RLQ analysis

RLQ analysis performs a double inertia analysis of two arrays (R and Q) with a link expressed by a contingency table (L). The rows of L correspond to the rows of R and the columns of L correspond to the rows of Q.

Usage

rlq(dudiR, dudiL, dudiQ, scannf = TRUE, nf = 2)
# S3 method for rlq
print(x, ...)
# S3 method for rlq
plot(x, xax = 1, yax = 2, ...)
# S3 method for rlq
summary(object, ...)
# S3 method for rlq
randtest(xtest,nrepet = 999, modeltype = 6,...)

Arguments

dudiR: a duality diagram providing from one of the functions dudi.hillsmith, dudi.pca, ...
dudiL: a duality diagram of the function dudi.coa
dudiQ: a duality diagram providing from one of the functions dudi.hillsmith, dudi.pca, ...
scannf: a logical value indicating whether the eigenvalues bar plot should be displayed
nf: if scannf FALSE, an integer indicating the number of kept axes
x: an rlq object
xax: the column number for the x-axis
yax: the column number for the y-axis
object: an rlq object
xtest: an rlq object
nrepet: the number of permutations
modeltype: the model used to permute data(2: permute rows of R, 4: permute rows of Q, 5: permute both, 6: sequential approach, see ter Braak et al. 2012)
...: further arguments passed to or from other methods

Value

Returns a list of class 'dudi', sub-class 'rlq' containing:

call: call
rank: rank
nf: a numeric value indicating the number of kept axes
RV: a numeric value, the RV coefficient
eig: a numeric vector with all the eigenvalues
lw: a numeric vector with the rows weigths (crossed array)
cw: a numeric vector with the columns weigths (crossed array)
tab: a crossed array (CA)
li: R col = CA row: coordinates
l1: R col = CA row: normed scores
co: Q col = CA column: coordinates
c1: Q col = CA column: normed scores
lR: the row coordinates (R)
mR: the normed row scores (R)
lQ: the row coordinates (Q)
mQ: the normed row scores (Q)
aR: the axis onto co-inertia axis (R)
aQ: the axis onto co-inertia axis (Q)

References

Doledec, S., Chessel, D., ter Braak, C.J.F. and Champely, S. (1996) Matching species traits to environmental variables: a new three-table ordination method. Environmental and Ecological Statistics, 3, 143--166.

Dray, S., Pettorelli, N., Chessel, D. (2002) Matching data sets from two different spatial samplings. Journal of Vegetation Science, 13, 867--874.

Dray, S. and Legendre, P. (2008) Testing the species traits-environment relationships: the fourth-corner problem revisited. Ecology, 89, 3400--3412.

ter Braak, C., Cormont, A., Dray, S. (2012) Improved testing of species traits-environment relationships in the fourth corner problem. Ecology, 93, 1525--1526.

Author

Stéphane Dray stephane.dray@univ-lyon1.fr

WARNING

IMPORTANT : row weights for dudiR and dudiQ must be taken from dudiL.

Note

A testing procedure based on the total coinertia of the RLQ analysis is available by the function randtest.rlq. The function allows to deal with various analyses for tables R and Q. Means and variances are recomputed for each permutation (PCA); for MCA, tables are recentred and column weights are recomputed.The case of decentred PCA (PCA where centers are entered by the user) for R or Q is not yet implemented. If you want to use the testing procedure for this case, you must firstly center the table and then perform a non-centered PCA on the modified table.

Examples

   data(aviurba)
   coa1 <- dudi.coa(aviurba$fau, scannf = FALSE, nf = 2)
   dudimil <- dudi.hillsmith(aviurba$mil, scannf = FALSE, nf = 2, row.w = coa1$lw)
   duditrait <- dudi.hillsmith(aviurba$traits, scannf = FALSE, nf = 2, row.w = coa1$cw)
   rlq1 <- rlq(dudimil, coa1, duditrait, scannf = FALSE, nf = 2)
   plot(rlq1)
#> Error in s.label(dfxy = rlq1$lR, xax = 1, yax = 2, plot = FALSE, storeData = TRUE,     pos = -3, psub = list(text = "R row scores"), plabels = list(        cex = 1.25)): non convenient selection for dfxy (can not be converted to dataframe)
   summary(rlq1)
#> RLQ analysis
#> 
#> Class: rlq dudi
#> Call: rlq(dudiR = dudimil, dudiL = coa1, dudiQ = duditrait, scannf = FALSE, 
#>     nf = 2)
#> 
#> Total inertia: 0.7278
#> 
#> Eigenvalues:
#>      Ax1      Ax2      Ax3      Ax4      Ax5 
#> 0.478283 0.141851 0.074261 0.023929 0.005514 
#> 
#> Projected inertia (%):
#>     Ax1     Ax2     Ax3     Ax4     Ax5 
#> 65.7131 19.4894 10.2031  3.2877  0.7576 
#> 
#> Cumulative projected inertia (%):
#>     Ax1   Ax1:2   Ax1:3   Ax1:4   Ax1:5 
#>   65.71   85.20   95.41   98.69   99.45 
#> 
#> (Only 5 dimensions (out of 8) are shown)
#> 
#> 
#> Eigenvalues decomposition:
#>         eig     covar      sdR      sdQ      corr
#> 1 0.4782826 0.6915798 1.558312 1.158357 0.3831293
#> 2 0.1418508 0.3766308 1.308050 1.219367 0.2361331
#> 
#> Inertia & coinertia R (dudimil):
#>     inertia      max     ratio
#> 1  2.428337 2.996911 0.8102800
#> 12 4.139332 5.345110 0.7744148
#> 
#> Inertia & coinertia Q (duditrait):
#>     inertia      max     ratio
#> 1  1.341791 2.603139 0.5154512
#> 12 2.828648 4.202981 0.6730098
#> 
#> Correlation L (coa1):
#>        corr       max     ratio
#> 1 0.3831293 0.6435487 0.5953384
#> 2 0.2361331 0.5220054 0.4523576
   randtest(rlq1)
#> Error in eval(expr, p): object 'dudimil' not found
   fourthcorner.rlq(rlq1,type="Q.axes")
#> Error in eval(expr, p): object 'dudimil' not found
   fourthcorner.rlq(rlq1,type="R.axes")
#> Error in eval(expr, p): object 'dudimil' not found