Constrained Analysis of Principal Coordinates

Question

I used a Constrained Analysis of Principal Coordinates (CAP) to evaluated if my independent variables could explain the ordination of assemblage composition based on species abundances. However, the output only gave me the proportion explained (60.5%) and the F (2.3023) and P (0.003) values for the whole model. But I don't know how to get the deviance explained by each variable and its p-value. I hope someone can help me.

vare.cap <- capscale(ponds ~ PD + PS + COV100 + COV500,
                      dist="bray")
vare.cap
summary(vare.cap)
anova(vare.cap)
anova(vare.cap, by = "axis")

Results:

Partitioning of squared Bray distance:
              Inertia Proportion
Total           2.714     1.0000
Constrained     1.643     0.6055
Unconstrained   1.071     0.3945

 Df          Variance   F    Pr(>F)   
Model     4   1.6435 2.3023  0.003 **
Residual  6   1.0708

Answer 1

If the variance explained is OK, then this is given by anova(model, by = "margin"). This tests marginal effects, hence this is a test of the amount of additional variance explained by adding that variable to a model that already contains all the other variables.

library("vegan")
data(varespec, varechem)
vare.cap <- capscale(varespec ~ N + P + K + Condition(Al), data = varechem,
                     dist="bray")

anova(vare.cap, by = "margin")

This gives us:

> anova(vare.cap, by = "margin")
Permutation test for capscale under reduced model
Marginal effects of terms
Permutation: free
Number of permutations: 999

Model: capscale(formula = varespec ~ N + P + K + Condition(Al), data = varechem, distance = "bray")
         Df Variance      F Pr(>F)  
N         1  0.27719 1.8616  0.093 .
P         1  0.20764 1.3945  0.191  
K         1  0.22249 1.4943  0.151  
Residual 19  2.82904                
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

If you want the marginal variance explained, then you need to divide those values by with(vare.cap, tot.chi - CA$imaginary.chi) if you have imaginary eigenvalues or vby with(vare.cap, tot.chi) if you don't. For example

> set.seed(1)
> marg <- anova(vare.cap, by = "margin")
> marg
Permutation test for capscale under reduced model
Marginal effects of terms
Permutation: free
Number of permutations: 999

Model: capscale(formula = varespec ~ N + P + K + Condition(Al), data = varechem, distance = "bray")
         Df Variance      F Pr(>F)  
N         1  0.27719 1.8616  0.065 .
P         1  0.20764 1.3945  0.201  
K         1  0.22249 1.4943  0.150  
Residual 19  2.82904                
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> marg$Variance / with(vare.cap, tot.chi - CA$imaginary.chi)
[1] 0.05770665 0.04322702 0.04631963 0.58896598

That last line contains the proportions of variance explained out of the total variance. If you want to ignore (or don't have any) imaginary eigenvalues, then you can just do

> marg$Variance / with(vare.cap, tot.chi)
[1] 0.06099498 0.04569025 0.04895909 0.62252739