Kendall's W
Encyclopedia
Kendall's W is a non-parametric statistic. It is a normalization of the statistic of the Friedman test
, and can be used for assessing agreement among raters. Kendall's W ranges from 0 (no agreement) to 1 (complete agreement).
Suppose, for instance, that a number of people have been asked to rank a list of political concerns, from most important to least important. Kendall's W can be calculated from these data. If the test statistic W is 1, then all the survey respondents have been unanimous, and each respondent has assigned the same order to the list of concerns. If W is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of W indicate a greater or lesser degree of unanimity among the various responses.
While tests using the standard Pearson correlation coefficient assume normally distributed values and compare two sequences of outcomes at a time, Kendall's W makes no assumptions regarding the nature of the probability distribution
and can handle any number of distinct outcomes.
W is linearly related to the mean value of the Spearman's rank correlation coefficient
s between all pairs of the rankings over which it is calculated.
and the mean value of these total ranks is
The sum of squared deviations, S, is defined as
and then Kendall's W is defined as
If the test statistic W is 1, then all the judges or survey respondents have been unanimous, and each judge or respondent has assigned the same order to the list of objects or concerns. If W is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of W indicate a greater or lesser degree of unanimity among the various judges or respondents.
Legendre discusses a variant of the W statistic which accommodates ties in the rankings and also describes methods of making significance tests based on W.
The effect of ties is to reduce the value of W; however, this effect is small unless there are a large number of ties. To correct for ties, assign ranks to tied values as above and compute the correction factors
where ti is the number of tied ranks in the ith group of tied ranks, (where a group is a set of values having constant (tied) rank,) and gj is the number of groups of ties in the set of ranks (ranging from 1 to n) for judge j. Thus, Tj is the correction factor required for the set of ranks for judge j, i.e. the jth set of ranks. Note that if there are no tied ranks for judge j, Tj equals 0.
With the correction for ties, the formula for W becomes
where Ri is the sum of the ranks for object i, and
is the sum of the values of Tj over all m sets of ranks .
Friedman test
The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row together, then...
, and can be used for assessing agreement among raters. Kendall's W ranges from 0 (no agreement) to 1 (complete agreement).
Suppose, for instance, that a number of people have been asked to rank a list of political concerns, from most important to least important. Kendall's W can be calculated from these data. If the test statistic W is 1, then all the survey respondents have been unanimous, and each respondent has assigned the same order to the list of concerns. If W is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of W indicate a greater or lesser degree of unanimity among the various responses.
While tests using the standard Pearson correlation coefficient assume normally distributed values and compare two sequences of outcomes at a time, Kendall's W makes no assumptions regarding the nature of the probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
and can handle any number of distinct outcomes.
W is linearly related to the mean value of the Spearman's rank correlation coefficient
Spearman's rank correlation coefficient
In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho or as r_s, is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can...
s between all pairs of the rankings over which it is calculated.
Definition
Suppose that object i is given the rank ri,j by judge number j, where there are in total n objects and m judges. Then the total rank given to object i isand the mean value of these total ranks is
The sum of squared deviations, S, is defined as
and then Kendall's W is defined as
If the test statistic W is 1, then all the judges or survey respondents have been unanimous, and each judge or respondent has assigned the same order to the list of objects or concerns. If W is 0, then there is no overall trend of agreement among the respondents, and their responses may be regarded as essentially random. Intermediate values of W indicate a greater or lesser degree of unanimity among the various judges or respondents.
Legendre discusses a variant of the W statistic which accommodates ties in the rankings and also describes methods of making significance tests based on W.
Correction for ties
When tied values occur, they are each given the average of the ranks that would have been given had no ties occurred. For example, the data set {80,76,34,80,73,80} has values of 80 tied for 4th, 5th, and 6th place; since the mean of {4,5,6} = 5, ranks would be assigned to the raw data values as follows: {5,3,1,5,2,5}.The effect of ties is to reduce the value of W; however, this effect is small unless there are a large number of ties. To correct for ties, assign ranks to tied values as above and compute the correction factors
where ti is the number of tied ranks in the ith group of tied ranks, (where a group is a set of values having constant (tied) rank,) and gj is the number of groups of ties in the set of ranks (ranging from 1 to n) for judge j. Thus, Tj is the correction factor required for the set of ranks for judge j, i.e. the jth set of ranks. Note that if there are no tied ranks for judge j, Tj equals 0.
With the correction for ties, the formula for W becomes
where Ri is the sum of the ranks for object i, and
is the sum of the values of Tj over all m sets of ranks .