Some researchers prefer a statistic called Kendall’s T_{b} (also written т_{ь} and pronounced tau-b) instead of gamma for bivariate tables of ordinal data because G ignores tied pairs in the data. The formula for T_{b} is:

where n_{s} is the number of same-ranked pairs, n_{o} is the number of opposite-ranked pairs, n_{td} is the number of pairs tied on the dependent variable, and n_{ti} is the number of pairs tied on the independent variable. The formula for calculating the tied pairs is:

where R refers to the row marginals (the dependent variable) and C refers to the column marginals (the independent variable). In table 21.10:

We already have the numerator for T_{b} in this case (we calculated the number of same- ranked and opposite-ranked pairs in figure 21.3), so:

This confirms the weak, negative association we saw from the results of the G test. Kendall’s T_{b} will usually be smaller than G because G ignores tied pairs, while T_{b} uses almost all the data (it ignores the relatively few pairs that are tied on both variables).

YULE'S Q: G FOR 2 X 2 TABLES

Yule’s Q is the equivalent of G for 2 X 2 tables of ordinal variables, like high versus low prestige, salary, education, religiosity, and so on. Yule’s Q can be calculated on frequencies or on percentages. The formula is:

Yule’s Q is an easy-to-use statistic. A good rule of thumb for interpreting Q is given by J. A. Davis (1971): When Q is 0, the interpretation is naturally that there is no association between the variables. When Q ranges from 0 to — 0.29, or from 0 to + 0.29, you can interpret this as a negligible or small association. Davis interprets a Q value of ± 0.30 to ±0.49 as a ‘‘moderate’’ association; a value of ±0.50 to ±0.69 as a “substantial” association; and a value of ± 0.79 or more as a ‘‘very strong’’ association.

Rutledge (1990) was interested in the effect of one- or two-parent families on children’s relations with their mothers and fathers. She surveyed African American, college- aged women, mostly from Chicago. One of the questions she asked was: ‘‘When you were growing up, how close were you to your father? Were you considerably close, moderately close, or not close at all?’’ I’ve collapsed Rutledge’s data into two response categories, close and not close, in table 21.11.

Table 21.11 Family Structure and Self-Reported Closeness to Parents

Close to father?

Two parents

One parent

Total

Yes

135

36

171

No

13

31

44

Total

148

67

215

SOURCE: E. M. Rutledge, ''Black Parent-Child Relations: Some Correlates,'' Journal of Comparative Family Studies, Vol. 21, pp. 369-78,1990. Abstracted from data in table 2. Reprinted by permission.

Here is the calculation of Yule’s Q for these data:

Yule’s Q for these data is .80. Most of the women (135/148 = .91) who come from two- parent homes are close to their fathers, compared to fewer than half who come from one- parent homes (31/67 = .46) are not. The reason is obvious: Overwhelmingly, one-parent homes are headed by mothers, not by fathers.