>I have been trying to prepare data tables for RLQ analysis. A weighted COA
>of my R table (394 rows, 3 columns) using a file.fcpl produced from COA of
>the L table repeatedly fails. Introducing a dummy fourth column allows the
>weighted COA of R to proceed. So the problem appears to be related to
>column number in R. Why?
>We have got round the problem by weighting the R table in Excel and
>proceeding with an unweighted COA. Is this valid?
>The attached file contains more information on the data matrices.
It seems that analysing your table R with a weighted COA is a wrong option.
This table contain quantitative data (decimals and integers) and it should
be analyzed by PCA (normalized if the variance of the three variables are
too different) and weighted by file.fcpl as indicated. Did you try PCA on
This question is the occasion for reminding some general aspects of what is
called RLQ analysis in ADE-4 which were proposed in a former mail.
Let F be a faunistic table with n rows-samples and p columns-species
Let M be an environmental table with n rows-samples and v columns-variables
Let B be a table that contains some biological information on the species
and and p columns-species
alors il existe un tableau Q croisant les trois avec t lignes-traits et p
If M and B contains dummy variables the matrix Q = BFtM simply contains the
contingency table that crosses the modalities of the environmental
variables and the modalities of the biological traits.
This particular situation was addressed by P. Legendre and coauthors with
the so-called fourth-corner problem (Legendre, P., Galzin, R. &
Harmelin-Vivien, M.L. (1997) Relating behavior to habitat: Solutions to the
fourth-corner problem. Ecology, 78, 547-562). The authors further proposed
some permutation tests to check for the significance of the link between M
and B through F.
In ADE-4, RLQ analysis helps to explore the relationship between M and B
through F using a broader numerical context since M and B may contain
quantitative, dummy or fuzzy variables. Before performing a RLQ analysis,
each table should be processed as follows:
F must be processed by a correspondence analysis (COA)
M can be analysed by any type of one-table analysis (PCA, MCA, FCA) using
the row weight of F
B can be analysed by any type of one-table analysis (PCA, MCA, FCA) using
the column weight of F
RLQ analysis then incorporates the following sequence:
RLQ : Prepare -> to match the three previous analyses
RLQ : RLQ test - Fixed L -> to test the relationship between M and B
through F using random permutations. Note that F remains fixed. Another
option should be to fix M and B and permute randomly F (See Legendre et al.
RLQ : Diagonalize -> to analyse the fourth corner matrix
RLQ : Coinertia analysis -> to display the scores and various
The name RLQ is thus justified by using R for a table analysed in R-mode
and Q for a table analysed in Q-mode. L stands for Link (Link table).
All the general theory is available in Dolédec, S., Chessel, D., Ter Braak,
C.J.F. & Champely, S. (1996) Matching species traits to environmental
variables: a new three-table ordination method. Environmental and
Ecological Statistics : 3, 143-166. In this general theory the matching of
two MCAs through a COA (as used in the example of the paper) is considered
a particular case of RLQ thus meaning that this approach is far from being
All the very best,
Université Claude Bernard - Lyon 1
43 Bd du 11 novembre
Bat 401C - 2ème étage
F-69622 Villeurbanne CEDEX
Tel : +33 4 72 43 13 63
Fax : +33 4 72 43 11 41
ADE-4 package is available on the Internet
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