Kendall coefficient of concordance

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

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 p.adjust and in addition "sidak" (see Details). The Bonferroni correction is overly conservative; it is not recommended. It is included to allow comparisons with the other methods.

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 = \(3, 4, 5, \dots\) 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 mult. Shown only if there are more than one group.
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 mult. Shown only if there are more than one group.
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.

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.

Author

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

Examples

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)
#> $Concordance_analysis
#>               Group.1
#> W          0.44160305
#> F          2.37252221
#> Prob.F     0.04403791
#> Chi2      15.89770992
#> Prob.perm  0.12000000
#> 
#> attr(,"class")
#> [1] "kendall.global"
kendall.post(mite.small, mult="holm", nperm=49)
#> $A_posteriori_tests
#>                     TVEL      ONOV      SUCT   Trhypch1
#> Spearman.mean  0.3265678 0.3965503 0.4570402 -0.1681251
#> W.per.species  0.4949258 0.5474127 0.5927802  0.1239061
#> Prob           0.0800000 0.0200000 0.0200000  0.7200000
#> Corrected prob 0.1600000 0.0800000 0.0800000  0.7200000
#> 
#> $Correction.type
#> [1] "holm"
#> 
#> attr(,"class")
#> [1] "kendall.post"

# 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)
#> $Concordance_analysis
#>                          Group.1      Group.2
#> W                   3.097870e-01 2.911888e-01
#> F                   1.032305e+01 4.108130e+00
#> Prob.F              1.177138e-85 4.676566e-22
#> Corrected prob.F    2.354275e-85 4.676566e-22
#> Chi2                5.130073e+02 2.210123e+02
#> Prob.perm           2.000000e-02 2.000000e-02
#> Corrected prob.perm 4.000000e-02 4.000000e-02
#> 
#> $Correction.type
#> [1] "holm"
#> 
#> attr(,"class")
#> [1] "kendall.global"
kendall.post(mite.hel, group=group, mult="holm", nperm=49)
#> $A_posteriori_tests_Group
#> $A_posteriori_tests_Group[[1]]
#>                   Brachy      PHTH     RARD      SSTR   Protopl      MEGR
#> Spearman.mean  0.1851177 0.4258111 0.359058 0.2505486 0.1802160 0.2833298
#> W.per.species  0.2190711 0.4497357 0.385764 0.2817757 0.2143736 0.3131911
#> Prob           0.0200000 0.0200000 0.020000 0.0200000 0.0400000 0.0200000
#> Corrected prob 0.7000000 0.7000000 0.700000 0.7000000 0.7000000 0.7000000
#>                      MPRO      HMIN     HMIN2      NPRA      TVEL      ONOV
#> Spearman.mean  0.09248024 0.2444656 0.4138494 0.1263751 0.4177343 0.3301159
#> W.per.species  0.13029357 0.2759462 0.4382723 0.1627761 0.4419954 0.3580278
#> Prob           0.14000000 0.0200000 0.0200000 0.0400000 0.0200000 0.0200000
#> Corrected prob 0.70000000 0.7000000 0.7000000 0.7000000 0.7000000 0.7000000
#>                     SUCT Oribatl1      PWIL  Galumna1  Stgncrs2      HRUF
#> Spearman.mean  0.2185421 0.421216 0.2574779 0.4180699 0.3623428 0.1250230
#> W.per.species  0.2511028 0.445332 0.2884163 0.4423170 0.3889118 0.1614804
#> Prob           0.0200000 0.020000 0.0200000 0.0200000 0.0200000 0.0600000
#> Corrected prob 0.7000000 0.700000 0.7000000 0.7000000 0.7000000 0.7000000
#>                     PPEL      SLAT      FSET  Lepidzts  Eupelops  Miniglmn
#> Spearman.mean  0.2188216 0.3016159 0.4217606 0.2577037 0.1108022 0.2301430
#> W.per.species  0.2513707 0.3307153 0.4458539 0.2886327 0.1478521 0.2622203
#> Prob           0.0200000 0.0200000 0.0200000 0.0200000 0.0600000 0.0200000
#> Corrected prob 0.7000000 0.7000000 0.7000000 0.7000000 0.7000000 0.7000000
#> 
#> $A_posteriori_tests_Group[[2]]
#>                     HPAV      TVIE      LCIL  Ceratoz1  Trhypch1      NCOR
#> Spearman.mean  0.1222579 0.2712078 0.1906408 0.1375601 0.1342409 0.3342345
#> W.per.species  0.2020527 0.3374616 0.2642189 0.2159637 0.2129463 0.3947586
#> Prob           0.0400000 0.0200000 0.0200000 0.0200000 0.0400000 0.0200000
#> Corrected prob 0.7000000 0.7000000 0.7000000 0.7000000 0.7000000 0.7000000
#>                     LRUG     PLAG2  Ceratoz3  Oppiminu  Trimalc2
#> Spearman.mean  0.3446561 0.1833099 0.3188922 0.1764232 0.2498877
#> W.per.species  0.4042328 0.2575544 0.3808111 0.2512938 0.3180797
#> Prob           0.0200000 0.0200000 0.0200000 0.0200000 0.0200000
#> Corrected prob 0.7000000 0.7000000 0.7000000 0.7000000 0.7000000
#> 
#> 
#> $Correction.type
#> [1] "holm"
#> 
#> attr(,"class")
#> [1] "kendall.post"

# NOTE: 'nperm' argument usually needs to be larger than 49.
# It was set to this low value for demonstration purposes.