# Multi-arm extension to Simon's single-arm two-stage design?

For adaptive phase II clinical trials with binary endpoint, the Simon's two stage design is well-known (Simon 1989) and easy to implement. However, I was wondering, whether there are multi-arm two-stage extensions to this design and if there exist any implementations in statistical software packages (preferably in R)?

I have found the paper by Logan (2005 , link) on optimal two-stage randomized phase II clinical trials, which sounds promising but is a bit too concise to be honest. Does anyone happen to know additional literature or software packages?

Many thanks!

In a sense both the standard Simon's two-stage design for a single arm trial and a similar randomized trial with a control group fall within group-sequential methods. In particular, we are talking about group-sequential methods, where you may stop early at an interim analysis due to rejecting the null hypothesis or because it is unlikely you will be able to do so in the end ("futility"). In the case of Simon's two-stage design the null hypothesis is that that a proportion $$p_1$$ in the single treatment group $$i=1$$ is $$\leq p_0$$ (and the alternative hypothesis $$p_1>p_0$$), while for a two group study it might be $$p_1 \leq p_2$$, where i=1 refers to the test group and i=2 to the control group (and the alternative hypothesis $$p_1>p_2$$). One special condition here is that we make the futility stopping rule binding and by doing so can "regain" some of the type I error that can no longer occur in those hypothetical outcomes where you stop at the interim for futility. With binomial outcomes in one or two groups, one can relatively easily do the necessary adjustments using enumeration of all cases.
Under a normal approximation there is plenty of software that will implement standard group-sequential methods (e.g. East, various R packages etc.) with binding stopping rules and give you operating characteristics. If your sample size is large (=normal approximation is fine), you could simply use such software to set your stopping criteria. You probably want different approaches for efficacy (e.g. one-sided $$\alpha=0.025$$ and O'Brien-Fleming alpha-spending) and futility (e.g. some more permissive choice such as one-sided $$\alpha=0.05$$ or 0.1 and perhaps Pocock style alpha-spending).
If you have a smallish sample size, then you can use the output of such a software as a starting point and then adjust the interim criteria by doing enumeration of all possible outcomes under the binomial distribution to make sure you do not violate your one-sided type I error for efficacy, but also fully exhaust the type I error rate. E.g. with 50 patients per arm and an interim after 25/arm, that's 456,976 scenarios (26*26 (first stage outcomes) * 26*26 (second stage outcomes)) that you need to check for a fine grid across the domain of $$p_1$$ and $$p_2$$ (double-check this, but I assume you can simplify this and need to only check for the boundary between the null- and alternative-hypothesis, so only for $$p_1=p_2$$).