Phi Coefficient

Setup and Calculation of a Phi Coefficient

To compute a phi coefficient, two variables must be used, each with dichotomous response options. This produces what is called a 2 × 2 (two variables, each with two response options) contingency table, which displays the variables, their frequencies, and the totals. Here is an example of such a table, with “yes” and “no” response options for both variables:

By examining the frequencies of the 2 × 2 contingency table, the direction of the relationship (i.e., positive or negative) can be surmised.[Page 1231] Specifically, if the bulk of responses fall in the a and d cells, the relationship will be positive; conversely, if the responses largely fall in cells b and c, the relationship will be negative. In addition to the requirement that both variables are measured dichotomously, the row and column totals of the 2 × 2 contingency table must be the same (i.e., [a + b] + [c + d] = [a + c] + [b + d]). Only after these two criteria have been met can a phi coefficient be calculated, which is done using the following equation:

$$\mathrm{\phi }=\frac{ad+bc}{\sqrt{\left(a+b\right)\left(c+d\right)\left(a+c\right)\left(b+d\right)}}.$$

After the phi coefficient is calculated (a number which should range between −1 and +1), a chi-square statistic must be used to test for significance (i.e., did the association occur by chance or is it systematic?). The phi coefficient must be converted to a chi-square (χ2) using the following equation: χ2 = Nφ2, where N is the total number of cases (see the example 2 × 2 contingency table). In addition, the degrees of freedom (df) for the test are needed, and are calculated using the following equation: (r − 1)(c − 1), where r is the number of rows and c is the number of columns. For a 2 × 2 contingency table, the df will always equal 1, as (2 − 1) (2 − 1) = 1, and a phi coefficient cannot be utilized on variables with more than two response options. After the phi coefficient has been converted to a chi-square and the df obtained, the chi-square value can be compared to the critical value for the test, and a conclusion about statistical significance made.

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