Single-Arm Phase II Survival Trial Design

Jianrong Wu

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Print ISBN:
  9780367653491
  9780367653453
• eBook ISBN:
  9781000422481
• Edition: 1
• Topic: Medicine
• Subtopic: Biostatistics

Single-Arm Phase II Survival Trial Design provides a comprehensive summary to the most commonly- used methods for single-arm phase II trial design with time-to-event endpoints. Single-arm phase II trials are a key component for successfully developing advanced cancer drugs and treatments, particular for target therapy and immunotherapy in which time-to-event endpoints are often the primary endpoints. Most test statistics for single-arm phase II trial design with time-to-event endpoints are not available in commercial software.

Key Features:

• Covers the most frequently used methods for single-arm phase II trial design with time-to-event endpoints in a comprehensive fashion.
• Provides new material on phase II immunotherapy trial design and phase II trial design with TTP ratio endpoint.
• Illustrates trial designs by real clinical trial examples
• Includes R code for all methods proposed in the book, enabling straightforward sample size calculation.

1

Introduction of Single-Arm Phase II Trial Design

1.1 Why a Single-Arm Phase II Trial?

Cancer clinical trials investigate the efficacy and toxicity of experimental cancer therapies through four phases of clinical trials, designated phase I, II, III and IV. A phase III cancer clinical trial is usually a large-scale randomized study to compare the efficacy of a new or experimental treatment to that of the best current standard treatment or placebo, while a phase II cancer clinical trial is often conducted as a single-arm study to determine whether a new treatment has sufficient antitumor activities to warrant further investigation in a large-scale randomized phase III trial. However, a single-arm trial compares to a reference group from historical data and doesn't have a concurrent control group. Thus, the results from a single-arm phase II trial could be biased. However, a single-arm trial requires relatively small sample size and all patients are treated with the new treatment. Furthermore for patients with a rare life-threatening disease, the use of a control arm may be unethical or unfeasible. Therefore, single-arm phase II trials are still frequently used in cancer studies to provide some preliminary efficacy and toxicity assessment for new or experimental treatments.

1.2 Primary Endpoint of a Single-Arm Phase II Trial

The antitumor activity for evaluating cytotoxic compounds might be quantified by tumor response which is often categorized to be a binary endpoint as responder if the patient achieved a complete response (CR) or partial response (PR) or non-responder if the patient had stable disease (SD) or progressive disease (PD), where the response evaluation criteria is often based on response evaluation criteria in solid tumor (RECIST). For the trial design with tumor response rate as the primary endpoint, investigators identify the response rate (p₀) (CR or PR) of the standard treatment from historical or literature data as the null hypothesis, e.g., $p_0=30\%$. Therefore, for single-arm phase II trial design, p₀ is assumed known and is not subject to variation even though it may be estimated from historical or literature data or a reference level selected by the investigators. This is a fundamental difference between a single-arm phase II trial and a randomized phase II trial.

1.3 Hypothesis of a Single-Arm Phase II Trial

For a single-arm phase II trial with tumor response rate as the primary endpoint, the research hypothesis is if the new or experimental treatment can improve the tumor response rate for a target patient group. Therefore the research hypothesis is typically set to be as a one-sided hypothesis

$$\begin{array}{l}{H}_0:p\le p_{0}vs{H}_1:p>p_0,\hfill \end{array}$$

where p is the true response rate of the new treatment which is unknown. If the true response rate of the new treatment is less than or equal to p₀, the new treatment is not promising for further investigation and if the true response rate of the new treatment is greater than p₀, the new treatment is promising for further investigation in the future in a phase III trial.

