Converting effect size to probability of success

Question

I am trying to simulate a data set for a power analysis based on the probability of success of binomial outcomes for two groups/conditions ($A$ & $B$). I've based my predictions for the probabilities for one group ($A$) on results of a similar study. For the other group ($B$), I have a meta-analysis that gives geenral effect sizes for the two conditions. For group $A$, $M = .352$ and for group $B$, $M = .232$.

Is there a way to convert the effect size to a probability of success so that I can simulate the data set for group $B$?

Answer 1

Let $p_A$ and $p_B$ be the probabilities that you are trying to derive using previous studies' results. I understand you to be saying you have a value of $p_A$ already derived but still need a derived value for $p_B$. But you do have an 'effect size' for both groups. To convert these to a derived value of $p_B$, you need to first determine what the underlying model is that the effect sizes correspond to. For example, are these effect sizes log-odds-ratios relative to some third reference category, say, C? If so, then you would take their difference, i.e. 0.232 - 0.352 = -0.12, to determine the the log-odds-ratio for group B relative to A. Then, using this and the probability for group A that you've already derived, your derived value of $p_B$ would solve the following equation

$$\mathrm{logit}(p_B) = \mathrm{logit}(p_A) -0.12$$

where $\mathrm{logit}(x) = \log(x/(1-x))$. If $p_A=0.30$, for example, then then means that $p_B = 0.275$ (and, as aside, in this case you would have no hope of properly powering this study)

In summary, you need to figure out what the underlying model is corresponding to your effect sizes.