Basic Concepts
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
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%).
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Lowry, R. (2023) The power of the chi-square “goodness of fit” test
http://vassarstats.net/chi_beta.html
Faul, F., Erdfelder, E., Buchner, A., & Lang, A. G. (2009). Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41, 1149-1160.
http://link.springer.com/article/10.3758/BRM.41.4.1149
STAT (2015)Â Power and sample size reference manual, release 13
http://www.stata.com/manuals13/pss.pdf
Oyeyemi, G. M. (2010) On the estimation of power and sample size in test of independence
https://scialert.net/fulltext/?doi=ajms.2010.139.146#google_vignette
Would the same formula be used for the phi-coefficient? Or is this just for Cramer’s w? I suppose for V the df would always be 2 and this just allows it to be greater than that?
Nikolas,
w is phi. It is not Cramer’s V.
Charles
Thanks, could you explain how to caculate the power of a Wilcoxon Signed Rank Test?
Venus,
I have not yet focused on this issue, although I expect to turn my attention to it shortly. I will probably use a simulation approach.
Charles
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.
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
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
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
What is “x” in this your reply?
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
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
Yes
Yes, you use the NVHISQ_NCP, as shown in Figure 1 of the referenced webpage.
Charles