Overview When categorical measures have an inherent order (ex., strongly agree to strongly disagree; high, medium, low), additional information is available beyond that possessed by merely nominal variables. A variety of effect size measures have been proposed for such ordinal data, including gamma, Kendall's tau-b, Kendall's tau-c, and Somers' d.

Contents Key concepts and terms Gamma Kendall's tau-b Kendall's tau-c Somers' d Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• Gamma, also called Goodman and Kruskal's gamma, is a symmetric measure which varies from +1 to -1, based on the difference between concordant pairs (P) and discordant pairs (Q). The concept of pairs is discussed separately in the section on association. That is, gamma is computed as (P - Q)/(P + Q).

  City Size/Arenas	Small	Medium	Large
  Weak Mayor	a = 10	b = 5	c = 2
  Strong Mayor	d = 10	e = 15	f = 20

  P = a(e+f) + bf = 10(15+20) + 5*20 = 450
  Q = c(d+e) + bd = 2(10+15) + 5*10 = 100
  gamma = (P - Q)/(P + Q) = (450-100)/(450 + 100) = .636

  Since gamma has a known sampling distribution approximating normal for large samples it is possible to compute its standard error and significance. SPSS and other major packages report the significance level of the computed phi value. Computation of the variance of gamma is given in Liebetrau (1983: 75-76).

  • Interpretation: Gamma is the surplus of concordant pairs over discordant pairs, as a percentage of all pairs ignoring ties. This can be given a PRE (proportionate reduction in error) interpretation. If we ignore tied pairs and are guessing the ranking of two pairs based on knowledge of the independent (column) variable x, then if we are presented with the x values for two randomly selected pairs, we will predict that if the second x is more than the first, then the rank of the second y value will be greater than the rank of the first y value. If gamma is .636, we may say that knowing the the independent variable reduces our errors in predicting the rank (not value) of the dependent variable by 63.6%.
  • Meaning of association: Gamma defines perfect association as weak monotonicity (see discussion in the section on association). Under statistical independence, gamma will be 0, but it can be 0 at other times as well (whenever concordant minus discordant pairs are 0).
  • Symmetricalness: Gamma is a symmetrical measure. It will compute to the same coefficient value regardless of which is the independent (column) variable. Somers' d, discussed below, may be viewed as an asymmetric extension of gamma.
  • Data level: Gamma assumes both variables are ordinal or higher.
  • Other features: For 2-by-2 tables, gamma equals Yule's Q. If one dichotomizes ordinal or higher data, Q will ordinarily be lower than gamma is for the undichotomized data since dichotomization loses information (particularly information on the tail distributions) and attenuates the association. Gamma can be computed even when cell count is low or zero. Gamma cannot be computed if all cases are in a single row or a single column.

  
  

• Kendall's tau-b is a measure of association often used with but not limited to 2-by-2 tables. It is computed as the excess of concordant over discordant pairs (P - Q), divided by a term representing the geometric mean between the number of pairs not tied on x (X₀) and the number not tied on y (Y₀):
  tau-b = (P - Q)/ SQRT[((P + Q + Y₀)(P + Q + X₀))]

  CITY SIZE BY RIOT INTENSITY	City Size 1	City Size 2	Row Totals
  Riot Size 1	1	0	1
  Riot Size 2	3	4	7
  Column Totals	4	4	8

  tau-b = (4 - 0) / SQRT[(4 + 0 + 12)(4 + 0 + 3)] = 4/SQRT[112] = .378

  Since tau-b has a known sampling distribution it is possible to compute its standard error and significance. SPSS and other major packages report the significance level of the computed tau-b value. The formula for the variance of tau-b is given in Liebetrau (1983: 70).

  • Interpretation: There is no well-defined intuitive meaning for tau-b, which is the surplus of concordant over discordant pairs as a percentage of concordant, discordant, and approximately one-half of tied pairs. The rationale for this is that if the direction of causation is unknown, then the surplus of concordant over discordant pairs should be compared with the total of all relevant pairs, where those relevant are the concordant pairs, the discordant pairs, plus either the X-ties or Y-ties but not both, and since direction is not known, the geometric mean is used as an estimate of relevant tied pairs.
  • Meaning of association: Tau-b defines perfect association as strict monotonicity, as discussed in the section on association. Although it requires strict monotonicity to reach 1.0, it does not penalize ties as much as some other measures. It defines null relationship as statistical independence.
  • Symmetricalness: Tau-b is a nondirectional coefficient. Somers' d is an asymmetric analogue to tau-b.
  • Data level: Tau-b requires binary or ordinal data.
  • Other features Tau-b reaches 1.0 (or -1.0 for negative relationships) only for square tables when all entries are on one diagonal. Tau-b is 0 under statistical independence for both square and non-square tables. Tau-c is used for non-square tables.

  
  

• Kendall's tau-c, also called Stuart's tau-c or Kendall-Stuart tau-c, is a variant of tau-b for larger tables. It equals the excess of concordant over discordant pairs times another term representing an adjustment for the size of the table.
  tau-c = (P - Q)*[2m/(n²(m-1))]
  where m is the number of rows or columns, whichever is smaller, and n is sample size.

