fourthcorner: Functions to compute the fourth-corner statistic

In ade4: Analysis of Ecological Data: Exploratory and Euclidean Methods in Environmental Sciences

Functions to compute the fourth-corner statistic

Description

These functions allow to compute the fourth-corner statistic for abundance or presence-absence data. The fourth-corner statistic has been developed by Legendre et al (1997) and extended in Dray and Legendre (2008). The statistic measures the link between three tables: a table L (n x p) containing the abundances of p species at n sites, a second table R (n x m) containing the measurements of m environmental variables for the n sites, and a third table Q (p x s) describing s species traits for the p species.

Usage

fourthcorner(tabR, tabL, tabQ, modeltype = 6, nrepet = 999, tr01 =
FALSE, p.adjust.method.G = p.adjust.methods, p.adjust.method.D =
p.adjust.methods, p.adjust.D = c("global", "levels"), ...)
fourthcorner2(tabR, tabL, tabQ, modeltype = 6, nrepet = 999, tr01 =
FALSE, p.adjust.method.G = p.adjust.methods, ...)
## S3 method for class '4thcorner'
print(x, varQ = 1:length(x$varnames.Q), varR =
1:length(x$varnames.R), stat = c("D", "D2"), ...)
## S3 method for class '4thcorner'
summary(object,...)
## S3 method for class '4thcorner'
plot(x, stat = c("D", "D2", "G"), type = c("table",
"biplot"), xax = 1, yax = 2, x.rlq = NULL, alpha = 0.05, col =
c("lightgrey", "red", "deepskyblue", "purple"), ...)
fourthcorner.rlq(xtest, nrepet = 999, modeltype = 6, typetest =
c("axes", "Q.axes", "R.axes"), p.adjust.method.G = p.adjust.methods,
p.adjust.method.D = p.adjust.methods, p.adjust.D = c("global",
"levels"), ...)

Arguments

tabR	a dataframe containing the measurements (numeric values or factors) of m environmental variables (columns) for the n sites (rows).
tabL	a dataframe containing the abundances of p species (columns) at n sites (rows).
tabQ	a dataframe containing numeric values or factors describing s species traits (columns) for the p species (rows).
modeltype	an integer (1-6) indicating the permutation model used in the testing procedure (see details).
nrepet	the number of permutations
tr01	a logical indicating if data in tabL must be transformed to presence-absence data (FALSE by default)
object	an object of the class 4thcorner
x	an object of the class 4thcorner
varR	a vector containing indices for variables in tabR
varQ	a vector containing indices for variables in tabQ
type	results are represented by a table or on a biplot (see x.rlq)
alpha	a value of significance level
p.adjust.method.G	a string indicating a method for multiple adjustment used for output tabG, see p.adjust.methods for possible choices
p.adjust.method.D	a string indicating a method for multiple adjustment used for output tabD/tabD2, see p.adjust.methods for possible choices
p.adjust.D	a string indicating if multiple adjustment for tabD/tabD2 should be done globally or only between levels of a factor ("levels", as in the original paper of Legendre et al. 1997)
stat	a character to specify if results should be plotted for cells (D and D2) or variables (G)
xax	an integer indicating which rlq axis should be plotted on the x-axis
yax	an integer indicating which rlq axis should be plotted on the y-axis
x.rlq	an object created by the rlq function. Used to represent results on a biplot (type should be "biplot" and object created by the fourthcorner functions)
col	a vector of length 4 containing four colors used for the graphical representations. The first is used to represent non-significant associations, the second positive significant, the third negative significant. For the 'biplot' method and objects created by the fourthcorner.rlq function, the second corresponds to variables significantly linked to the x-axis, the third for the y-axis and the fourth for both axes
xtest	an object created by the rlq function
typetest	a string indicating which tests should be performed
...	further arguments passed to or from other methods

Details

For the fourthcorner function, the link is measured by a Pearson correlation coefficient for two quantitative variables (trait and environmental variable), by a Pearson Chi2 and G statistic for two qualitative variables and by a Pseudo-F and Pearson r for one quantitative variable and one qualitative variable. The fourthcorner2 function offers a multivariate statistic (equal to the sum of eigenvalues of RLQ analysis) and measures the link between two variables by a square correlation coefficient (quant/quant), a Chi2/sum(L) (qual/qual) and a correlation ratio (quant/qual). The significance is tested by a permutation procedure. Different models are available:

