Russel-Research-Method-in-Anthropology

618 Chapter 20
of opposite-ranked pairs is calculated by multiplying each cell by the sum of
all cells below it and to its left. This is diagramed in figure 20.3.
In Table 20.10,
the number of same-ranked pairs is:
16 (7 + 7 + 4 + 27) = 720
+ 7 (7 + 27) = 238
+ 7 (4 + 27) = 217
+ 7 (27)
= 189
Total
1,364
Figure 20.3. Calculating gamma.
The number of opposite-ranked pairs is:
61 (7 + 7 + 13 + 4) = 1,891
+ 7 (7 + 13) = 140
+ 7 (4 + 13)
+ 7 (13)
=
=
119
91
Total
2,241
Gamma for table 20.10, then, is:
G 1,3642,241
1,3642,241 877
3,605 .24
Gamma tells us that the variables are associated negatively—people who
agree with either of the statements tend to disagree with the other, and vice
versa—but it also tells us that the association is relatively weak.
Is Gamma Significant?
How weak? If you have more than 50 elements in your sample, you can
test for the probability that gamma is due to sampling error using a procedure
developed by Goodman and Kruskal (1963). A useful presentation of the procedure
is given by Loether and McTavish (1993:598, 609). First, the gamma
value must be converted to a z-score, or standard score. The formula for converting
gamma to a z-score is:
z Gn s n o /2n1G 2 Formula 20.14
where G is the sample gamma, is the gamma for the population, n is the size
of your sample, n s is the number of same-ranked pairs, and n o is the number of
opposite-ranked pairs.
As usual, we proceed from the null hypothesis and assume that for the
entire population is zero—that is, that there really is no association between
the variables we are studying. If we can reject that hypothesis, then we can
assume that the gamma value for our sample probably approximates the
gamma value, , for the population. Using the data from figure 20.3 and the
gamma value for table 20.10: