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

592 CHAPTER 17 | The Chi-Square Statistic: Tests for Goodness of Fit and Independence
that is predicted by H 0
. The test determines how
well the observed frequencies (sample data) fit the
expected frequencies (data predicted by H 0
).
3. The expected frequencies for the goodness-of-fit test
are determined by
expected frequency = f e
= pn
where p is the hypothesized proportion (according to
H 0
) of observations falling into a category and n is the
size of the sample.
4. The chi-square statistic is computed by
chi-square = χ 2 = S s f o 2 f e d2
where f o
is the observed frequency for a particular
category and f e
is the expected frequency for that
category. Large values for χ 2 indicate that there is a
large discrepancy between the observed ( f o
) and the
expected ( f e
) frequencies and may warrant rejection
of the null hypothesis.
5. Degrees of freedom for the test for goodness of fit are
df = C – 1
where C is the number of categories in the variable.
Degrees of freedom measure the number of categories
for which f e
values can be freely chosen. As can be
seen from the formula, all but the last f e
value to be
determined are free to vary.
6. The chi-square distribution is positively skewed
and begins at the value of zero. Its exact shape is
determined by degrees of freedom.
7. The test for independence is used to assess the relationship
between two variables. The null hypothesis
states that the two variables in question are independent
of each other. That is, the frequency distribution
for one variable does not depend on the categories
of the second variable. On the other hand, if a relationship
does exist, then the form of the distribution
for one variable depends on the categories of the
other variable.
f e
8. For the test for independence, the expected frequencies
for H 0
can be directly calculated from the marginal
frequency totals,
f e
5 f c f r
n
where f c
is the total column frequency and f r
is the
total row frequency for the cell in question.
9. Degrees of freedom for the test for independence are
computed by
df = (R – 1)(C – 1)
where R is the number of row categories and C is the
number of column categories.
10. For the test of independence, a large chi-square value
means there is a large discrepancy between the f o
and
f e
values. Rejecting H 0
in this test provides support for
a relationship between the two variables.
11. Both chi-square tests (for goodness of fit and
independence) are based on the assumption that each
observation is independent of the others. That is, each
observed frequency reflects a different individual, and
no individual can produce a response that would be
classified in more than one category or more than one
frequency in a single category.
12. The chi-square statistic is distorted when f e
values
are small. Chi-square tests, therefore, should not be
performed when the expected frequency of any cell is
less than 5.
13. Cohen’s w is a measure of effect size that can be used
for both chi-square tests. For a chi-square test for
independence, effect size is measured by computing a
phi-coefficient, which is equivalent to Cohen’s w, for
data that form a 2 × 2 matrix or computing Cramér’s
V for a matrix that is larger than 2 × 2.
phi 5Î x2
n
Cramér’s V = Î x2
nsdf*d
where df* is the smaller of (R – 1) and (C – 1). Both
phi and Cramér’s V are evaluated using the criteria in
Table 17.10.
KEY TERMS
parametric test (561)
nonparametric test 561)
chi-square test for goodness
of fit (562)
observed frequencies (564)
expected frequencies (565)
chi-square statistic (566)
chi-square distribution (568)
test for independence (574)
Cohen’s w (583)
phi-coefficient (584)
Cramér’s V (585)