1.4 Test Statistic and Study Design

To calculate the required sample size, investigators have to choose a response rate of the new treatment (p₁), e.g., $p_1=50\%$, where $p_1-p_0=20\%$ is a minimum clinical meaningful effect size to detect for the sample size calculation. For the study design, typically, the reference response rate p₀ is chosen to be a value of the probability of tumor response to the standard treatment for the same disease group, and p₁ or effect size $p_1-p_0$ is selected to identify a minimum clinically meaningful improvement over the historical value. To design the study, we have to select a test statistic to make the inference to the research hypothesis, e.g., reject null hypothesis H₀. A nature test statistic is $\widehat{p}=X/n$ which is an estimated probability of response rate for the experimental treatment, where X is the number of subjects who experienced response (CR or PR) and n is the total number of subjects in the study. However the distribution of $\widehat{p}$ is often skewed when sample size is small. To make the distribution more normal, we take an arcsin square root transformation to be $\text{arcsin}\sqrt{\widehat{p}}$. Using delta method, it can be shown that

$$\begin{array}{l}\text{var}(\text{arcsin}\sqrt{\widehat{p}})\simeq \frac{1}{4n},\hfill \end{array}$$

then, the standardized test statistic Z is given by

$$\begin{array}{l}Z=\sqrt{4n}(\text{arcsin}\sqrt{\widehat{p}}-\text{arcsin}\sqrt{p_0}),\hfill \end{array}$$

which is approximately standard normal distributed under the null hypothesis. A large observed value of Z indicates a large treatment effect, therefore, given a type I error rate α, we reject null hypothesis H₀ if $Z>z_{1-\alpha }$, where $z_{1-\alpha }={\Phi }^{-1}(1-\alpha )$ and $\Phi (·)$ is the cumulative distribution function (CDF) of standard normal.

To design a study, we need to calculate the sample size (n) or number of subjects required to have sufficient statistical power ($1-\beta$) to detect the treatment effect. The statistical power is the probability to reject null hypothesis H₀ when the alternative hypothesis H₁ is true. Under the alternative hypothesis $H_1:p=p_1(>p_0)$, the test Z is approximately normal with mean $\sqrt{4n}(\text{arcsin}\sqrt{p_1}-\text{arcsin}\sqrt{p_0})$ and unit variance. Thus, the study power $1-\beta$ satisfies the following equation:

$$\begin{array}{lll}1-\beta \hfill & =\hfill & P(Z>z_{1-\alpha }|H_1)\hfill \\ \hfill & \simeq \hfill & \Phi \{\sqrt{4n}(\text{arcsin}\sqrt{p_1}-\text{arcsin}\sqrt{p_0})-z_{1-\alpha }\}.\hfill \end{array}$$

By inverting the CDF $\Phi (·)$ from the above equation, we have

$$\begin{array}{l}\sqrt{4n}(\text{arcsin}\sqrt{p_1}-\text{arcsin}\sqrt{p_0})=z_{1-\alpha }+z_{1-\beta }.\hfill \end{array}$$

Solving n, we obtain the following sample size formula for the Z test

$$\begin{array}{l}n=\frac{{(z_{1-\alpha }+z_{1-\beta })}^2}{4{(\text{arcsin}\sqrt{p_1}-\text{arcsi...}}^{}}\hfill \end{array}$$

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. Introduction of Single-Arm Phase II Trial Design
• 2. Phase II Trial Design under Parametric Model
• 3. One-Stage Design Evaluating Survival Probabilities
• 4. Two-Stage Design Evaluating Survival Probabilities
• 5. One-Stage Design Evaluating Survival Distributions
• 6. Two-Stage Design Evaluating Survival Distributions
• 7. Phase II Immunotherapy Trial Design
• 8. Phase II Trial Design with GMI Endpoint
• 9. Bayesian Single-Arm Phase II Trial Design
• A. Probability of Failure under Uniform Accrual
• B. Asymptotic Distribution of Nelson-Aalen Estimate of the Cumulative Hazard
• C. Derivation Asymptotic Distribution of the OSLRT
• D. Derivation of Equations (6.8) and (6.9)
• E. Crossing Point for the Mixture Cure Model
• F. Derivation Asymptotic Distribution of the Score Test Q
• G. Generate Random Variables from GBVE Model
• H. Derivation Censoring Survival Distribution of GMI
• I. Proof of Monotonicity of the Posterior Probability
• J. Relationship between Frequentist and Bayesian Type I Error Rates
• K. R Code
• Bibliography
• Index