Permutation Test for weighted redundancy analysis

anova.wrda performs residual predictor permutation for weighted redundancy analysis (wRDA), which is robust against differences in the weights (ter Braak, 2022). The arguments of the function are similar to those of anova.cca, but more restricted.

Usage

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

Arguments

object: an object from dc_CA.
...: unused.
permutations: a list of control values for the permutations as returned by the function how, or the number of permutations required (default 999), or a permutation matrix where each row gives the permuted indices.
by: character "axis" which sets the test statistic to the first eigenvalue of the RDA model. Default: NULL which sets the test statistic to the weighted variance fitted by the predictors (=sum of all constrained eigenvalues). The default is quicker computationally 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 with two elements with names table and eigenvalues. The table is as from anova.cca and eigenvalues gives the wrda eigenvalues.

Details

The algorithm is based on published R-code for residual predictor permutation in weighted redundancy analysis (ter Braak, 2022), but using QR-decomposition instead of ad-hoc least-squares functions.

References

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$Sites
response <- dune_trait_env$comm[, -1]  # must delete "Sites"

w <- rep(1, 20) 
w[1:10] <- 8 
w[17:20] <- 0.5

object <- wrda(formula = response ~ A1 + Moist + Mag + Use + Condition(Manure),
               data = dune_trait_env$envir, 
               weights = w)
object # Proportions equal to those Canoco 5.15
#> 
#> Call: wrda(formula = response ~ A1 + Moist + Mag + Use + Condition(Manure),
#> data = dune_trait_env$envir, weights = w)
#> 
#>               Inertia Proportion Rank
#> Total         65.7007     1.0000     
#> Conditional    7.3839     0.1124    2
#> Constrained   36.4952     0.5555    6
#> Unconstrained 21.8217     0.3321   19
#> 
#> Inertia is weighted variance
#> 
#> Eigenvalues for constrained axes:
#>  wRDA1  wRDA2  wRDA3  wRDA4  wRDA5  wRDA6 
#> 18.133  8.233  4.047  3.155  2.238  0.689 
#> 
#> Eigenvalues for unconstrained axes:
#> wPCA1 wPCA2 wPCA3 wPCA4 wPCA5 wPCA6 wPCA7 wPCA8 
#>    NA    NA    NA    NA    NA    NA    NA    NA 
#> (Showing 8 of 19 unconstrained eigenvalues)
#> 
#> mean, sd, VIF and canonical coefficients with their optimistic [!] t-values:
#>           Avg    SDS     VIF   Regr1   tval1
#> Manure 2.4659 1.2056 10.5656 -0.2010 -0.1344
#> A1     4.1466 1.3786  1.4841  0.1013  0.1807
#> Moist  2.1761 1.4762  1.4883 -4.1077 -7.3195
#> MagBF  0.1932 0.3948  6.8216  0.7873  0.6553
#> MagHF  0.4545 0.4979  5.8054  1.8392  1.6594
#> MagNM  0.0455 0.2083  4.8878  0.1327  0.1304
#> Use    1.9205 0.7107  2.6787  0.2101  0.2791
#> 

mod_scores <- scores(object, display = "all")
scores(object, which_cor = c("A1", "X_lot"), display = "cor")
#>            wRDA1      wRDA2
#> A1    -0.2365172 -0.3510007
#> X_lot -0.4987793  0.4250562
#> attr(,"meaning")
#> [1] "inter set correlation, correlation between environmental variables and the sites scores (CWMs)"
anova(object)
#> $table
#> Permutation test for weighted reduncancy analysis
#> Model: wrda(formula = response ~ A1 + Moist + Mag + Use + Condition(Manure), data = dune_trait_env$envir, weights = w) 
#> Residualized predictor permutation
#> Permutation: free
#> Number of permutations: 999
#> 
#>          df Variance      R2      F Pr(>F)   
#> wRDA      6   36.495 0.62581 3.3449  0.006 **
#> Residual 12   21.822                         
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> $eigenvalues
#>      wRDA1      wRDA2      wRDA3      wRDA4      wRDA5      wRDA6 
#> 18.1328543  8.2332883  4.0467247  3.1546579  2.2383295  0.6893052 
#>