Appendix D -- Computation of Measures of Association

There
are many measures of association used to measure the strength of relationship.
Each has advantages and disadvantages. In this module, we have used two--Cramer's
V and Goodman and Kruskal's Gamma. This appendix describes how to compute these
measures. However, you will use a statistical package such as SPSS for Windows
to do the actual computations for the exercises.

Cramer's V
is one of several measures based on chi square. Chi square itself is not a
measure of association, but a test of the hypothesis that two variables are
unrelated. V is equal to the square root of the following value--chi square
divided by the product of the number of cases in the table and the smaller
of two values--the number of rows minus one and the number of columns minus
one. The chi square for table 3.2 was equal to 12.52. V would be equal to
12.52 divided by the square root of the product of 900 and 1 or 12.52 divided
by 30 or 0.42.

Cramer's V should
be used when one or both of the variables consist of an unordered set of categories.
It varies from 0 to 1. The closer it is to 0, the weaker the relationship
and the closer to 1, the stronger the relationship. V can never be negative.

Gamma
assumes that both of the variables consist of ordered categories. To understand
the computation of Gamma, you must think of pairs of cases. Imagine four individuals.
We'll just call them A, B, C, and D. For each person, we know their income
and education which has been categorized as low, medium, and high. The table
below displays these values.

Education of Respondent
Income	High	Medium	Low
High	A	.	.
Medium	.	C	.
Low	B	.	D

These four individuals
form six possible pairs. A can be paired with B, A with C, A with D, B with
C, B with D, and C with D. Notice that the AB pair is the same as the BA pair
since they involve the same two individuals.

A has more education
and income than C. This is what we call a concordant pair. A is higher
on both variables (i.e., income and education) than C. The AD pair is also
concordant. A is higher on both education and income than D. And the CD pair
is also concordant. C has more education and income than D.

However, C has
more income than B, but C has less education than B. This is what we call
a discordant pair. C is higher on one variable (i.e., education), but
lower on the other variable (i.e., income).

When we look
at the AB pair we see another possibility. A has more income than B, but A
and B are tied on education. This is what we call a tied pair. The BD pair
is also tied. B has more education than D, but B and D have the same income.

Gamma ignores
all tied pairs. Since two of the six possible pairs are tied, Gamma would
be based on the remaining four untied pairs. Gamma is equal to the number
of concordant pairs (C) minus the number of discordant pairs (D) divided by
the sum of the number of concordant pairs and discordant pairs.

In this example,
Gamma would equal (3-1)/(3+1) or 2/4 or 0.50. Since there are more concordant
pairs than discordant pairs, we can observe that it is more common for pairs
to have the same order on both variables than to have different orders. In
other words, large amounts of education tend to go with large amounts of income
and small amounts of education tend to go with small amounts of income. This
is what we call a positive relationship. Gamma has a positive sign
if the relationship is positive [Note 1].

If the number
of discordant pairs had been greater than the number of concordant pairs,
the relationship would have been negative and the sign of Gamma would
have been negative. This would have meant that large values of one variable
would tend to go with small values of the other variable and that small values
of one variable would tend to go with large values of the other variable.

The numerical
value of Gamma tells us the strength of the relationship. The closer the value
of Gamma to 0, the weaker the relationship and the closer to 1, the stronger
the relationship [Note 2].

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Notes

1.
This will be true only if the columns are arranged from high to low (left
to right) and the rows are arranged from high to low (top to bottom).

2.
The fact that Gamma ignores all tied pairs tends to inflate the value of Gamma.
For this reason, Gamma produces a larger measure of association than other
measures.

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REFERENCES
AND SUGGESTED READING

Cramer's V

Norusis, Marija
J. 1997. SPSS 7.5 Guide to Data Analysis. Upper Saddle River, New Jersey:
Prentice Hall.

Gamma

Knoke, David,
and George W. Bohrnstedt. 1991. Basic Social Statistics. Itesche. IL.:
Peacock.