Measures of Association for Categorical Data
Effect size after a chi-square test
A significant chi-square test establishes that an association exists, but it says nothing about how strong the association is. Effect-size measures fill this gap. The phi coefficient is used for 2x2 tables, Cramér V for larger tables, the contingency coefficient as a compatible alternative for either table type, and the point-biserial correlation when one variable is binary and the other is continuous. These measures rescale the association onto a standard metric so that practical importance, not just statistical significance, can be assessed and communicated.
The Concept and Its Logic
The chi-square statistic is sensitive to sample size: the same underlying table can yield a significant result with a large sample and a non-significant one with a small sample. Effect-size measures remove this dependence by rescaling the chi-square so that the result reflects only the strength of association in the table itself. The phi coefficient is computed as phi = sqrt(chi2 / n) and is guaranteed to stay between 0 and 1 only for 2x2 tables. Cramér V corrects phi for table dimensions using V = sqrt(chi2 / (n * (k-1))), where k is the smaller of the number of rows or columns. Both measures range from 0 (no association) to 1 (perfect association).
Computing and Interpreting the Measures
Most statistical software packages such as SPSS, R, and Stata report phi and Cramér V automatically alongside chi-square output. Cohen (1988) defined phi = 0.10 as a small effect, 0.30 as medium, and 0.50 as large for 2x2 tables. For Cramér V the benchmarks shift with table dimensions; for a 3x3 table, a medium effect is approximately 0.21. The contingency coefficient is computed as C = sqrt(chi2 / (chi2 + n)); however, its theoretical maximum is less than 1 and varies with table size, which limits direct comparisons across studies with different table dimensions.
Common Misuses and Misconceptions
The most frequent error is equating statistical significance with practical importance. With large samples, even a very small phi or V can reach p < 0.001, signaling only a large sample, not a strong association. A second common mistake is applying the phi coefficient to tables larger than 2x2, which can produce phi values greater than 1, making the statistic uninterpretable. A third error occurs when one categorical variable is actually ordinal: in that case Spearman correlation is more informative than Cramér V because it preserves rank information that a nominal-level measure discards.
Why It Matters and How to Report It
APA publication standards and most peer-reviewed journals now require an effect-size statistic to be reported alongside chi-square. An example reporting format is: chi-square(2, N = 250) = 14.32, p = .001, Cramér V = .24. Reporting effect size serves three practical goals: it gives readers the information needed to judge the practical significance of the finding, it enables results to be pooled in meta-analyses across studies, and it provides the basis for a power analysis when replication is planned. In short, reporting a chi-square test without an effect-size measure leaves the finding incompletely described.