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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