[R-sig-eco] Anderson and Willis 2003, CAP squared canonical correlations of delta^2

[Kari Lintulaakso]

Dear list,

I'm trying to follow the CAP analysis described in Anderson and Willis
2003: Canonical Analysis of Principal Coordinates: A Useful Method of
Constrained Ordination for Ecology
For this I'm using CAPdiscrim (instead of capscale) as it seems to
follow the original description.
I'm using a data set with n different biomes. Each biome has several
sites and each site has species counts listed.

I use the dune data set to describe my questions which are in the comments.

require(BiodiversityR)
require(vegan)
data(dune)
data(dune.env)
# Transform variables
dune.trs <- decostand(dune,"log")

# Calculate dissimilarities between each pair of observations, Bray-Curtis
dune.bray <- vegdist(dune.trs, method = "bray")

# Canonical Analysis of Principal Coordinates (CAP):
# This is done for Management which acts like class data
dune.cap <- CAPdiscrim(dune.bray ~ Management, dune.env
,dist="bray",axes=4,m=0,permutations=9)

# In Anderson and Willis 2003, page 518:
# "... The canonical analysis (CAP) yielded two canonical axes,
# with squared canonical correlations of delta1^2 = 0.610 and delta1^2
= 0.478..."
#
# It seems that those values come from Eigenvalues (Correlations) of
0.78101 and 0.69142
http://www.stat.auckland.ac.nz/~mja/prog/CAP_UserNotes.pdf
# QUESTION 1: How do I get similar values using CAPdiscrim?
# The only Eigenvalue related value I find is dune.cap$tot

# And later in anderson and Willis 2003:
# "The two canonical test statistics were highly significant (P =
0.0001 for both tests, using 9999 permutations)"
# QUESTION 2: I did only 9 permutations in my example, but do I get
similar results by using  dune.cap$manova => Pr(>F) 0.002224 ?

Instead of CAPdiscrim should I use capscale in some form to achieve
similar results as they did in A&W 2003?
I'm still out of my comfort area here, so any help will be valuable.
Cheers,
-Kari

Kari Lintulaakso, M.Sc.(Biosciences)

Doctoral student
Paleontology and Paleoecology
Department of Geosciences and Geography
University of Helsinki