Statistical Inference: A Short Course

13.7 A Measure of Strength of Association in a Contingency Table

We noted in Section 13.5 that the Pearson chi-square statistic is used to test for association between two attributes A and B. To conduct the test, we hypothesize independence among the A and B factors and then, for a given level of significance α, we see if we can reject the null hypothesis of independence in favor of the alternative hypothesis that the A and B characteristics are not independent but, instead, exhibit statistical association. If we detect statistical association between A and B, then the next logical step in our analysis of these attributes is to assess how strong the association between the A and B categories happens to be. That is, we need a measure of the strength of association between factors A and B.

One device for depicting statistical association in an r × c contingency table is Cramer's phi-squared statistic

(13.19)

obtained by dividing Pearson's chi-square statistic U by its theoretical maximum value n (q − 1), where n is the sample size and q = min{r,c}. Here Φ² measures the overall strength of association, that is, for Φ² = 0, the sample exhibits complete independence between the A and B attributes; and when Φ² = 1, there exists complete dependence between these factors.

Example 13.7

Given the chi-square test results determined in Example 13.5 (we had a 4 × 3 contingency table involving ...