Selecting Statistics  


Statistical Test

Kendall's tau a, tau b or tau c,

Or Goodman And Kruskal's Gamma,

Or Kim's d

Statistical Measure

For N greater than 10 refer the critical ratio of S to a table of the unit normal curve; for N less than or equal to 10, refer these statistics to a table of the critical values of S.

Notes

These statistics differ with respect to how they treat pairs of cases that fall in the same category on one or both of the variables. Except in extreme cases (i.e., where any of the statistics equals 0 or 1) the absolute value of Gamma will be the highest of the five statistics, tau a will be the smallest, and tau b, tau c, and Kim's d will be intermediate. This ordering is because Gamma ignores all ties (when present in the data - as is usually the case), whereas the other four statistics penalize for ties in the sense of reducing the absolute value of the statistic obtained. Unlike tau b or Kim's d, tau c can attain plus or minus 1 even if the two variables do not have the same number of categories. If there are no ties on either variable the five measures are identical. See Goodman and Kruskal (1954), Kendall (1970), Kendall and Stuart (1961), Stuart (1953), and Kim (1971).

Kendall's tau a: Kendall, M. G. Rank Correlation Methods. Fourth edition. London: Griffin, 1970. p. 5

Standard Error of S assuming no ties: Kendall, M. G. Rank Correlation Methods. Fourth edition. London: Griffin, 1970. p. 51

Table of critical values of S assuming no ties: Kendall, M. G. Rank Correlation Methods. Fourth edition. London: Griffin, 1970. p. 173

Kendall's tau b: Kendall, M. G. Rank Correlation Methods. Fourth edition. London: Griffin, 1970. p. 35

Kendall's tau c: Kendall, M. G. Rank Correlation Methods. Fourth edition. London: Griffin, 1970. p. 47

Goodman and Kruskal's Gamma: Hays, W. L. Statistics for the Social Sciences. Second edition. New York: Holt, Rinehart, and Winston, 1973. p. 800

Kim's d: Kim, J. Predictive measures of ordinal association. American Journal of Sociology 76 (1971): 891-907. p. 899

Kendall's tau a:

SPSS: NONPAR CORR

Kendall's tau b:

SPSS: CROSSTABS

SAS: FREQ, CORR

Kendall's tau c:

SPSS: CROSSTABS

SAS: FREQ (refered to as Stuart's tau c)

Goodman & Kruskal Gamma:

SPSS: CROSSTABS

SAS: FREQ