Calculates Kendall’s Coefficient of Concordance; synonym `CONCORD`

(S.J. Welham, N.M. Maclaren & H.R. Simpson).

### Options

`PRINT` = string tokens |
Output required (`test` , `ranks` ): `test` produces the relevant test statistics, `ranks` produces the vector of mean ranks and the ranks for each sample; default `test` |
---|---|

`GROUPS` = factor |
Defines the variable stored in each unit if only one variate is specified by `DATA` |

`STATISTIC` = scalar |
Scalar to save the coefficient of concordance |

`CHISQUARE` = scalar |
Scalar to save the chi-square approximation to the coefficient (calculated only if the sample size is at least 8) |

`MEANRANKS` = variate |
Variate to save the mean ranks for individuals over variables |

`DF` = scalar |
Scalar to save the degrees of freedom for `CHISQUARE` |

### Parameters

`DATA` = variates |
List of variables to be compared, or a single variate containing the data for all the variables (the `GROUPS` option must then be set to indicate the variable recorded in each unit belongs) |
---|---|

`RANKS` = variates |
Save the ranks of the variables |

### Description

Kendall’s Coefficient of Concordance is a measure of association between *K* rankings on *N* individuals, i.e. a set of *N* individuals are ranked on each of *K* variables in turn, and these rankings are to be compared. The variables can be stored in separate variates and the `DATA`

parameter set to list them all. Alternatively, all the data can be stored in a single variate, and the `GROUPS`

option set to a factor to indicate which variable is recorded in each unit of the variate. (`KCONCORDANCE`

then assumes that the individuals are recorded in the same order for each variable.)

Concord calculates the chi-square approximation to the statistic if the sample sizes are large enough (i.e. 8 or more). Otherwise, for 2<*K*<21 and 2<*N*<8, `KCONCORDANCE`

looks up the probability from a stored table. The results of these calculations can be printed using the `test`

setting of `PRINT`

, or saved using the options `STATISTIC`

(for the coefficient), `CHISQUARE`

(for the chi-square statistic) and `DF`

(degrees of freedom). The `ranks`

setting of `PRINT`

causes the vector of mean ranks (over all variates) and the ranks for each variate individually to be displayed, and these can be saved using the `MEANRANKS`

option and the `RANKS`

parameter.

Options: `PRINT`

, `GROUPS`

, `STATISTIC`

, `CHISQUARE`

, `MEANRANKS`

, `DF`

.

Parameters: `DATA`

, `RANKS`

.

### Method

Kendall’s Coefficient of Concordance, *KC*, is built up from the sum of ranks over the *K* variables for each individual, *R _{j}* ;

*j*=1…

*N*:

*KC* = sum{ (*R _{j}*–

*R*)×(

*R*–

_{j}*R*) ;

*j*=1…

*N*} / {

*K*×

*K*×

*N*×(

*N*×

*N*-1)/12 }

where *R* is the mean of the set { *R _{j}* ;

*j*=1…

*N*}.

If ties are present in the data, then the denominator of *KC* must be modified to avoid bias in the statistic. The adjusted denominator is:

{ *K*×*N*×(*N*×*N*-1)/12 – *K*×sum{ *T _{j}* ;

*j*=1…

*N*} }

where *T _{j}* = is the sum over all ranks

*k*in group

*j*of ( (

*t*

_{k}^{3})-

*t*)/12, and

_{k}*t*is the number of observations in the group with rank

_{k}*k*. (See e.g. Siegel 1956, pages 229-238.)

The chi-square approximation for this statistic (valid only when *N*≥8) is *K*×(*N*-1)×*KC* with *N*-1 degrees of freedom.

### Action with `RESTRICT`

If any of the variates in `DATA`

is restricted, the statistic is calculated only for the units not excluded by the restriction.

### Reference

Siegel S. (1956). *Nonparametric Statistics for the Behavioural Sciences*. McGraw-Hill, New York.

### See also

Procedures: `CMHTEST`

, `FCORRELATION`

, `KTAU`

, `LCONCORDANCE`

, `SPEARMAN`

.

Commands for Basic and nonparametric statistics.

### Example

CAPTION 'KCONCORDANCE example',!t(\ 'Data from Siegel (1956), Nonparametric Statistics, p. 234.',\ 'Ten objects are ranked on three variables: X, Y, Z.'); STYLE=meta,plain VARIATE [VALUES=1,4.5,2,4.5,3,7.5,6,9,7.5,10] X & [VALUES=2.5,1,2.5,4.5,4.5,8,9,6.5,10,6.5] Y & [VALUES=2,1,4.5,4.5,4.5,4.5,8,8,8,10] Z PRINT X,Y,Z; DECIMALS=1 CAPTION !T('Calculate the coefficient, print out test results,',\ 'rank vectors, and save all the results.') KCONCORDANCE [PRINT=test,ranks; STATISTIC=W; CHISQUARE=Chi2;\ MEANRANKS=MeanRanks] X,Y,Z; RANKS=RX,RY,RZ PRINT W,Chi2 & MeanRanks,RX,RY,RZ; DECIMALS=2