Objective
We describe how to estimate confidence intervals for Cohen’s d* and Hedges’ g*, effect sizes for two independent samples where the variances are not assumed to be equal. These effect sizes are described in Two Sample t-test with Unequal Variances. We start by reviewing these effect sizes.
Cohen’s d* and Hedges’ g*
Cohen’s d* is defined by
where
Hedges’ g* is a less biased version of d*, and is defined by
where m = df*/2 and
Confidence Interval
We estimate confidence intervals for d* and g* using the same approach as for d and g (see CI Functions for Effect Sizes d and g). In particular, first we estimate the noncentrality parameter (ncp*) by
Next, we find the confidence interval for ncp* by using the Real Statistics NT_NCP function. The end points of this CI are =NT_NCP(α/2, df*, ncp*) and =NT_NCP(1-α/2, df*, ncp*).
We then use this confidence interval to find a CI for d*. This is done by noting that
Thus, once we have a CI for ncp* we can multiply the end points of this CI by the factor on the right side of the above expression to obtain a CI for d*. Finally, we multiply the end points of the d* CI by the usual factor (involving the gamma function) to obtain a CI for g*.
Example
Example 1: Find the 95% confidence interval of the effect size for Example 2 of Two-Sample t-Test with Unequal Variances.
We show the results of the t-test on the left side of Figure 1 to obtain the needed descriptive statistics. The results on the right side of the figure shows how to estimate the desired confidence intervals.
Figure 1 – 95% confidence interval for d* and g* effect sizes
Worksheet Functions
Real Statistics Functions: The Real Statistics Resource Pack provides the following array functions.
T_EFFECT3(m1, m2, s1, s2, n1, n2, lab, alpha, iter) = column array with the values Cohen’s d*, Hedges’ g*, and the lower and upper confidence interval limits for d* and g* based on a two independent sample t-test with unequal variances for sample 1 with mean m1, standard deviation s1 and sample size n1, and sample 2 with mean m2, standard deviation s2 and sample size n2.
TT_EFFECT3(R1, R2, lab, alpha, iter, iter0, prec) = T_EFFECT3(m1, m2, s1, s2, n1, n2, lab, alpha, iter, iter0, prec) where m1 = AVERAGE(R1), s1 = STEV.S(R1), n1 = COUNT(R1), m2 = AVERAGE(R2), s2 = STEV.S(R2) and n2 = COUNT(R2).
alpha is the significance level (default .05). If lab = TRUE (default FALSE) then an extra column of labels is appended to the output.
The last three arguments are as for the NT_NCP function (see Noncentral t Distribution), except that if iter = 0 then the Hedges and Olkin estimate of the confidence interval is employed (iter0 and prec are not used), while if iter > 0 (default 1000) then the estimate of the confidence interval described above based on the noncentrality parameter is used. In this case, iter, iter0, and prec are as for the NT_NCP function (see Noncentral t Distribution).
Example
Referring to the data on the left side of Figure 1, we see that the array formula
=T_EFFECT3(B4,C4,SQRT(B5),SQRT(C5),B6,C6,TRUE)
produces the output shown in Figure 2, which agrees with the results shown on the right side of Figure 1.
Figure 2 – T_EFFECT3 output
Examples Workbook
Click here to download the Excel workbook with the examples described on this webpage.
References
Delacre, M., Lakens, D., Ley, C., Liu, L., Leys, C. (2021) Why Hedges’ g*s based on the non-pooled standard deviation should be reported with Welch’s t-test
https://psyarxiv.com/tu6mp/download
Howell (2010) Confidence intervals on effect size
https://www.uvm.edu/~statdhtx/methods8/Supplements/MISC/Confidence%20Intervals%20on%20Effect%20Size.pdf
Lecoutre B., (2007) Another look at the confidence intervals for the noncentral t distribution
https://digitalcommons.wayne.edu/jmasm/vol6/iss1/11/
Steiger, J. H., Fouladi, R. T. (1997) Noncentrality interval estimation and the evaluation of statistical models
https://statpower.net/Steiger%20Biblio/Steiger&Fouladi97.PDF
Hedges, L. V. and Olkin, I. (1985) Statistical methods for meta-analysis. Academic Press
https://www.researchgate.net/publication/216811655_Statistical_Methods_in_Meta-Analysis