I often run into randomized experiments (and am guilty myself) where treatment is much more expensive than control, but sample sizes in the two groups are equal. Has been said before but bears repeating: when c(T) > c(~T), the optimal sample size ratio T/~T is sqrt((c(~T)/c(T)).
Implications: 1. With fixed budget B and sigma1=sigma0, the power-maximizing group sizes are n(~T)*=B/ (sqrt(c(~T)*c(T))+c(~T)) and n(T)*=B/(sqrt(c(~T)*c(T))+c(T)).
2. With desired effect size delta & power beta: n(~T)*=t_MDE^2/delta^2 *(1+sqrt(c(T)/c(~T))); n(T)*=t_MDE^2/delta^2 *(1+sqrt(c(~T) /c(T))), where t_MDE=t_{alpha/2}+t_beta
