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Making Inference from a Simon's Two-stage Design in Phase II Clinical Trials

by Tatsuki Koyama

Introduction Background Technical Details Reference


Two-stage designs are commonly used in phase II clinical trials (especially in cancer clinical trials). Simon (1989) proposed two criteria (minimax and optimal) for selecting sample sizes and critical values for these two-stage designs. Simon's two stage designs are very popular; the paper by Simon has been cited for more than 500 times. Despite its popularity, a two-stage design lacks easy-to-use tools for making inference once the data are collected. Although these two-stage designs are primarily for decision making and estimation is usually not a primary objective of a phase II trial it may be preferable to compute a pvalue, a confidence interval and an unbiased estimate of the true response rate at the termination of the trial. We are currently developing R / Splus programs (still improving - 7/5/05) and web-based programs (under construction - 6/3/05 programmer Will Gray) to compute an "honest" pvalue, confidence interval and unbiased estimate from a two-stage design in phase II clinical trial.

Introduction Background Technical Details Reference


In a typical two-stage trail, $n_1$ patients are enrolled in stage I. If the number of responses is fewer than or equal to prespecified $r_1$, the trial is terminated for lack of efficacy. Otherwise an additional $n_2$ patients are enrolled in stage II. The total number of patients is $n_t \equiv n_1 + n_2$. If the cumulative number of responses are fewer than or equal to $r_t$, then lack of efficacy is concluded. Otherwise, it is concluded that the treatment is sufficiently effective for further investigation.

A two-stage design is indexed by four numbers, $n_1,\,r_1,\,n_t,\,r_t$. These numbers are chosen so that the probability of concluding efficacy when the treatment is not effective is less than $\alpha$ and the probability of concluding futility when the treatment is effective is less than $\beta$.

For a given $p_0,\,p_1$, there are many designs, i.e., $n_1,\, r_1,\, n_t,\, r_t$ that satisfy $\alpha$ and $\beta$ conditions. Following are candidates for good designs among them.
  1. The optimal design has the minimum expected sample size under $H_0: p=p_0$.
  2. The minimax design has the smallest $n_t$.
  3. The balanced design has $n_1 = n_2$. (reference Ye and Shyr)
There are softwares capable of finding these designs given $p_0,\,p_1,\,\alpha,\,\beta$. One such software is developed and maintained by Fei Ye ( and freely available here. (Vanderbilt-Ingram Cancer Center, Biostatistics Shared Resource)

Jung et al. (2004) proposed admissible designs that lie somewhere between the optimal and minimax designs and have favorable characteristics in terms of both the expected and maximum sample sizes. They have also created a software that is capable of finding the optimal, minimax and admissible

Introduction Background Technical Details Reference

Technical Details

See here.


  1. Simon R. Optimal two-stage designs for phase II clinical trials. Controlled Clinical Trials 1989; 10: 1 - 10.
  2. Jung SH, Lee T, Kim KM, George SL. Admissible two-stage designs for phase II cancer clinical trials. Statistics in Medicine 2004; 23: 561 - 569.
  3. Armitage P. Resticted Sequential Procedures. Biometrika 1957; 44: 9 - 56.
  4. Jung SH, Kim KM. On the estimation of the binomial probability in multistage clinical triasl. Statistics in Medicine 2004; 23: 881 - 896.
  5. Whitehead J. On the bias of maximum likelihood estimation following a sequential test. Biometrika 1986; 73: 573 - 581.
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Topic revision: r1 - 06 Dec 2006, TatsukiKoyama

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