# 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: