Russel-Research-Method-in-Anthropology

Bivariate Analysis: Testing Relations 617
There are several ways that pairs of observations can be distributed if they
are ranked on two ordinal variables.
1. They can be ranked in the same order on both variables. We’ll call these ‘‘same.’’
2. They can be ranked in the opposite order on both variables. We’ll call these
‘‘opposite.’’
3. They can be tied on either the independent or dependent variables, or on both.
We’ll call these ‘‘ties.’’
In fact, in almost all bivariate tables comparing ordinal variables, there are
going to be a lot of pairs with tied values on both variables. Gamma, written
G, is a popular measure of association between two ordinal variables because
it ignores all the tied pairs. The formula for gamma is:
G
No. of same-ranked pairs No. of opposite-ranked pairs
No. of same-ranked pairs No. of opposite-ranked pairs
Formula 20.13
Gamma is an intuitive statistic; it ranges from 1.0 (for a perfect negative
association) to 1.0 (for a perfect positive association), through 0 in the middle
for complete independence of two variables.
If there are just two ordinal ranks in a measure, and if the number of opposite-ranked
pairs is 0, then gamma would equal 1. Suppose we measure
income and education in a Malaysian community ordinally, such that: (1)
Anyone with less than a high school diploma is counted as having low education,
and anyone with at least a high school diploma is counted as having high
education; and (2) Anyone with an income of less than RM34,000 a year is
counted as having low income, while anyone with at least RM34,000 a year is
counted as having high income. (RM is the symbol for the Malaysian ringgit.
RM34,000, or about $9,000 U.S. dollars, was the average per capita GDP in
Malaysia in 2005.)
Now suppose that no one who had at least a high school diploma earned
less than RM34,000 a year. There would be no pair of observations, then, in
which low income and high education (an opposite pair) co-occurred.
If the number of same-ranked pairs is zero, then gamma would equal 1.0.
Suppose that no one who had high education also had a high income. This
would be a perfect negative association, and gamma would be 1.0. Both
1.0 and 1.0 are perfect correlations.
Calculating the Pairs for Gamma
The number of same-ranked pairs in a bivariate table is calculated by multiplying
each cell by the sum of all cells below it and to its right. The number