ES <- 0.3
s <- 1
n_E <- 30
n_C <- 30
k <- 30
di <- rnorm(k, ES, s * (1/n_E + 1/n_C))Intro
The paper by Viechtbauer (2005) provide a clear way to simulate standardized (SMD) and unstandardized (UMD) effect sizes for simulation studies.
Unstandardized effect sizes
We define two independent groups as:
\[ X^C_{ij} \sim N(\mu^C_i, \sigma^2_i) \\ X^E_{ij} \sim N(\mu^E_i, \sigma^2_i) \]
and the UMD as:
\[ ES_i = \mu^E_i - \mu^C_i \]
With sampling variance:
\[ \sigma^2_{\epsilon_i} = s^2_i\left(\frac{1}{n^E_i} + \frac{1}{n^C_i}\right) \]
Where \(s^2_i\) is the typical pooled within-group variance.
In this case we simulate participant-level data: