Single-Arm Phase II Survival Trial Design Under the Proportional Hazards Model

Abstract

For designing single-arm phase II trials with time-to-event endpoints, a sample size formula is derived for the modified one-sample log-rank test under the proportional hazards model. The derived formula enables new methods for designing trials that allow a flexible choice of the underlying survival distribution. Simulation results showed that the proposed formula provides an accurate estimation of sample size. The sample size calculation has been implemented in an R function for the purpose of trial design.

Two‐stage screened selection designs for randomized phase II trials with time‐to‐event endpoints

“…To derive the sample size for the SSD, we assume that a proportional hazard (PH) model $$S_g(t)=[S_{0}(t)]^{\delta _g}$$ is satisfied for each treatment arm $$g=A, B$$, where $$\delta _g$$ is the hazard ratio of treatment arm g versus the historical control arm. Under the local alternative hypothesis, it has been shown that the OSLRT $$Z_{g}=(E_g-O_g)/\sqrt {E_g}$$ is asymptotically normally distributed with mean $$\mu _g=-\sqrt {np}\log \delta _g$$ and unit variance, where n is the sample size per arm and p is the failure probability under the null hypothesis (Wu, 2017). Under the PH model, the hazard ratio $$\delta _g$$ can be estimated as follows (Wu, 2022): $$\begin{equation} \hat{\delta }_g = \frac{O_g}{E_g}, \;\;\; g=A, B.$$…”

“…To derive the sample size for the SSD, we assume that a proportional hazard (PH) model 𝑆 𝑔 (𝑡) = [𝑆 0 (𝑡)] 𝛿 𝑔 is satisfied for each treatment arm 𝑔 = 𝐴, 𝐵, where 𝛿 𝑔 is the hazard ratio of treatment arm 𝑔 versus the historical control arm. Under the local alternative hypothesis, it has been shown that the OSLRT 𝑍 𝑔 = (𝐸 𝑔 − 𝑂 𝑔 )∕ √ 𝐸 𝑔 is asymptotically normally distributed with mean 𝜇 𝑔 = − √ 𝑛𝑝 log 𝛿 𝑔 and unit variance, where 𝑛 is the sample size per arm and 𝑝 is the failure probability under the null hypothesis (Wu, 2017). Under the PH model, the hazard ratio 𝛿 𝑔 can be estimated as follows (Wu, 2022):…”

Two‐stage phase II survival trial design

“…The OSLRT was first introduced by Breslow, and its applications to the single‐arm phase II trial designs were discussed by Sun et al, Kwak and Jung, Wu, Schmidt et al, and Belin et al The study design based on the OSLRT requires each patient to be followed until an event occurs or until the end of study. In real practice, however, the full follow‐up information for a phase II trial is often difficult to obtain in the late period of trial, particularly when the accrual duration is long; obtaining the status of each patient within a restricted period is more realistic …”

“…Table also gives results for the comparison between proposed design to Belin's design under the exponential model. The proposed design provided an accurate power while Belin's design did not preserve power for some scenarios because Belin's design uses an approximate variance estimate that could either underestimate or overestimate the exact variance of the OSLRT …”

“…Furthermore, none of these designs are fully efficient. Finkelstein et al, Sun et al, Wu, and Schmidt et al proposed single‐stage designs by using the one‐sample log‐rank test (OSLRT) which evaluates the treatment effect based on the entire survival distribution. Kwak and Jung and Belin et al developed an optimal two‐stage design without and with restricted follow‐up, respectively.…”