[R-sig-eco] Correct Interpretation of summary(rlq) in ADE4

[Andrew Park]

Dear ListServers,
The following may well be a trivial problem, but I am having some problems with an rlq analysis.  The summary table comparing the inertia of the separate analyses appears to produce some eigenvalues for the second and subsequent axes that are larger than those of the unconstrained PCAs.  That is, if you want to work out the variance explained by Axis 2 of rlq to the Environment, the data below give you the following:
 Inertia & coinertia R (Env.pca):    inertia      max     ratio1  2.224835 3.384093 0.657439212 4.970223 5.926633 0.8386251
The row "12" is the cumulative inertia of axis 1 and 2.  Now 4.970223 - 2.224835 = 2.745388; and 5.926633 - 3.384093 = 2.54254; So apparently axis 2 of the rql explains more of the residual variation than axis 2 of the unconstrained environment PCA.  I'll admit that I do not understand all the mathematical details of the method, but I was under the impression that because RLQ maximizes the trait-environment covariation as constrained by species scores, that there was no way for an RLQ axis to explain more variance than the unconstrained environmental PCA?  Or am I completely misunderstanding what is going on in this table?  
Here is the data structure and the analysis code:
> dim(Spec2)[1]  37 151
> dim(Base2)[1] 37 10
> dim(T3)[1] 151  35
Coa.sp <- dudi.coa(Spec2, scannf = FALSE, nf=2)
# Correspondence analysis of species
Env.pca <- dudi.hillsmith(Base2, scannf = FALSE, row.w=Coa.sp$lw, nf=2)
# PCA of Trait Data.
Trt.pca <- dudi.pca(T3, scannf = FALSE, row.w=Coa.sp$cw, nf=2)
# DO PCA ON THE IMPUTED DATA FRAME 
# GENERATED BY IMPUTE in missMDA 
#(Takes account of NAs)
Tr.rlq <- rlq(Env.pca, Coa.sp, Trt.pca,scannf = F,nf=2)
# BASIC RLQ 
This yields the following results
Total inertia: 2.054
Eigenvalues:    Ax1     Ax2     Ax3     Ax4     Ax5 1.54256 0.32543 0.06390 0.05465 0.02599 
Projected inertia (%):    Ax1     Ax2     Ax3     Ax4     Ax5  75.105  15.845   3.111   2.661   1.265 
Eigenvalues decomposition:        eig     covar      sdR      sdQ      corr1 1.5425622 1.2419993 1.491588 2.408579 0.34570972 0.3254322 0.5704667 1.656921 1.548147 0.2223906
Inertia & coinertia R (Env.pca):    inertia      max     ratio1  2.224835 3.384093 0.657439212 4.970223 5.926633 0.8386251
Inertia & coinertia Q (Env.pca):    inertia      max     ratio1  5.801251 6.315691 0.918545712 8.198008 9.597220 0.8542066
Correlation L (Coa.sp):       corr       max     ratio1 0.3457097 0.7470330 0.46277702 0.2223906 0.6212101 0.3579958

In the event that I increase nf. to 8, I get the partial results shown below. The last two columns are the cumulative variation for axis N minus cumulative variation of axis N-1, and therefore should represent the unique contribution of those axes right?  But over half the RQL axes are larger than the unconstrained Environment axes.  Not only that but he RQL eventually explains 100% of the environmental variation:
I've looked at both Dray et al's (2014)  paper and the accompanying tutorial, but they do not hint at this problem of interpretation.
Any help appreciated,
Sincerely
Andy Park

“You may never know what results come of your action, but if you do nothing there will be no result.”  Gandhi
 		 	   		  
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