y,g
31.9444012014312,1
20.5406532971108,1
13.4639000249002,1
...
20.3628306367208,2
29.7063904254911,2
26.5223485727072,2
...
The first row of the data can be variable names. In this case, the option "With variable names" should be selected.
As for the case with the summary statistics, one can also specify the confidence limit, choose different types of CI, and change the number of bootstraps.
==== Output ====
A sample output is given below:
The estimated effect size is:-0.3224
The associated 95% CI is: [-0.8305 , 0.1884]
t-test results
t-value = -1.249
Degrees of freedom = 58
p-value = 0.2168
The first part is the estimated effect size and the confidence interval. The second part provides information on the two-sided t-test.
----
===== Methods =====
==== Point estimate ====
The most widely used effect size measure for independent two-sample t-test is Cohen's d. It can be 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 two different conditions, respectively.
Cohen's d is a biased estimator of the population effect size. An unbiased estimator, often referred to Hedge's g, was developed by Hedge (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 ====
Two types of confidence intervals are available - one is based on Algina and Keselman (2003) and the other is based on the bootstrap method.
=== Algina and Keselman (2003) ===
We implemented the method by Algina, J., & Keselman, H. J. (2003). Approximate confidence intervals for effect sizes. //Educational and Psychological Measurement//, //63//, 537-553.
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
$$
t = \frac{\bar{y}_1 - \bar{y}_2}{\sqrt{\frac{(n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2}}{n_{1}+n_{2}-2}} \times \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}} .
$$
Now the confidence interval is given by
$$
\left[\lambda_L \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}, \lambda_U \sqrt{\frac{1}{n_{1}}+\frac{1}{n_{2}}}\right]
$$
=== Bootstrap method ===
When the summary statistics are provided, a parametric bootstrap method is used. In this case, the bootstrap samples are generated from normal distributions $N(\mu_j, \sigma_j^2), j=1,2$ in which $\mu_j$ and $\sigma_j^2$ are replaced by the sample statistics $\bar{y}_j$ and $s_j^2$.
When the raw data are provided, a non-parametric bootstrap method is used. In this case, the bootstrap samples are generated by sampling the raw data with replacement.