Minimax Design vs . Simon’s Optimal Design for Single-Arm Two-Stage Phase II Cancer Clinical Trials

A phase II trial is to determine if an experimental therapy has promising efficacy worthy of a further investigation. Traditionally, phase II clinical trials have been conducted using single-arm two-stage designs with a futility interim test by recruiting about 50 patients. Since Rich Simon proposed optimal design and minimax design for two-stage phase II trials in 1986, either of them has been chosen for most of phase II clinical trials. When designing a phase II trial, however, clinical trialists almost always question which design to select. In this article, we compare these two designs as an effort to give them some directions.


Introduction
Cancer clinical trials are to investigate the efficacy and toxicity of experimental cancer therapies. If an appropriate dose level of an experimental drug is determined from a phase I trial, the drug's anticancer activity is assessed through phase II clinical trials using a short term efficacy endpoint, such as tumor response. Phase II clinical trials are to screen out inefficacious experimental therapies before they proceed to further investigation through large scale phase III trials. For ethical reasons, clinical trials use multi-stage designs for possible early termination for futility or efficacy. Phase II trials generally require shorter study periods than phase III trials, so that two-stage designs are commonly used for phase II cancer clinical trials. Usually, phase II trials adopt a futility stopping rule only since there is no ethical issue to continue treating patients with efficacious therapy and we want to collect as much data as possible to be used when designing a future phase III trial on the experimental therapy.
Let p denote the true response rate of the experimental therapy and p 0 that of a historical control. We want to test the null hypothesis  a) If the cumulative number of responders X 1 + X 2 is less than or equal to a, then reject the experimental therapy.
b) Otherwise, accept the experimental therapy for further investigation.
Let p 1 (> p 0 ) denote the response rate of the experimental therapy with a clinically meaningful improvement from p 0 . Usually, we assume that p 1 -p 0 = 0.15 to 0.2. For a two-stage design defined by (n 1 , n 2 , a 1 , a), we can calculate the 1-sided type I error rate α at p = p 0 and power 1β at p = p 1 based on the fact that X 1 and X 2 are independent binomial random variables. The sample size for a trial with a two-stage design is either n 1 or n = n 1 + n 2 . For the true response rate p, we can also calculate the expected sample size, EN(p), refer to Simon [1]. Let α * denote the upper limit of type I error rate and 1β * the lower limit we allow for a phase II trial. All two-stage designs with a type I error rate no larger than α * and a power no smaller than 1β * are called candidate designs. Among the candidate two-stage designs, the Simon's (1986) minimax design has the smallest maximal sample size n and the optimal design has the smallest expected sample size under H 0 : p = p 0 , EN(p 0 ).

Minimax Design vs. Optimal Design
When designing a two-stage phase II trial with input parameters (p 0 , p 1 , α * , 1β * ), one often questions which design to choose between the minimax design and the optimal design. By its definition, the optimal design is to stop the trial early when the study therapy does not seem to be efficacious based on the data from stage 1 without concerning about the required maximal sample size. So, one may choose the optimal design if the maximal sample size is not much of a concern and the protection against the worst scenario (i.e. when H 0 is true is the primary concern). Oftentimes, however, the maximal sample size is regarded as the sample size required for a two-stage trial, so that it is not easy to ignore it.
This article is to advocate the minimax design when the two designs are very different in terms of the maximal sample size.
Let's compare the properties of these two designs for a specific However, the optimal design sacrifices such efficiency, so that it should increase the maximal sample size n to recover the loss of power by the interim analysis. Table 1 displays the expected sample sizes of the minimax design and the optimal design for response rates between p 0 = 0.1 and p 1 = 0.3. The optimal design has the smallest EN for p = p 0 = 0.1 among the candidate designs, so that it has a smaller EN than the minimax design for p = 0.1. However, its EN quickly increases as the true response rate p increases, so that its EN is larger than that of the minimax design for p > 0. 1

. As such,
if the experimental therapy is more efficacious than the historical control, then the strength of the optimal design over the minimax design disappears. Based on these comparisons, we may select the minimax design over the optimal design. However, both of the minimax and optimality criteria have their own nice properties, so that one may not want to give up the optimality criterion by just selecting the minimax design. Oftentimes, in this case, there exist some designs that closely satisfy both criteria.   So far, we have investigated the performance of minimax design, optimal and admissible designs under a specific design setting.
But, this kind of comparisons hold over the wide range of design settings. As an optimality criterion, we have considered minimizing the expected sample size under H 0 by a futility interim test. If we want to save the EN of two-stage phase II trials for efficacious experimental therapies, then we will have to conduct a superiority early stopping with or without a futility stopping. Change, et al. [4] proposed designs with both futility and superiority interim tests.
We can carry out similar comparisons between different optimality criteria for such designs too.