Confidence Intervals for Noncentrality Parameters

In recent years, it has become a notion to not only report point estimates of effect sizes, but also confidence intervals for said effect sizes. I have created a small R script to calculate the bounds of such a confidence interval in the case of t- and F-distributions. 

Steiger and Foulandi have explained the mathematical approach to the calculation of the confidence interval limits in Harlow, Mulaik & Steiger (1997): the basis is to calculate the confidence interval for the noncentrality parameter of the noncentral t-, F- or Chi-square distribution. With simple calculations the noncentrality parameters can then be transformed into the effect size measure of choice, including Cohen’s d or Cohen’s f^2 (see Steiger, 2004, for details in the case of F-tests).

The R script calculates the lower and upper bounds for a 100(alpha)% confidence interval for the noncentrality parameter of a noncentral t- and F-distribution, from which the confidence interval for an effect size can be derived. It uses R’s built in optimize function and is indeed similar to other approaches, such as conf.limits.ncp.t from the MBESS package. If you do not want to load the whole package and are looking only for a quick way to calculate the CIs, my snippet might be of help.

A small example is given at the GitHub repository, where you can also find the script:

Download script & see examples

Feedback and remarks are appreciated – either here as a comment or as an issue at the GitHub page.


  • Harlow, L. L., Mulaik, S. A., & Steiger, J. H. (Eds.). (1997). What If There Were No Significance Tests? Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
  • Steiger, J. H. (2004). Beyond the F Test: Effect Size Confidence Intervals and Tests of Close Fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9(2), 164–182.

Leave a Reply

Your email address will not be published. Required fields are marked *