# Proper inference from Simon's two-stage designs

by

Tatsuki Koyama,

Heidi Chen and

Will Gray
### Introduction

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]).

### Notation

#### Response Rate

- Response rate under the null hypothesis
- Response rate under the alternative hypothesis

#### Stage 1

- Planned sample size for stage 1.
- Critical value for stage 1. Note if there are or more ''successes'' in stage 1, the trial continues to stage 2.

#### Stage 2 (planned)

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

#### Stage 2 (actual)

- Actual sample size for stage 2.

#### Data

- Number of successes in stage 1.
- Number of successes in stage 2 (out of or ).