CHAPTER 15
NOMINAL MEASURES OF CORRELATION: PHI, THE CONTINGENCY COEFFICIENT,
AND CRAMER'S V
Chapters 13 and 14 introduced and explained the use of a set of statistical tools that
researchers use to measure and evaluate the degree of association that exists between interval
variables and ordinal variables. This chapter concludes the discussion of correlational statistics
by providing three new measures which can be applied in research situations where an individual
wishes to determine the degree of association that exists between nominal variables. Three
statistics are introduced in this chapter: Phi
Cramer's V
, the Contingency Coefficient
, and
. All three statistics are simple and easy to calculate. Each begins with the
calculation of the Chi-Square statistic using the methods outlined in Chapter 11 of this text.
Given the nature of the nominal data used to calculate each of these statistics, the obtained values
for each statistic will always fall along a range from a low of 0 to a high of 1. Negative
correlations with each of these statistics are mathematically impossible.
The choice of which statistic to employ in a given research situation is determined by the
size of the data matrix and whether or not the two nominal variables under consideration have the
same number of possible values. The Phi
statistic is used when both of the nominal
variables under consideration have exactly two possible values. When this is true, the data matrix
will always have a simple 2x2 design. The Contingency Coefficient
is used when there are
3 or more values for each nominal variable, as long as there are an equal number of possible
values leading to the construction of a data matrix that has an equal number of rows and columns
is used when the number of possible values for the two
(3x3, 4x4, etc). Cramer's V
variables is unequal, yielding a different number of rows and columns in the data matrix (2x3,
3x5, etc).
Taking an example from Chapter 11, a Chi-Square statistic is calculated as follows (using
Yate's Correction because expected values for two of the cells were below 10).
Figure 15:1
Chi-Square Statistic: Gender and Income
Men
High Income
Women
15
25
(19.66)
(20.34)
Low Income
14
5
(9.34)
(9.66)
Total 29
30
Total
40
19
59
Row
Column
1
1
15
19.66
-4.66
4.16
17.31
.88
1
2
25
20.34
4.66
4.16
17.31
.85
2
1
14
9.34
4.66
4.16
17.31
1.85
2
2
5
9.66
-4.66
4.16
17.31
1.79
The result of the calculations yielded a value of 5.37 for Chi-Square. A consultation of the table
in Appendix H indicates that there is a significant difference between the groups (at .05) that
suggests women are more likely to be found in the high income classification than men. Once
this initial set of calculations is complete, Phi
can be calculated using the following formula:
Using the obtained value of 5.37 for Chi-Square, and the value for n of 59 obtained from the total
in the data matrix, Phi is calculated:
The obtained value for Phi suggests the presence of a moderate correlation between the two
variables.
The next measure to be discussed in this chapter is the Contingency Coefficient
This statistic is calculated using the fomula:
.
. In the way of an example, assume
that a significant chi-square value of 9.68 was obtained from a comparison of two variables that
each had three possible values. The data matrix would be 3x3 in this case, indicating that the
Contingency Coefficient
would be the most appropriate measure of association. Assuming
an n of 60 for this research scenario, the calculation of the Contingency Coefficient proceeds as
follows:
As in the first example, the calculated value for this statistic suggests the presence of a moderate
correlation between the two variables.
The final statistic commonly employed by those measuring association between nominal
variables is Cramer's V
. It is calculated using the formula:
. To determine
the value of k in the formula, look at the number of possible values of each variable (the number
of rows and columns in the data matrix). The smaller of the two numbers is used to represent the
variable k. Assuming once again that a researcher has conducted a Chi-Square test on a sample
with an n of 60 and obtained a significant value of 9.68 for Chi-Square using a data set where
variable X had 5 possible values and variable Y had 7 possible values (5x7 data matrix),
calculation of Cramer's V proceeds as follows:
The obtained value of .2 in this case indicates the presence of a weak correlation between the two
variables under consideration.
In conclusion, remember that the appropriate measure of correlation when working with
nominal data is based on the characteristics of the data and can be determined by the structure of
the data matrix used to calculate the chi-square statistic. When the data matrix is 2x2, the Phi
statistic is used. When the number of rows and columns in the data matrix is the same (3x3, 4x4,
22x22), the Contingency Coefficient is employed. Cramer's V is used when the number of rows
and columns is unequal (2x3, 3x5, 5x7).
Exercises – Chapter 15
1.
Compute a chi-square statistic and the appropriate nominal correlation statistic using the
following data. Draw statistical and research conclusions. Show all work
Under
$10,000
$20,001$55,000
Over
$55,000
Total
Whites
30
40
10
70
150
Blacks
50
60
10
20
140
80
100
20
90
290
Total
2.
$10,000 $20,000
A pollster for a candidate wishes to determine whether there is a relationship between an
individual's voting patterns and their television watching habits. A random sample of 350
voters was taken to address this issue. The results of the sampling yielded the data below.
Calculate value for Chi-Square and determine if it is significant. Determine the
appropriate nominal measure of correlation and apply it to this situation. Draw statistical
and research conclusions.
Television
Viewing Time
Democrats
Republicans
Light
45
5
70
120
Moderate
50
10
68
128
Heavy
60
10
32
102
155
25
170
350
Total
Independents
Total