Testing for Homogeneity

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Chapter: Biostatistics for the Health Sciences: Categorical Data and Chi-Square Tests

A chi-square test for homogeneity is used in empirical investigations when the marginal totals for one condition have been fixed at certain values and the totals for the other condition may vary at random.


TESTING FOR HOMOGENEITY

A chi-square test for homogeneity is used in empirical investigations when the marginal totals for one condition have been fixed at certain values and the totals for the other condition may vary at random. This situation might occur when an investigator has assigned a fixed number of subjects to a study design and then determines how the subjects are distributed according to a second variable, such as an exposure factor for a disease.

Table 11.6 provides an example of the possible association between smoking and chronic cough. Suppose that a researcher who is studying adult factory workers recruits 250 smokers and a comparison group of 300 nonsmokers. 

TABLE 11.5. Values of |O – E|2/E for the Association between Race and Cancer Stage


TABLE 11.6. The Association between Smoking and Chronic Cough


The researcher then refers the employees to a medical exam that assesses the presence of lung dis-eases; chronic cough is included in the review of symptoms. The data are charted in Table 11.6.

The expected frequencies are computed in the same way as in a 2 × 2 table. (Refer back to Section 11.3 for the formulas.) Note also that the frequencies shown in cells b and d can be determined by subtraction. That is, if you know only the total number of smokers and the number of cases of chronic cough among smokers, you can determine the number of smokers who do not have chronic cough by subtrac-tion (250 – 99).

Chi-square= [(99–52.73)2 / 52.73] + [(17–63.27)2 / 63.27] + [(151–197.27)2 / 197.27]  + [(283–236.73)2 / 236.73] = 94.33

This is a significant chi-square for df = 1 and suggests that the proportions of per-sons with chronic cough are not equally distributed between smokers and nonsmokers.

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