• model 1 (modeltype=1): Permute values for each species independently (i.e., permute within each column of table L)

• model 2 (modeltype=2): Permute values of sites (i.e., permute entire rows of table L)

• model 3 (modeltype=3): Permute values for each site independently (i.e., permute within each row of table L)

• model 4 (modeltype=4): Permute values of species (i.e., permute entire columns of table L)

• model 5 (modeltype=5): Permute values of species and after (or before) permute values of sites (i.e., permute entire columns and after (or before) entire rows of table L)

• model 6 (modeltype=6): combination of the outputs of models 2 and 4. Dray and Legendre (2008) and ter Braak et al. (20012) showed that all models (except model 6) have inflated type I error.

Note that the model 5 is strictly equivalent to permuting simultaneously the rows of tables R and Q, as proposed by Doledec et al. (1996).

The function summary returns results for variables (G). The function print returns results for cells (D and D2). In the case of qualitative variables, Holm's corrected pvalues are also provided.

The function plot produces a graphical representation of the results (white for non significant, light grey for negative significant and dark grey for positive significant relationships). Results can be plotted for variables (G) or for cells (D and D2). In the case of qualitative / quantitative association, homogeneity (D) or correlation (D2) are plotted.

Value

The fourthcorner function returns a a list where:

tabD is a krandtest object giving the results of tests for cells of the fourth-corner (homogeneity for quant./qual.). tabD2 is a krandtest object giving the results of tests for cells of the fourth-corner (Pearson r for quant./qual.). tabG is a krandtest object giving the results of tests for variables (Pearson's Chi2 for qual./qual.).

The fourthcorner2 function returns a list where:

tabG is a krandtest object giving the results of tests for variables. trRLQ is a krandtest object giving the results of tests for the multivariate statistic (i.e. equivalent to randtest.rlq function).

Author(s)

Stéphane Dray [e-mail]

References

Doledec, S., Chessel, D., ter Braak, C.J.F. and Champely, S. (1996) Matching species traits to environmental variables: a new three-table ordination method. Environmental and Ecological Statistics, 3, 143–166.

Legendre, P., R. Galzin, and M. L. Harmelin-Vivien. (1997) Relating behavior to habitat: solutions to the fourth-corner problem. Ecology, 78, 547–562.

Dray, S. and Legendre, P. (2008) Testing the species traits-environment relationships: the fourth-corner problem revisited. Ecology, 89, 3400–3412.

ter Braak, C., Cormont, A., and Dray, S. (2012) Improved testing of species traits-environment relationships in the fourth corner problem. Ecology, 93, 1525–1526.

Dray, S., Choler, P., Doledec, S., Peres-Neto, P.R., Thuiller, W., Pavoine, S. and ter Braak, C.J.F (2014) Combining the fourth-corner and the RLQ methods for assessing trait responses to environmental variation. Ecology, 95, 14–21. doi:10.1890/13-0196.1

See Also

rlq, combine.4thcorner, p.adjust.methods

Examples

data(aviurba)

## Version using the sequential test (ter Braak et al 2012)
## as recommended in Dray et al (2013), 
## using Holm correction of P-values (only 99 permutations here)
four.comb.default <- fourthcorner(aviurba$mil,aviurba$fau,aviurba$traits,nrepet=99)
summary(four.comb.default)
plot(four.comb.default, stat = "G")

## using fdr correction of P-values
four.comb.fdr <- fourthcorner(aviurba$mil, aviurba$fau, aviurba$traits,
nrepet = 99, p.adjust.method.G = 'fdr', p.adjust.method.D = 'fdr')  
summary(four.comb.fdr)
plot(four.comb.fdr, stat = "G")

## Explicit procedure to combine the results of two models
## proposed in Dray and Legendre (2008);the above does this implicitly
four2 <- fourthcorner(aviurba$mil,aviurba$fau,aviurba$traits,nrepet=99,modeltype=2)
four4 <- fourthcorner(aviurba$mil,aviurba$fau,aviurba$traits,nrepet=99,modeltype=4)
four.comb <- combine.4thcorner(four2, four4)
summary(four.comb)
plot(four.comb, stat = "G")

ade4 documentation built on Feb. 16, 2023, 7:58 p.m.