is based on the distribution of
. Exact Tests reports both the asymptotic and exact
p
values.
The formulas for computing the three contingency coefficients are given below. The
formula for each measure involves taking the square root of a function of
. The
positive root is always selected. For a more detailed discussion of these measures of as-
sociation, see Liebetrau (1983).
The phi contingency coefficient is given by the formula
Equation 15.1
The minimum value assumed by
is 0, signifying no association. However, its upper
bound is not fixed but depends on the dimensions of the contingency table. Therefore, it
is not a very suitable measure for arbitrary
tables. For the special case of the
table, Gibbons (1985) shows that
is identical to the absolute value of Kendall’s
co-
efficient and is evaluated by the formula
Equation 15.2
Notice from Equation 15.2 that, for the
contingency table,
could be either
positive or negative, which implies a positive or negative association in the
table.
The Pearson contingency coefficient is given by the formula
Equation 15.3
This contingency coefficient assumes a minimum value of 0, signifying no association.
It is bounded from above by 1, signifying perfect association. However, the maximum
value attainable by
CC
is
, where
. Thus, the range of this
contingency coefficient still depends on the dimensions of the
table. Cramér’s
V
coefficient ranges between 0 and 1, with 0 signifying no association and 1 signifying
perfect association. It is given by
Equation 15.4
Exact Tests reports the point estimate of the contingency coefficient. The formulas for
these asymptotic standard errors are fairly complicated. These formulas are described in
the algorithms manual available on the Manuals CD and also available by selecting
Algorithms
on the Help menu.
CH y
( )
CH x
( )
φ
CH x
( )
N
---------------
=
φ
r
c
×
2
2
×
φ
τ
b
φ
x
11
x
22
x
12
x
21
–
m
1
m
2
n
1
n
2
------------------------------------
=
2
2
×
φ
2
2
×
CC
CH x
( )
CH x
( )
N
+
--------------------------
=
q
1
–
(
)
q
⁄
q
min
r c
,
(
)
=
r
c
×
V
CH x
( )
N q
1
–
(
)
--------------------
=

Measures of Association for Nominal Data
187
These measures may be used to analyze an unordered contingency table given in Sie-
gel and Castellan (1988). The data consist of a crosstabulation of three possible responses
(
completed
,
declined
,
no response
) to a questionnaire concerning the financial account-
ing standards used by six different organizations responsible for maintaining such stan-
dards. These organizations are identified only by their initials (
AAA, AICPA, FAF, FASB,
FEI,
and
NAA
). The crosstabulated data are shown in Figure 15.1.
First, these data are analyzed using only the first three columns of Figure 15.1. For this
subset of the data, Figure 15.2 shows the results for the contingency coefficients. The
exact two-sided
p
value for testing the null hypothesis that there is no association is also
reported. Its value is 0.090, slightly lower than the asymptotic
p
value of 0.092.

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