# Power calculation for logistic regression in G*Power

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? Thank you very much in advance! Irene

## 1 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:

$$P(Y=1 |X = S_x) = Odds_{1SD} / (1+ Odds_{1SD})$$