Chapter 2 Phase II design

Simon’s two-stage design

Simon’s two-stage design is used to minimize the expected sample size when the true response is less than some pre-determined uninterested level $\pi_0$ (Simon 1989).

The null hypothesis is that $H_0: \pi = \pi_0$ and the alternative hypothesis is $H_1: \pi = \pi_1$ where $\pi_1$ is some desirable level that warrant further development. Suppose the Type I and Type II errors are $\alpha$ and $\beta$.

Let $n_1$ patients be given treatment in the first stage. If $r_1$ or less respond or more than $r$ respond, stop stage 1. Otherwise, let $n_2$ patients be given treatment in the second stage. $n = n_1 + n_2$.

Suppose $X_1 \sim Bin(n_1, \pi)$ and $X_2 \sim Bin(n_2, \pi)$. We declare the new drug a

• Failure if $\xi_F = X_1 \leq r_1 \text{ OR } \{X_1 > r_1 \text { AND } X_1 + X_2 \leq r \}$
• Success if $\xi_S = \{X_1 > r_1 \text{ AND } X_1 + X_2 > r \}$

Therefore, $$P(\xi_F \mid \pi \leq \pi_0) \leq \alpha, \quad P(\xi_S \mid \pi \geq \pi_1) \geq 1- \beta$$

Moreover, we have $$\begin{aligned} P(\xi_S \mid \pi) & = \sum_{m_1 > r_1, m_1 + m_2 > r} b(m_1; n_1, \pi)b(m_2; n_2, \pi) \\ P(\xi_F \mid \pi) & = 1- P(\xi_S \mid \pi) \\ & = B(r_1; n_1, \pi) + \sum_{x = r_1 + 1}^{\min\{n_1, r\}}b(x; n_1, \pi)B(r-x; n_2, \pi) \end{aligned}$$

The expected sample size $EN(\pi_0)$ is given by $$n_1\left[P(X_1 \leq r_1\mid \pi = \pi_0) + P(X_1 > r\mid \pi = \pi_0) \right] + nP(r_1 + 1 \leq X_1 \leq r \mid \pi = \pi_0)$$ The probability of early rejection in stage 1 is $$PET(\pi_0) = P(X_1 \leq r_1 \mid \pi = \pi_0)$$

There are several choices to select $(r_1, n_1, r, n)$:

• Optimal design: the one has the smallest expected sample size when $\pi = \pi_0$.
• Minimax design: the one has the smallest total sample size for the whole trial when $\pi = \pi_0$.
• Admissible design: compromise the minimax and the optimal design. (Jung et al. 2004)

library(clinfun)
ph2simon(pu = 0.2, pa = 0.4, ep1 = 0.1, ep2 = 0.2)

## 
##  Simon 2-stage Phase II design 
## 
## Unacceptable response rate:  0.2 
## Desirable response rate:  0.4 
## Error rates: alpha =  0.1 ; beta =  0.2 
## 
##         r1 n1 r  n EN(p0) PET(p0)
## Optimal  2 12 7 25  17.74  0.5583
## Minimax  2 14 7 24  19.52  0.4481

library(ph2mult)
binom.design(type = "admissible", p0 = 0.15, p1 = 0.3, 
             signif.level = 0.05, power.level = 0.9)

##              r1 n1  r  n   EN.p0.   PET.p0.      error     power
## Optimal       5 30 17 82 45.05006 0.7105757 0.04609244 0.9007424
## Admissible    5 31 16 76 45.28032 0.6826597 0.04694758 0.9037415
## Admissible.1  6 36 15 70 45.86191 0.7099439 0.04654875 0.9000510
## Minimax       6 42 14 64 51.80052 0.5545216 0.04845876 0.9002785

References

Jung, Sin-Ho, Taiyeong Lee, KyungMann Kim, and Stephen L George. 2004. “Admissible Two-Stage Designs for Phase II Cancer Clinical Trials.” Statistics in Medicine 23 (4): 561–69.

Simon, Richard. 1989. “Optimal Two-Stage Designs for Phase II Clinical Trials.” Controlled Clinical Trials 10 (1): 1–10.