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, Rj ; j=1…N :
KC = sum{ (Rj–R)×(Rj–R) ; j=1…N } / { K×K×N×(N×N-1)/12 }
where R is the mean of the set { Rj ; 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{ Tj ; j=1…N } }
where Tj = is the sum over all ranks k in group j of ( (tk3)-tk )/12, and tk is the number of observations in the group with rank 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