Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

584 CHAPTER 17 | The Chi-Square Statistic: Tests for Goodness of Fit and Independence
The Role of Sample Size You may have noticed that the formula for computing w
does not contain any reference to the sample size. Instead, w is calculated using only the
sample proportions and the proportions from the null hypothesis. As a result, the size of
the sample has no influence on the magnitude of w. This is one of the basic characteristics
of all measures of effect size. Specifically, the number of scores in the sample has little or
no influence on effect size. On the other hand, sample size does have a large impact
on the outcome of a hypothesis test. For example, the data in Example 17.5 produce
χ 2 = 4.00. With df = 3, the critical value for a = 0.5 is 7.81 and we conclude that there
are no significant preferences among the four pizza shops. However, if the number of
individuals in each category is doubled, so that the observed frequencies become 12, 24,
16, and 28, then the new x 2 = 8.00. Now the statistic is in the critical region so we reject
H 0
and conclude that there are significant preferences. Thus, increasing the size of the
sample increases the likelihood of rejecting the null hypothesis. You should realize, however,
that the proportions for the new sample are exactly the same as the proportions for
the original sample, so the value of w does not change. For both sample sizes, w = 0.316.
Chi-square and w Although the chi-square statistic and effect size as measured by
w are intended for different purposes and are affected by different factors, they are algebraically
related. In particular, the portion of the formula for w that is under the square
root, can be obtained by dividing the formula for chi-square by n. Dividing by the sample
size converts each of frequencies (observed and expected) into a proportion, which produces
the formula for w. As a result, you can determine the value of w directly from the
chi-square value by the following equation:
w 5Î x2
n
(17.7)
For the data in Example 15.3, we obtained x 2 = 4.00 and w = 0.316. Substituting in the
formula, produces
w 5Î x2
n
Î 5 4.00 5 Ï0.10 5 0.316
40
Although Cohen’s w statistic also can be used to measure effect size for the chi-square test
for independence, two other measures have been developed specifically for this hypothesis
test. These two measures, known as the phi-coefficient and Cramér’s V, make allowances
for the size of the data matrix and are considered to be superior to w, especially with very
large data matrices.
The value of x 2 is
already a squared value.
Do not square it again.
■ The Phi-Coefficient and Cramér’s V
In Chapter 15 (p. 518), we introduced the phi-coefficient as a measure of correlation for
data consisting of two dichotomous variables (both variables have exactly two values).
This same situation exists when the data for a chi-square test for independence form a
2 × 2 matrix (again, each variable has exactly two values). In this case, it is possible
to compute the correlation phi (ϕ) in addition to the chi-square hypothesis test for the
same set of data. Because phi is a correlation, it measures the strength of the relationship,
rather than the significance, and thus provides a measure of effect size. The value for the
phi-coefficient can be computed directly from chi-square by the following formula:
f5Î x2
(17.8)
n