Biostatistics

12.5 TESTS OF HOMOGENEITY 631
TABLE 12.5.1 A Contingency Table for Data for a
Chi-Square Test of Homogeneity
Population
Variable Category 1 2 3 Total
A n A1 n A2 n A3 n A:
B n B1 n B2 n B3 n B:
C n C1 n C2 n C3 n C:
Total n.1 n.2 n.3 n
homogeneous, or, equivalently, if the samples are all drawn from the same population, with
respect to the categories A, B, and C, our best estimate of the proportion in the combined
population who belong to category A is n A: =n. By the same token, if the three populations
are homogeneous, we interpret this probability as applying to each of the populations
individually. For example, under the null hypothesis, n A . is our best estimate of the
probability that a subject picked at random from the combined population will belong to
category A. We would expect, then, to find n.1ðn A: =nÞ of those in the sample from population
1 to belong to category A, n.2ðn A: =nÞ of those in the sample from population 2 to belong to
category A, and n.3ðn A: =nÞ of those in the sample from population 3 to belong to category A.
These calculations yield the expected frequencies for the first row of Table 12.5.1. Similar
reasoning and calculations yield the expected frequencies for the other two rows.
We see again that the shortcut procedure of multiplying appropriate marginal totals
and dividing by the grand total yields the expected frequencies for the cells.
From the data in Table 12.5.1 we compute the following test statistic:
" #
X 2 ¼ Xk ðO i E i Þ 2
i¼1
EXAMPLE 12.5.1
Narcolepsy is a disease involving disturbances of the sleep–wake cycle. Members of the
German Migraine and Headache Society (A-8) studied the relationship between migraine
headaches in 96 subjects diagnosed with narcolepsy and 96 healthy controls. The results
are shown in Table 12.5.2. We wish to know if we may conclude, on the basis of these data,
E i
TABLE 12.5.2
Frequency of Migraine Headaches by Narcolepsy Status
Reported Migraine Headaches
Yes No Total
Narcoleptic subjects 21 75 96
Healthy controls 19 77 96
Total 40 152 192
Source: The DMG Study Group, “Migraine and Idiopathic Narcolepsy—A Case-Control Study,”
Cephalagia, 23 (2003), 786–789.