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 class 'rlq'
print(x, ...)
# S3 method for class 'rlq'
plot(x, xax = 1, yax = 2, ...)
# S3 method for class 'rlq'
summary(object, ...)
# S3 method for class '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