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

586 CHAPTER 17 | The Chi-Square Statistic: Tests for Goodness of Fit and Independence
independence has df = (R – 1)(C – 1), where R is the number of rows in the table and C is the
number of columns. For Cramér’s V, the value of df* is the smaller of either (R – 1) or (C – 1).
Cohen (1988) also suggested standards for interpreting Cramér’s V that are shown in
Table 17.10. Note that when df* = 1, as in a 2 × 2 matrix, the criteria for interpreting V are
exactly the same as the criteria for interpreting a regular correlation or a phi-coefficient.
TABLE 17.10
Standards for interpreting
Cramér’s V as proposed
by Cohen (1988).
Small
Effect
Medium
Effect
Large
Effect
For df* = 1 0.10 0.30 0.50
For df* = 2 0.07 0.21 0.35
For df* = 3 0.06 0.17 0.29
In a research report, the measure of effect size appears immediately after the results of
the hypothesis test. For the study in Example 17.4, for example, we obtained χ 2 = 6.381 for
a sample of n = 200 participants. Because the data form a 2 × 2 matrix, the phi-coefficient
is the appropriate measure of effect size and the data produce
f5Î x2
n
Î 5 6.381
200 5 0.179
For these data, the results from the hypothesis test and the measure of effect size would be
reported as follows:
The results showed a significant relationship between parents’ rules about alcohol and subsequent
alcohol-related problems, χ 2 (1, n = 200) = 6.381, p < .05, ϕ = 0.179. Specifically, teenagers
whose parents allowed supervised drinking were more likely to experience problems.
■ Assumptions and Restrictions for Chi-Square Tests
To use a chi-square test for goodness of fit or a test of independence, several conditions
must be satisfied. For any statistical test, violation of assumptions and restrictions casts
doubt on the results. For example, the probability of committing a Type I error may be distorted
when assumptions of statistical tests are not satisfied. Some important assumptions
and restrictions for using chi-square tests are the following.
1. Independence of Observations This is not to be confused with the concept of
independence between variables, as seen in the chi-square test for independence
(Section 17.3). One consequence of independent observations is that each observed
frequency is generated by a different individual. A chi-square test would be
inappropriate if a person could produce responses that can be classified in more
than one category or contribute more than one frequency count to a single category.
(See p. 244 for more information on independence.)
2. Size of Expected Frequencies A chi-square test should not be performed when
the expected frequency of any cell is less than 5. The chi-square statistic can be
distorted when f e
is very small. Consider the chi-square computations for a single
cell. Suppose that the cell has values of f e
= 1 and f o
= 5. Note that there is a
4-point difference between the observed and expected frequencies. However,
the total contribution of this cell to the total chi-square value is
cell 5 s f o 2 f e d2
f e
5
s5 2 1d2
1
5 42
1 5 16