# Confidence Intervals for Power and Effect Size for Chi-square Tests

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%).

### 9 Responses to Confidence Intervals for Power and Effect Size for Chi-square Tests

1. Nikita Khromov-Borisov says:

Dear Dr. Ziontz,
please, help me to understand is it possible and how to calculate confidence interval for the effect size w using RealStatistics
Thank you in advance, best regards, Nikita

• Charles says:

Yes

• Charles says:

Yes, you use the NVHISQ_NCP, as shown in Figure 1 of the referenced webpage.
Charles

2. Nikita Khromov-Borisov says:

Using calculator for the noncentral chi-square distribuition
http://keisan.casio.com/exec/system/1180573184
for your above example I have recieved values 14.79 and 58.80 for the lower and upper limits for lambda which differ from yours.

• Charles says:

What you have calculated is not the lower and upper limits for lambda, but the lower and upper limits for x. In fact NCHISQ_INV(.025,4,30) = 14.79 and NCHISQ_INV(.975,4,30) = 58.80, the same as the values you have shown.
Charles

• Nikita Khromov-Borisov says:

I cannot understand where is this procedure in Real Statistics. In the list of Data Analysis Tools there is only Staistical Power and Sample Size. I mark it and then mark Chi-Square test and in the menu there are only Size Effect, Sample size, df, alpha and Sum Count (BTW, what is this?). However, there is no option for the CI calculation and no table like your Fig. 1

• Charles says:

Nikita,
I have not implemented this capability yet in the Real Statistics software. You can calculate it yourself as described on the referenced webpage.
Sum Count = the number of terms used from the theoretically infinite sum required to calculate the noncentral chi-square distribution value. You don’t really need to worry about this and simply use the default.
Charles

• Nikita Khromov-Borisov says:

• Charles says:

Nikita,
x was simply my way of referencing the noncentral chi-square statistic value. The main thing is that NCHISQ_INV(.025,4,30) = 14.79 and NCHISQ_INV(.975,4,30) = 58.80.
Charles