I am performing a sample size calculation in G*Power for a logistic regression with continuous predictor. I am requested to estimate a value for Pr(Y=1 | X=1) H0, i.e. the probability that my binary outcome (group) =1, when my main predictor is at its mean, and to estimate a value for Pr(Y=1 | X=1)H1, i.e. the probability that my binary outcome (group) =1, when my main predictor is one SD unit above its mean.
The paper I am extracting the estimates from reports means and SDs of the two groups (which are my predicted variable).
How can I estimate the probability values requested by G*Power?
Best Answer
Under $H_0$, $P(Y=1|X=x)$ does not change for any value of $x$ d/t no association. If $X$ is standardized, the mean response is $X=0$. So $P(Y=1|X=0)$ under $H_1$ or $H_0$ would be the proportion of the sample with the response. For ease, let's just call this $p$.
Hopefully table 1 gives you the SD for the predictor $X$. Call this $S_x$. If not, you can't do this.
If you get the odds ratio from the logistic output in the paper you reference, call this $OR$. Then you can find the predicted proportion with response for a one-SD higher group of $X$ by first calculating this
$$ Odds_{1SD} = p/(1-p) * OR^{S_x}$$
Odds-1SD is the odds of response at one SD higher value from the "center". Convert back to probability with:
$$P(Y=1 |X = S_x) = Odds_{1SD} / (1+ Odds_{1SD})$$