Obtains Hedges' g estimate and confidence interval of effect size.
hedgesg(tstat, m, ntilde, cilevel = 0.95)A data frame with the following variables:
tstat: The value of the t test statistic.
m: The degrees of freedom for the t-test.
ntilde: The normalizing sample size to convert the
standardized treatment difference to the t-test statistic.
g: Hedges' g effect size estimate.
varg: Variance of g.
lower: The lower confidence limit for effect size.
upper: The upper confidence limit for effect size.
cilevel: The confidence interval level.
The value of the t-test statistic for comparing two treatment conditions.
The degrees of freedom for the t-test.
The normalizing sample size to convert the
standardized treatment difference to the t-test statistic, i.e.,
tstat = sqrt(ntilde)*meanDiff/stDev.
The confidence interval level. Defaults to 0.95.
Kaifeng Lu, kaifenglu@gmail.com
Hedges' \(g\) is an effect size measure commonly used in meta-analysis to quantify the difference between two groups. It's an improvement over Cohen's \(d\), particularly when dealing with small sample sizes.
The formula for Hedges' \(g\) is $$g = c(m) d,$$ where \(d\) is Cohen's \(d\) effect size estimate, and \(c(m)\) is the bias correction factor, $$d = (\hat{\mu}_1 - \hat{\mu}_2)/\hat{\sigma},$$ $$c(m) = 1 - \frac{3}{4m-1}.$$ Since \(c(m) < 1\), Cohen's \(d\) overestimates the true effect size. \(\delta = (\mu_1 - \mu_2)/\sigma.\) Since $$t = \sqrt{\tilde{n}} d,$$ we have $$g = \frac{c(m)}{\sqrt{\tilde{n}}} t,$$ where \(t\) has a noncentral \(t\) distribution with \(m\) degrees of freedom and noncentrality parameter \(\sqrt{\tilde{n}} \delta\).
The asymptotic variance of \(g\) can be approximated by $$Var(g) = \frac{1}{\tilde{n}} + \frac{g^2}{2m}.$$ The confidence interval for \(\delta\) can be constructed using normal approximation.
For two-sample mean difference with sample size \(n_1\) for the treatment group and \(n_2\) for the control group, we have \(\tilde{n} = \frac{n_1n_2}{n_1+n_2}\) and \(m=n_1+n_2-2\) for pooled variance estimate.
Larry V. Hedges. Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics 1981; 6:107-128.
n1 = 7
n2 = 8
meanDiff = 0.444
stDev = 1.201
m = n1+n2-2
ntilde = n1*n2/(n1+n2)
tstat = sqrt(ntilde)*meanDiff/stDev
hedgesg(tstat, m, ntilde)
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