreviated nor omitted.

data

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

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".

scale

Should the columns of Y be standardized to unit variance

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. Use short names.

id.size

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

x

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.

...

Other parameters passed to functions.

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), 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 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.

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:

  • DfDegrees of freedom of numerator of the F-statistic for the fraction.

  • R.squareRaw R-squared. This is calculated only for fract and this is NA in other items.

  • Adj.R.squareAdjusted R-squared.

  • 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.

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-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 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. Adapted to vegan 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.

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.

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.

Examples

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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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