# Kendall coefficient of concordance

### Description

Function `kendall.global`

computes and tests the coefficient of
concordance among several judges (variables, species) through a
permutation test.

Function `kendall.post`

carries out *a posteriori* tests
of the contributions of individual judges (variables, species) to
the overall concordance of their group through permutation tests.

If several groups of judges are identified in the data table,
coefficients of concordance (`kendall.global`

) or a posteriori
tests (`kendall.post`

) will be computed for each group
separately. Use in ecology: to identify significant species
associations.

### Usage

1 2 | ```
kendall.global(Y, group, nperm = 999, mult = "holm")
kendall.post(Y, group, nperm = 999, mult = "holm")
``` |

### Arguments

`Y` |
Data file (data frame or matrix) containing quantitative or semiquantitative data. Rows are objects and columns are judges (variables). In community ecology, that table is often a site-by-species table. |

`group` |
A vector defining how judges should be divided into groups. See example below. If groups are not explicitly defined, all judges in the data file will be considered as forming a single group. |

`nperm` |
Number of permutations to be performed. Default is 999. |

`mult` |
Correct P-values for multiple testing using the
alternatives described in |

### Details

`Y`

must contain quantitative data. They will be transformed to
ranks within each column before computation of the coefficient of
concordance.

The search for species associations described in Legendre (2005) proceeds in 3 steps:

(1) Correlation analysis of the species. A possible method is to
compute Ward's agglomerative clustering of a matrix of correlations
among the species. In detail: (1.1) compute a Pearson or Spearman
correlation matrix (`correl.matrix`

) among the species; (1.2)
turn it into a distance matrix: `mat.D = as.dist(1-correl.matrix)`

;
(1.3) carry out Ward's hierarchical
clustering of that matrix using `hclust`

:
`clust.ward = hclust(mat.D, "ward")`

; (1.4) plot the dendrogram:
`plot(clust.ward, hang=-1)`

; (1.5) cut the dendrogram in two
groups, retrieve the vector of species membership:
`group.2 = cutree(clust.ward, k=2)`

. (1.6) After steps 2 and 3 below,
you may
have to come back and try divisions of the species into k = *3, 4, 5, …*
groups.

(2) Compute global tests of significance of the 2 (or more) groups
using the function `kendall.global`

and the vector defining the
groups. Groups that are not globally significant must be refined or
abandoned.

(3) Compute a posteriori tests of the contribution of individual
species to the concordance of their group using the function
`kendall.post`

and the vector defining the groups. If some
species have negative values for "Spearman.mean", this means that
these species clearly do not belong to the group, hence that group
is too inclusive. Go back to (1.5) and cut the dendrogram more
finely. The left and right groups can be cut separately,
independently of the levels along the dendrogram; write your own
vector of group membership if `cutree`

does not produce the
desired groups.

The corrections used for multiple testing are applied to the list of
P-values (P); they take into account the number of tests (k) carried
out simultaneously (number of groups in `kendall.global`

, or
number of species in `kendall.post`

). The corrections are
performed using function `p.adjust`

; see that function
for the description of the correction methods. In addition, there is
Šidák correction which defined as
*P_{corr} = 1 -(1 - P)^k*.

### Value

A table containing the following information in rows. The columns
correspond to the groups of "judges" defined in vector "group". When
function `Kendall.post`

is used, there are as many tables as
the number of predefined groups.

`W ` |
Kendall's coefficient of concordance, W. |

`F ` |
F statistic. F = W*(m-1)/(1-W) where m is the number of judges. |

`Prob.F ` |
Probability associated with the F statistic, computed from the F distribution with nu1 = n-1-(2/m) and nu2 = nu1*(m-1); n is the number of objects. |

`Corrected prob.F ` |
Probabilities associated with F, corrected
using the method selected in parameter |

`Chi2 ` |
Friedman's chi-square statistic (Friedman 1937) used in the permutation test of W. |

`Prob.perm ` |
Permutational probabilities, uncorrected. |

`Corrected prob.perm ` |
Permutational probabilities corrected
using the method selected in parameter |

`Spearman.mean ` |
Mean of the Spearman correlations between the judge under test and all the other judges in the same group. |

`W.per.species ` |
Contribution of the judge under test to the overall concordance statistic for that group. |

### Author(s)

F. Guillaume Blanchet, University of Alberta, and Pierre Legendre, Université de Montréal

### References

Friedman, M. 1937. The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association 32: 675-701.

Kendall, M. G. and B. Babington Smith. 1939. The problem of m rankings. Annals of Mathematical Statistics 10: 275-287.

Legendre, P. 2005. Species associations: the Kendall coefficient of concordance revisited. Journal of Agricultural, Biological, and Environmental Statistics 10: 226-245.

Legendre, P. 2009. Coefficient of concordance. In: Encyclopedia of Research Design. SAGE Publications (in press).

Siegel, S. and N. J. Castellan, Jr. 1988. Nonparametric statistics for the behavioral sciences. 2nd edition. McGraw-Hill, New York.

### See Also

`cor`

, `friedman.test`

,
`hclust`

, `cutree`

, `kmeans`

,
`cascadeKM`

, `indval`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | ```
data(mite)
mite.hel <- decostand(mite, "hel")
# Reproduce the results shown in Table 2 of Legendre (2005), a single group
mite.small <- mite.hel[c(4,9,14,22,31,34,45,53,61,69),c(13:15,23)]
kendall.global(mite.small, nperm=49)
kendall.post(mite.small, mult="holm", nperm=49)
# Reproduce the results shown in Tables 3 and 4 of Legendre (2005), 2 groups
group <-c(1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,2,1,2,1,1,1,1,2,1,2,1,1,1,1,1,2,2,2,2,2)
kendall.global(mite.hel, group=group, nperm=49)
kendall.post(mite.hel, group=group, mult="holm", nperm=49)
# NOTE: 'nperm' argument usually needs to be larger than 49.
# It was set to this low value for demonstration purposes.
``` |