The same approach used to calculate a confidence interval for the effect size of a t test (see Confidence Intervals for Power and Effect Size of t Test) can be employed to create a confidence interval for a noncentrality parameter, and in turn Cohen’s effect size and statistical power, for a chi-square goodness of fit or independence test.
Example 1: Find the 95% confidence interval for the effect size w and power of a chi-square test of independence for a 3 × 3 contingency table with sample size 500 when χ2 = 30.
Figure 1 – Confidence intervals for effect size and power
We see from Figure 1 that the 95% confidence interval for the noncentrality parameter is (9.98, 51.81). The corresponding confidence interval for the effect size w of .24 is (.14, .32) and the confidence interval for power of 99.7% is (71.5%, 99.99%).