====== Hedges' g for independent two-sample design ======

In independent two-sample design, the most popular effect size measure is defined as 

$$ \delta = \frac{\mu_1 - \mu_2}{\sigma} $$

where $\mu_1$ and $\mu_2$ are the population means of the two population and $\sigma$ is the population standard deviation which is assumed to be the same for the two populations.

==== Hedges' g ====

Although [[Cohen's d]] is the most widely used effect size estimator in samples, it is biased. An unbiased estimator, often referred to Hedges' g, was developed by Hedges (1981). It corrects the [[Cohen's d]] in the following way:

$$
g = d\times \frac{\Gamma(m/2)}{\sqrt{(m/2)} \Gamma[(m-1)/2]}
$$

where $m = n_1 + n_2 - 2$.

==== Confidence Intervals ====

Hedges (1981) showed that the variance of $g$ is 

$$
var(g) = \frac{m(1+\tilde{n} \delta^2)}{(m-2)\tilde{n}} - \frac{\delta^2}{c^2}
$$

where $\tilde{n}=n_1 n_2/(n_1+n_2)$ and 

$$c=\frac{\Gamma(m/2)}{\sqrt{(m/2)} \Gamma[(m-1)/2]}$$.

With it, a confidence interval for $g$ can be constructed as

$$g \pm \Phi^{-1}(1-\alpha/2) \sqrt{\widehat{var(g)}}$$


where $\sqrt{\widehat{var(g)}}$ is an estimate of $var(g)$ by replacing $\delta$ with Cohen's d.

==== Assumptions ====

  * Normally distributed data
  * Equal variances

==== Related effect size ====

  - [[Cohen's d]]
  - [[two-sample-t-test|Effect size of independent two samples]]

==== References ====

Hedges, L. V. (1981). Distribution theory for Glass’s estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107–128.