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 0.4481library(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