====== Effect size and confidence interval for paired-t-test ======

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

The sample effect size is calculated as 

$$
d = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{(s_1^2 + s_2^2)/2}}
$$

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

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 $\lambda_U$ with

$$\lambda_L = t^{-1}_{ncp}(t, df=n-1, 1-\alpha/2)$$

and 

$$\lambda_U = t^{-1}_{ncp}(t, df=n-1, \alpha/2)$$

where 

$$
t = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{(s_1^2 + s_2^2 - 2r s_1 s_2)/n}}
$$

with $r$ being the correlation. 

Now the confidence interval is given by

$$
\left[\lambda_L \sqrt{\frac{2(s_1^2 + s_2^2 - 2r s_1 s_2)}{n(s_1^2 + s_2^2)}}, \lambda_U \sqrt{\frac{2(s_1^2 + s_2^2 - 2r s_1 s_2)}{n(s_1^2 + s_2^2)}}\right]
$$