Example using Demidenko programme for a logistic regression power calculation
Now I see: you use binomial regression. Yes, you can use binomial data with Demidenko's calculator: just think that if the outcome y=7 (ie 7 0-1 responses given) then it means y=1,1,1,1,1,1,1 (the binary variable=1 repeated 7 times). Binomial data does not change the model, it just modifies the sample size (if you get n for regular regression than for a 8-category binomial logit you need 7 times smaller n). The only problem is that I do not use a mixed model but the regular logistic regression. However, if the variance of the random effect is not large, we can close eyes on this fact.
Run logistic regression with interaction, say, logit(y) = a’+a*x+b*z+g*x*z. Then you can set ORyx=exp(a);ORyz=exp(b) and Pry=Pr(y=1|x=0,z=0)=E/(1+E) where E=exp(a’).
Alternative OR (Odds Ratio) is exp(g) you want to detect with the sample size n. Sometimes, people take detectable/alternative OR=1.2. Remember to divide the n from my program by 7 (or whatever is the total possible number of positive outcomes e.g. number correct) because you have a binomial model.
So, we can use Demidenko's web calculator with binomial data, write all the quantities, and then finally, divide the sample size found by 7 (if n=7 is the number of 0-1 responses given per subject).