Optimal Two-Stage Designs for Phase II Clinical Trials¶

Author: Jiang Wenxin

Date: 2024 April 16

Introduction: Clinical Trials¶

Goal: Determine if a new test or treatment works and is safe.

Three main phases of clinical trials$^{[1]}$:

Phase Purpose Typical number of participants Success rate
I Safety and dosage 20-100 healthy volunteers or patients Approx. 70%
II Efficacy and side effects Up to several hundred patients Approx. 33%
III Efficacy and monitoring of adverse reactions 300 to 3,000 patients Approx. 25%-30%

Phase II Clinical Trials¶

  • Objective: Determine if the new treatment is effective enough to warrant further study in a larger phase III trial, as well as to further assess safety.
  • Endpoints: Binary (e.g., response vs. no response)
  • Hypothesis:

$$ \begin{equation*} \begin{aligned} H_0: p \leq p_0 \quad &\text{vs.} \quad H_1: p \geq p_1 \\ \text{type I error: } \alpha & = P(\text{rej. } H_0 | H_0) \\ \text{type II error: } \beta & = P(\text{rej. } H_1 | H_1) \end{aligned} \end{equation*} $$ where $p$ is the true response rate and $p_0$ and $p_1$ are the uninteresting level and the desirable level, respectively.

A Motivating Example

Let set $p_0 = .05$, $p_1 = .25$, $\alpha = .10$, and $\beta = .10$ and calculate the sample size under Simon's two-stage optimal design$^{[2]}$. Note: $n, n_i, r, r_i, x_i, i=1,2$ are integers.

Notation Definition Value
$n_1$ sample size in stage I 9
$x_1$ # responses in stage I ?
$r_1$ rejection points in stage I 0
$n_2$ sample size in stage II 15
$x_2$ # responses in stage II ?
$r$ total rejection points 2

Accept only if $x_1 \geq r_1$ and $x_1+x_2 \geq r$.

A Motivating Example (cont'd)

Question: Why we need a two-stage design?

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Fig 1: Flowchart of Simon's two-stage design

A Motivating Example (cont'd)

Notation Definition Value
$n$ total sample size 24
PET$(p_0)$ probability of early termination under $p_0$ 0.63
EN$(p_0)$ expected sample size under $p_0$ 14.5

$\text{PET}(p_0) = P(x_1 \leq r_1 | p=p_0) = \text{Bin}(r_1; n_1, p_0) = \sum_{i=0}^{r_1} \binom{n_1}{i} p_0^i (1-p_0)^{n_1-i} = .63$

$\text{EN}(p_0) = n_1 + \left(1-\text{PET}(p_0)\right)*n_2 = 14.5$

If early termination is allowed, the expected sample size EN is less than the total sample size $n$.

Phase II Clinical Trials: (cont'd)¶

  • Optimal design: Minimize EN$(p_0)$ when $p=p_0$ subject to the constraints of $\alpha$ and $\beta$.
  • Minimax design: Minimize $n$ when $p=p_0$ subject to the constraints of $\alpha$ and $\beta$.
  • Constraints: $$\begin{aligned} \alpha & = P(\text{rej. } H_0 | H_0) \\ \beta & = P(\text{rej. } H_1 | H_1) \end{aligned}$$

Calculation of Sample Size¶

Suppose # responses in stage I and stage II $X_1 \sim \text{Bin}\left(n_1, p\right)$ and $X_2 \sim \text{Bin}\left(n_2, p\right)$, respectively. We declare the new drug a

  • Failure if $\xi_F: X_1 \leq r_1$ OR ($X_1>r_1$ and $X_1+X_2 \leq r$)
  • Success if $\xi_S: X_1>r_1$ and $X_1+X_2>r$

Therefore, $$\begin{equation*} P\left(\xi_F \mid p \leq p_0\right) \leq \alpha, \quad P\left(\xi_F \mid p \geq p_1\right) \leq \beta \end{equation*} $$

Moreover, we have $$\begin{equation*} \begin{aligned} P\left(\xi_S \mid p\right) & =\sum_{x_1>r_1, x_1+x_2>r} b\left(x_1 ; n_1, p\right) b\left(x_2 ; n_2, p\right) \\ P\left(\xi_F \mid p\right) & =1-P\left(\xi_S \mid p\right) \\ & =B\left(r_1 ; n_1, p\right)+\sum_{x=r_1+1}^{\min \left\{n_1, r\right\}} b\left(x ; n_1, p\right) B\left(r-x ; n_2, p\right) \end{aligned} \end{equation*} $$

Calculation of Sample Size (cont'd)

The probability of early rejection in stage I is $$ \begin{equation*} \begin{aligned} \text{PET}(p_0)&=P\left(X_1 \leq r_1 \mid p=p_0\right)\\ & = \text{Bin}(r_1; n_1, p_0)\\ & = \sum_{i=0}^{r_1} \binom{n_1}{i} p_0^i (1-p_0)^{n_1-i} \end{aligned} \end{equation*} $$

The expected sample size EN$\left(p_0\right)$ is given by $$\text{EN}(p_0) =n_1+n_2 (1-\text{PET}(p_0))$$

Algorithm for Simon's Two-Stage Design¶

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Then, calculate PET$(p_0)$ and EN$(p_0)$. The minimax design chooses the pair $(r_1, r, n_1, n_2)$ that minimizes $n$, while the optimal design chooses the pair that minimizes EN$(p_0)$.

Results¶

 Simon 2-stage Phase II design 

Unacceptable response rate:  0.05 
Desirable response rate:  0.25 
Error rates: alpha =  0.1 ; beta =  0.1 

           r1 n1 r  n EN(p0) PET(p0)   qLo   qHi
Minimax     0 13 2 20  16.41  0.5133 0.523 1.000
Admissible  0 11 2 21  15.31  0.5688 0.332 0.523
Admissible  0 10 2 22  14.82  0.5987 0.119 0.332
Optimal     0  9 2 24  14.55  0.6302 0.000 0.119

Results (cont'd)

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Fig 2: Sample size calculation results. The 'M' stands for minimax design and the 'O' stands for optimal design.

Reference¶

  1. U.S. Food and Drug Administration (FDA). Step 3: Clinical Research. The Drug Development Process. https://www.fda.gov/patients/drug-development-process/step-3-clinical-research.
  2. Simon, Richard. 1989. “Optimal Two-Stage Designs for Phase II Clinical Trials.” Controlled Clinical Trials 10 (1): 1–10., doi: 10.1016/0197-2456(89)90015-990015-9).
  3. Hao Sun. Chapter 2 Phase II design: Simon’s two-stage design. Design Notebook. 2023.https://bookdown.org/eugenesun95/designbook/phase-ii-design.html#simons-two-stage-design