## Binding / Nonbinding futility rule - alpha control [Two-Stage / GS Designs]

Dear Ben,

» » Binding, nonbinding - does it have an impact on the alpha control? I think not, but are not totally sure.
» Non-binding: Type 1 error is protected, even if the futility criterion is ignored.

Was also my thought because I didn't find any relationship to a futility rule in the proof of alpha control in the paper of Maurer et al. Or do I err here?

» Binding: Type 1 error is protected only if the futility criterion will be adhered to. ('Binding' is not common practice, authorities don't want this).

Are you sure for the binding case?
I thought: If the TIE is controlled without adhering to any futility rule, then it is more than ever controlled also by applying a futility criterion. The probability of deciding BE is lowered by doing so, and therefore also the TIE.

Of course the power may be compromized.
Example (some sort of 'forced BE', whatever this is):
power.tsd.in(CV=0.25, theta0=0.9, GMR=0.9, n1=36)
gives pBE('empiric power')= 0.68452 (!). Increasing n1 doesn't help. Try it.
Empiric TIE (theta0=1.25) is: pBE= 0.034186.

Without the futility criterion w.r.t. the CI
power.tsd.in(CV=0.25, theta0=0.9, GMR=0.9, n1=36, fCrit="no")
you obtain pBE('empiric power')= 0.80815.
Power much more raised if you also forget the power futility rule:
power.tsd.in(CV=0.25, theta0=0.9, GMR=0.9, n1=36, fCrit="no", fCpower=1)
gives a pBE= 0.90658.
Empiric TIE (theta0=1.25) is: pBE= 0.050012. Nitpickers! Don't cry "alpha inflation"! The +0.000012 to 0.05 is the simulation error. Try setseed=F and you will get something like p(BE)= 0.049858 or in the next run p(BE)= 0.04982.

I think that your statement for the binding case is only valid if you make a further adaption of the local alpha / critical values taking the futility rule into consideration. But I don't know how this could be done. The implementation in Power2Stage anyhow doesn't make such an adaption, if I see it correctly.

Do you have any experinces for your statement 'Binding' is not common practice, authorities don't want this'.
If yes, what is/are the reason(s) given by authorities to abandon binding futility rule(s) or not to 'like' them?

Regards,

Detlew