## RLQ analysis

### Description

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 Q 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, ...)
```

### 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 `...` 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)

### 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.

### Author(s)

Stephane Dray dray@biomserv.univ-lyon1.fr

### 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.

`coinertia`

### 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)
summary(rlq1)
randtest.rlq(rlq1)
```

### Worked out examples

```
> ### Name: rlq
> ### Title: RLQ analysis
> ### Aliases: rlq print.rlq plot.rlq summary.rlq randtest.rlq
> ### Keywords: multivariate spatial
>
> ### ** 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)
>    summary(rlq1)

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:
inertia      max     ratio
1  2.428337 2.996911 0.8102800
12 4.139332 5.345110 0.7744148

Inertia & coinertia Q:
inertia      max     ratio
1  1.341791 2.603139 0.5154512
12 2.828648 4.202981 0.6730098

Correlation L:
corr       max     ratio
1 0.3831293 0.6435487 0.5953384
2 0.2361331 0.5220054 0.4523576
>    randtest.rlq(rlq1)
Monte-Carlo test
Call: randtest.rlq(xtest = rlq1)

Observation: 0.7278339

Based on 1000 replicates
Simulated p-value: 0.000999001
Alternative hypothesis: greater

Std.Obs  Expectation     Variance
12.218602634  0.195251670  0.001899898
```
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>
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>
```