  CITY SIZE BY RIOT INTENSITY	City Size 1	City Size 2	City Size 3	Row Totals
  Riot Size 1	4	2	0	6
  Riot Size 2	2	3	4	9
  Column Totals	6	5	4	15

  tau-c = (36 - 4)*[4/(15²(1))] = 128/225 = .57

  Since tau-c has a known sampling distribution it is possible to compute its standard error and significance. SPSS and other major packages report the significance level of the computed tau-c value. The formula for the variance of tau-c is given in Liebetrau (1983: 73-74).

  • Interpretation: There is no simply intuitable meaning which can be attached to tau-c. However, tau-c is tau-b expressed as a percentage of the maximum value tau-b may obtain given table size and thus tau-c can be viewed as having the same interpretation as tau-b, except devoid of table size effects. However, this same attribute means tau-c assumes different values depending on table size, which in turn may be a function of how data are subdivided into categories, causing Somers and others to reject tau-c as a useful measure of association.
  • Meaning of association: Perfect association is defined by tau-c in terms of strict monotonicity, as discussed in the section on association. Although it requires strict monotonicity to reach 1.0, it does not penalize ties as much as some other measures. Null association is defined in terms of statistical independence.
  • Symmetricalness: Tau-c is a symmetrical measure which will yield the same coefficient magnitude regardless of which of two variables is considered the independent (column) variable.
  • Data level: Tau-c is designed for use where at least one of two variables is ordinal in level. It may be used with higher level data but is considered inefficient for this purpose.
  • Other features Tau-c varies from -1 (for negative relationships) to +1, and was specifically designed to be able to attain these bounds for non-square tables.

  
  

• Somers' d is gamma modified to penalize for pairs tied on x only, in directional (asymmetric) hypotheses in which x causes of predicts y; and to penalize for pairs tied on y only, in hypotheses in which y causes of predicts x. Somers' d, d_yx = (P - Q)/(P + Q + Y₀) for the hypothesis that x causes or predicts y. For the hypothesis that y causes of predicts x, the formula is: d_xy = (P - Q)/(P + Q + X₀). As explained in the section on gamma, P is concordant pairs, Q is discordant pairs, Y₀ is pairs tied on Y, and X₀ is pairs tied on X. Computation of the variance of gamma is given in Liebetrau (1983: 80-82).

  CITY SIZE (X) BY RIOT INTENSITY (Y)	City Size 1	City Size 2	City Size 3	Row Totals
  Riot Size 1	6	0	0	6
  Riot Size 2	0	5	4	9
  Column Totals	6	5	4	15

  d_yx = (P - Q)/(P + Q + Y₀)
  d_yx = (54 - 0)/(54 + 0 + 20) = .73

  d_xy = (P - Q)/(P + Q + X₀)
  d_xy = (54 - 0)/(54 + 0 + 0) = 1.0

  • Interpretation: Somers' d is the surplus of concordant pairs as a percentage of concordant, discordant, and relevant tied pairs of observations. Given the condition that a randomly selected pair of observations are not tied on the independent variable, Somers' d is the conditional probability that the pair is concordant minus the conditional probability that the pair is discordant.
  • Meaning of association: Somers' d reaches a maximum of 1.0 (or a mimum of -1.0 for negative relationships) for perfect strict monotonic association, or for perfect ordered monotonic association, as discussed in the section on association. Null relation in Somers' d is independence.
  • Symmetricalness: Somers' d is an asymmetric statistic, but it can be made symmetric simply by averaging d_xy and d_yx. Somers' symmetric d is 1.0 only when both variables are strict monotonic functions of the other.
  • Data level: Somers' d is used with ordinal data.
  • Other features For 2-by-2 tables, Somers' d is equivalent to percent difference. For a given table, asymmetric Somers' d will be less than or equal to gamma or tau-c. For square tables, tau-b is the geometric mean between d_xy and d_yx.




Assumptions

• Ordinal data are assumed. While categorical interval data might also be used with these measures, information is lost in the categorization process and a better assessment of strength of relationship is obtained by using continuous data with an interval level measure such as Pearson's correlation, r.

• Note measures of association, unlike significance, do not assume randomly sampled data.




Frequently Asked Questions

• Where does one find these measures of association in SPSS?
  All are found in the SPSS CROSSTABS module. From the menu, select Statistics, Summarize, Crosstabs. In the "Crosstabs" dialog box, click the "Statistics" button, then in the "Crosstabs: Statistics" dialog box, check the measures you want.

Bibliography

• Agresti, Alan (1996). Introduction to categorical data analysis. NY: John Wiley and Sons.

• Goodman, Leo A. and W. H. Kruskal (1954, 1959, 1963, 1972). Measures for association for cross-classification, I, II, III and IV. Journal of the American Statistical Association. 49: 732-764, 54: 123-163, 58: 310-364, and 67: 415-421 respectively. The 1972 installment discusses the uncertainty coefficient.

• Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32.