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effectsize:cohen_s_d

# Cohen's d for independent two-sample design

For studying the standardized group mean difference in an 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 groups and $\sigma$ is the common standard deviation for the two populations.

### Cohen's d

One estimator of the population effect size is Cohen's d. It is calculated as

$$d = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}}}$$

where $n_1$ and $n_2$ are sample sizes, $\bar{y}_1$ and $\bar{y}_1$ are sample means, and $s_1^2$ and $s_2^2$ are sample variances under the two different conditions, respectively.

### Confidence Intervals

Algina and Keselman (2003) constructed a confidence interval for the population effect size based on a non-central t-distribution. In the method, one first gets the lower and upper bounds of the non-centrality parameter as $\lambda_L$ and $\lambda_U$ by solving

$$2[1-pt(d/\sqrt{1/n_{1}+1/n_{2}},n_{1}+n_{2}-2,\lambda_{L})]=1-\alpha$$

and

$$2[1-pt(d/\sqrt{1/n_{1}+1/n_{2}},n_{1}+n_{2}-2,\lambda_{U})]=\alpha$$

where $pt$ is the cdf of the t-distribution.

Now the confidence interval is given by

$$\left[\lambda_L \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}, \lambda_U \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}\right]$$

### Assumptions

• Normally distributed data
• Equal group variances

### Issues

Cohen's d is a biased estimator of the population effect size for standardized group mean difference. An alternative measure is Hedges' g.

Algina, J., & Keselman, H. J. (2003). Approximate confidence intervals for effect sizes. Educational and Psychological Measurement, 63, 537-553.

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