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Community- and Species-Level Permutation Test in Double Constrained Correspondence Analysis (dc-CA)

Source: R/anova.dcca.R
anova.dcca.Rd

anova.dcca performs the community- and species-level permutation tests of dc-CA and combines these in the 'max test', which takes the maximum of the P-values. The function arguments are similar to (but more restrictive than) those of anova.cca.

Usage

# S3 method for class 'dcca'
anova(
  object,
  ...,
  rpp = TRUE,
  permutations = 999,
  by = c("omnibus", "axis"),
  n_axes = "all"
)

Arguments

object

an object from dc_CA.

...

unused.

rpp

Logical indicating residual predictor permutation (default TRUE). When FALSE, residual response permutation is used.

permutations

a list of control values for the permutations for species and sites (species first, sites second, for traits and environment) as returned by the function how, or the number of permutations required (default 999, or a two-vector with the number for the species-level test first and that for the sites-level second), or a list of two permutation matrices (again, species first, sites second) where each row gives the permuted indices.

by

character "axis" which performs a series of tests, one for each axis, with the eigenvalue of the axis as test statistic. Default: NULL which sets the test statistic to the inertia (sum of all double constrained eigenvalues; named constraintsTE in the inertia element of dc_CA.

The interpretation of this inertia is, at the species-level, the environmentally constrained inertia explained by the traits (without trait covariates) and, at the community-level, the trait-constrained inertia explained by the environmental predictors (without covariates). The default (NULL) is computationally quicker as it avoids computation of an svd of permuted data sets.

n_axes

number of axes used in the test statistic (default: "all"). Example, the test statistic is the sum of the first two eigenvalues, if n_axes=2. With a numeric n_axes and model ~X + Condition(Z), the residuals of X with respect to Z are permuted with a test statistic equal to the sum of the first n_axes eigenvalues of the fitted Y in the model Y ~ X + Z, with Y the response in the model. In the default "all", the test statistic is all eigenvalues of the model Y ~ X|Z, i.e. the effects of X after adjustment for the effects on Y of Z. If by = "axis", the setting of n_axes is ignored.

Value

A list of 3 of structures as from anova.cca. The elements are c("species", "sites", "maxP")

Details

In the general case of varying site abundance totals (divideBySiteTotals = FALSE) both the community-level test and the species-level test use residualized predictor permutation (ter Braak 2022), so as to ensure that anova.dcca is robust against differences in species and site total abundance in the response (ter Braak & te Beest, 2022). The community-level test uses anova_sites. For the species-level test, anova_species is used.

With equal site weights, obtained with divide.by.site.total = TRUE, the community-level test is obtained by applying anova to object$RDAonEnv using anova.cca. This performs residualized response permutation which performs about equal to residualized predictor permutation in the equi-weight case. The function anova.cca is implemented in C and therefore a factor of 20 or so quicker than the native R-code used in anova_sites.

The n_axes argument is new, and can be considered experimental.

References

ter Braak, C.J.F. & te Beest, D.E. 2022. Testing environmental effects on taxonomic composition with canonical correspondence analysis: alternative permutation tests are not equal. Environmental and Ecological Statistics. 29 (4), 849-868. doi:10.1007/s10651-022-00545-4

ter Braak, C.J.F. (2022) Predictor versus response permutation for significance testing in weighted regression and redundancy analysis. Journal of statistical computation and simulation, 92, 2041-2059. doi:10.1080/00949655.2021.2019256

Examples

data("dune_trait_env")

