Proper inference from Simon's two-stage designs

by Tatsuki Koyama, Heidi Chen and Will Gray


There are many software available for constructing a Simon's design. These software usually give sample sizes and critical values for stages 1 and 2, therefore, hypothesis testing is simple to conduct. Analyzing the data beyond hypothesis testing, however, is not straightforward when a Simon's design is used. When the actual stage 2 sample size is different from the one originally planned, data analysis is more complicated. Many previously published clinical trials do not correctly account for such mid-course changes. We present a software that computes a preferable P value, a reasonable estimate and a confidence interval for the response rate.

The web-based software is not available any more. Pleases use the R-functions found on: twostage2020Web.R.

The method is based on "Proper inference from Simon's two-stage designs" by Koyama and Chen (Statistics in Medicine [2007]).


Response Rate

  • $p_0 \, \cdots$ Response rate under the null hypothesis
  • $p_1 \, \cdots$ Response rate under the alternative hypothesis

Stage 1

  • $n_1 \, \cdots$ Planned sample size for stage 1.
  • $R_1 \, \cdots$ Critical value for stage 1. Note if there are $R_1$ or more ''successes'' in stage 1, the trial continues to stage 2.

Stage 2 (planned)

  • $n_2 \, \cdots$ Planned sample size for stage 2.
  • $R_t \, \cdots$ Planned critical value at the end of stage 2. Note if there are $R_t$ or more ''successes'' in stage 1 and 2 combined, the null hyopthesis is rejected.

Stage 2 (actual)

  • $n^*_2 \, \cdots$ Actual sample size for stage 2.


  • $X_1 \, \cdots$ Number of successes in stage 1.
  • $X_2 \, \cdots$ Number of successes in stage 2 (out of $n_2$ or $n^*_2$).

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