I have read that in group sequential designs, having a maximum of $k$ stops, for the $j$-th stop $j < k$ there are 4 bounds chosen $a_j,b_j, c_j, d_j$ such that the three decisions are considered. A set of lecture notes supporting this concept can be found here.
- Reject $H_=, H_+$ in favor of $H_-$ if $T<a_j$ (reject futility, benefit in favor of harm).
- Reject $H_-, H_+$ in favor of $H_=$ if $b_j < T < c_j$ (reject harm, benefit in favor of futility).
- Reject $H_-, H_=$ in favor of $H_+$ if $T > d_j$ (reject harm, futility in favor of benefit).
Otherwise continue until the next stop.
What’s the point of having a futility hypothesis? It seems to me that it’s either a negative study outcome or a positive study outcome: thus, it should be specified as such a priori. If futility is “negative” it means the sponsor has no interest in moving on to market their drug because it’s no better than existing therapies. If that’s the case, then $T > a_j$ but $T< b_j$ seems like rationale to stop the study, because I don’t care too much whether there’s harm or futility: I’m not going to sell this drug.
If futility is “positive” it means the sponsor is interested in shifting their design strategy toward proving non-inferiority since their new drug is perhaps cheaper or better tolerated or has fewer interactions. In that case, it seems to me that they did the wrong type of study. They should have conducted a non-inferiority design to begin with.