# rownames are carried forward in results
rownames(dune_trait_env$comm) <- dune_trait_env$comm$Sites

mod <- dc_CA(formulaEnv = ~ A1 + Moist + Mag + Use + Manure,
             formulaTraits = ~ SLA + Height + LDMC + Seedmass + Lifespan,
             response = dune_trait_env$comm[, -1],  # must delete "Sites"
             dataEnv = dune_trait_env$envir,
             dataTraits = dune_trait_env$traits,
             verbose = FALSE)
anova(mod)
#> $species
#> Species-level permutation test using dc-CA
#> Model: dc_CA(formulaEnv = ~A1 + Moist + Mag + Use + Manure, formulaTraits = ~SLA + Height + LDMC + Seedmass + Lifespan, response = dune_trait_env$comm[, -1], dataEnv = dune_trait_env$envir, dataTraits = dune_trait_env$traits, verbose = FALSE) 
#> Residualized predictor permutation
#> 
#>          df ChiSquare     R2      F Pr(>F)  
#> dcCA      5   0.44542 0.3367 2.2335  0.019 *
#> Residual 22   0.87749                       
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> $sites
#>          Df ChiSquare      R2      F Pr(>F)    
#> dcCA1     7   0.44542 0.65733 3.2885  0.001 ***
#> Residual 12   0.23220                          
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> $maxP
#> Max test combining the community- and species- level tests 
#> Model: dc_CA(formulaEnv = ~A1 + Moist + Mag + Use + Manure, formulaTraits = ~SLA + Height + LDMC + Seedmass + Lifespan, response = dune_trait_env$comm[, -1], dataEnv = dune_trait_env$envir, dataTraits = dune_trait_env$traits, verbose = FALSE) 
#> 
#> Taken from the species-level test:
#> Residualized predictor permutation
#> Permutation: free
#> Number of permutations: 999
#> 
#>          df ChiSquare     R2      F Pr(>F)  
#> dcCA      5   0.44542 0.3367 2.2335  0.019 *
#> Residual 22   0.87749                       
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 

a_species <- anova_species(mod)
a_species
#> $table
#> Species-level permutation test using dc-CA
#> Model: dc_CA(formulaEnv = ~A1 + Moist + Mag + Use + Manure, formulaTraits = ~SLA + Height + LDMC + Seedmass + Lifespan, response = dune_trait_env$comm[, -1], dataEnv = dune_trait_env$envir, dataTraits = dune_trait_env$traits, verbose = FALSE) 
#> Residualized predictor permutation
#> Permutation: free
#> Number of permutations: 999
#> 
#>          df ChiSquare     R2      F Pr(>F)  
#> dcCA      5   0.44542 0.3367 2.2335  0.012 *
#> Residual 22   0.87749                       
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> $eigenvalues
#>      dcCA1      dcCA2      dcCA3      dcCA4      dcCA5 
#> 0.23680387 0.10903220 0.05933626 0.03791909 0.00232876 
#> 
# anova_species can be used for manual forward selection of
# trait variables, as done for environmental variables in the demo 
# dune_FS_dcCA.r, based on the first eigenvalue and its significance
# and adding the first axis of the constrained species scores from mod to 
# the Condition of a new mod.
(eig1_and_pval <- c(eig = a_species$eig[1], pval = a_species$table$`Pr(>F)`[1]))
#> eig.dcCA1      pval 
#> 0.2368039 0.0120000 
a_species$eig
#>      dcCA1      dcCA2      dcCA3      dcCA4      dcCA5 
#> 0.23680387 0.10903220 0.05933626 0.03791909 0.00232876 
anova_sites(mod)
#> $table
#> Community-level permutation test using dc-CA
#> Model: dc_CA(formulaEnv = ~A1 + Moist + Mag + Use + Manure, formulaTraits = ~SLA + Height + LDMC + Seedmass + Lifespan, response = dune_trait_env$comm[, -1], dataEnv = dune_trait_env$envir, dataTraits = dune_trait_env$traits, verbose = FALSE) 
#> Residualized predictor permutation
#> Permutation: free
#> Number of permutations: 999
#> 
#>          df ChiSquare      R2      F Pr(>F)    
#> dcCA      7   0.44542 0.65733 3.2885  0.001 ***
#> Residual 12   0.23220                          
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> $eigenvalues
#>      dcCA1      dcCA2      dcCA3      dcCA4      dcCA5 
#> 0.23680387 0.10903220 0.05933626 0.03791909 0.00232876 
#>