# The Special Case of 2 × 2 Contingency Table

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

Many situations in biomedical research call for the use of a 2 × 2 contingency table in which the researcher might be comparing two levels of a study condition, such as treatment and control, and two levels of an outcome, such as yes/no or dead/alive.

THE SPECIAL CASE OF THE 2 × 2 CONTINGENCY TABLE

Many situations in biomedical research call for the use of a 2 × 2 contingency table (Table 11.7) in which the researcher might be comparing two levels of a study condition, such as treatment and control, and two levels of an outcome, such as yes/no or dead/alive. By using algebra, the formula for chi-square has been greatly simpli-fied for easy computation. The calculation formula has many applications in epi-demiologic research settings.

In a 2 × 2 table we use an independent chi-square test, where chi-square = Σ(|O – E| – 1/2)2/E. The term “1/2” is called Yates’ correction and provides a more precise estimate of chi-square when there are only two rows and columns.

By algebra, the calculation formula for a 2 × 2 χ2 is:

χ2(df = 1) = (|ad –bc| – N/2)2N   /  (a + b)(c + d)(a + c)(b + d) where df = 1, χ2 critical = 3.84, and α = 0.05.

Now let us apply the calculation formula to a specific example. Data shown in Table 11.8 reflect the number of male and female smokers between two hypotheti-cal samples of males and females (n = 54 and n = 46, respectively).

If there is no association between gender and smoking, one would expect that the deviations between the observed and expected numbers of smokers and nonsmok-ers in each of the four cells are not statistically significant. If there is an association, some of the cells will have statistically significant deviations between the observed and expected frequencies, which would suggest an association between smoking and gender.

Whether this association is likely or not likely to be due to chance may be evalu-ated by the chi-square statistic. Using the data in the bivariate 2 × 2 contingency table (Table 11.8),

x2 = (|21 × 31 – 15 × 33| – 100/2)2 100   /  (36) (64) (54) (46)   = .196 Because the calculated χ2 does not exceed the critical value (3.84), gender does not appear to be related to smoking status.

### TABLE 11.7. General 2 × 2 Contingency Table ### TABLE 11.8. Bivariate 2 × 2 Contingency Table 