varpart: 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 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

varpart(Y, X, ..., data, transfo, scale = FALSE)
showvarparts(parts, labels, ...)
## S3 method for class 'varpart234':
plot(x, cutoff = 0, digits = 1, ...)

Arguments

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 beginning with ~ and then defining the model, (2) name of a single numeric variable, or (3) name of data

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"

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.

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.

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.YSum of squares of matrix Y.
nNumber of observations (rows).
nsetsNumber of explanatory tables
bigwarningWarnings on collinearity.
fractBasic fractions from all estimated constrained models.
indfractInvididual fractions or all possible subsections in the Venn diagram (see showvarparts).
contr1Fractions that can be found after conditioning on single explanatory table in models with three or four explanatory tables.
contr2Fractions 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:

{Degrees of freedom of numerator of the $F$-statististic for the fraction.} R.square{Raw R-squared. This is calculated only for fract and this is NA in other items.} Adj.R.square{Adjusted R-squared.} 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.}

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. A simplified, fast version of RDA is used (function simpleRDA2). The actual calculations are done in fucntions varpart2 to varpart4, but these are not intended to be called directly by the user.

The function primarily uses adjusted R squares to assess the partitions explained by the explanatory tables and their combinations, because this is the only unbiased method (Peres-Neto et al., submitted). The raw R squares 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 "partionining.pdf" (which can be read using vegandocs), 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 "partitioning.pdf" (readable with vegandocs) 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.

References

(a) References on variation partioning

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

Legendre, P. & L. Legendre. 1998. Numerical ecology, 2nd 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.

Peres-Neto, P., P. Legendre, S. Dray and D. Borcard. 2006. Variation partioning of species data matrices: estimation and comparison of fractions. Ecology 87: 2614--2625.

Examples

Run this code

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

## See detailed documentation:
vegandocs("partition")

# 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
showvarparts(2)
plot(mod)
# 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 RDA:
rda.result <- rda(decostand(mite, "hell"), mm, mite.pcnm)
anova(rda.result, step=200, perm.max=200)

# Three explanatory matrices 
mod <- varpart(mite, ~ SubsDens + WatrCont, ~ Substrate + Shrub + Topo,
   mite.pcnm, data=mite.env, transfo="hel")
mod
showvarparts(3)
plot(mod)
# 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]
rda.result <- rda(mite.hel, mm1, cbind(mm2, 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)
# Show values for all partitions by putting 'cutoff' low enough:
plot(mod, cutoff = -Inf, cex = 0.7)

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