The phi coefficient is a ratio of two quantities, determined as follows. We label the four cells in a contingency table a, b, c, and d, by convention, as shown in Table 11-2.
If there is no association between test and disease, the two ratios a/b and c/d would be equal and would also be equal to the proportion of people in the sample who have the disease. So the ratio of these two numbers would be 1, and if we subtract 1 from this ratio of ratios, we end up with a number related to the degree of association between the two variables. Playing around a bit with the algebra, we have:
Contingency Table Depicting Ratios
Actually, the phi coefficient uses a different denominator, made up of the product of the marginals:
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