Chapter 7 Tables 299
Goodman-Kruskal
gamma
A measure of association used when the row and column
values are ordinal variables. Gamma ranges from 21 to 1.
A negative value indicates negative association, a positive
value indicates positive association, and 0 indicates no
association between the variables.
Kendall’s tau-b Similar to gamma, except that tau-b includes a correction
for ties. Used only for ordinal variables.
Stuart’s tau-c Similar to tau-b, except that it includes a correction for
table size. Used only for ordinal variables.
Somers’ D A modifi cation of the tau-b statistic. Somers’ D is used for
ordinal variables in which one variable is used to predict
the value of the other variable. Somers’ D (R|C) is used
when the column variable is used to predict the value
of the row variable. Somers’ D (C|R) is used when the
row variable is used to predict the value of the column
variable.
Because the x^2 distribution is a continuous distribution and counts rep-
resent discrete values, some statisticians are concerned that the Pearson
chi-square statistic is not appropriate. They recommend using the continuity-
adjusted chi-square statistic instead. We feel that the Pearson chi-square statis-
tic is more accurate and can be used without adjustment.
Among the other statistics in Table 7-4, the likelihood ratio chi-square
statistic is usually close to the Pearson chi-square statistic. Many statisti-
cians prefer using the likelihood ratio chi-square because it is used in log-
linear modeling—a topic beyond the scope of this book.
All of the three test statistics shown in Figure 7-20 are significant at
the 5% level. The association between the Calculus Requirement and
Department variables ranges from 0.354 to 0.378 for the three measures of
association (Phi, Contingency, and Cramer’s V). The final four measures
of association (gamma, tau-b, tau-c, and Somers’ D) are used for ordinal data
and are not appropriate for nominal data.
Validity of the Chi-Square Test with Small
Frequencies
One problem you may encounter is that it might not be valid to use the
Pearson chi-square test on a table with a large number of sparse cells. A
sparse cell is defi ned as a cell in which the expected count is less than 5.
The Pearson chi-square test requires large samples, and this means that cells
with small counts can be a problem. You might get by with as many as one-
fi fth of the expected counts under 5, but if it’s more than that, the p value
