Sample size calculation for the one-sample log-rank test

Abstract

In this paper, an exact variance of the one-sample log-rank test statistic is derived under the alternative hypothesis, and a sample size formula is proposed based on the derived exact variance. Simulation results showed that the proposed sample size formula provides adequate power to design a study to compare the survival of a single sample with that of a standard population.

Single-arm Phase II cancer survival trial designs

“…Hence, we reject the null hypothesis H 0 with one-sided type I error α if L 1 = W /υ̂ < − z 1−α , where z 1−α is the 100(1 − α) percentile of the standard normal distribution. Simulation results showed, however, that the OSLRT L 1 is conservative, even when the sample size is relatively large (Kwak and Jung, 2013; Sun et al, 2011; and Wu, 2015). …”

“…The exact variance of W has been derived for the sample size calculation (Wu, 2015). Let the exact mean and variance of W at the alternative be $E_{H_1}false(Wfalse)=\sqrt{n}\omega$ and Var H 1 ( W ) = σ 2 , respectively, where ω and σ 2 are given below.…”

“…By the central limit theorem, $false(W-\sqrt{n}\omega false)/\sigma$ is approximately standard normal distributed under H 1 . Under the alternative hypothesis, $${\hat{\upsilon }}^2=n^{-1}\sum _{i=1}^n{\int }_0^{\infty }{Y}_{i}false(tfalse)d{\mathrm{\Lambda }}_{0}false(tfalse)\to {\upsilon }_0={\int }_0^{\infty }Gfalse(tfalse)S_{1}false(tfalse)d{\mathrm{\Lambda }}_{0}false(tfalse).$$ Thus, the power of the OSLRT L 1 = W /σ̂ should satisfy the following equations: $$1-\beta =Pfalse(L_1<-z_{1-\alpha }false|H_{1}false)\simeq \mathrm{normal\Phi }true(-\frac{\sqrt{{\upsilon }_0}}{\sigma }{z}_{1-\alpha }-\frac{\sqrt{n}\omega }{\sigma }true).$$ Therefore, the required sample size for the test statistic L 1 is given by $$n=\frac{{(\sqrt{{\upsilon }_0}{z}_{1-\alpha }+\sigma z_{1-\beta })}^2}{{\omega }^2},$$ where ω = υ 1 −υ 0 and ${\sigma }^2={\upsilon }_1-{\upsilon }_1^2+2{\upsilon }_{00}-{\upsilon }_0^2-2{\upsilon }_{01}+2{\upsilon }_0{\upsilon }_1$, with υ 0 , υ 1 , υ 00 and υ 01 being given by the following equations (the derivation is given by Wu (2015) and also see equations (A3) to (A6) in the Appendix): $${\upsilon }_0={\int }_0^{\infty }Gfalse(tfalse)S_{1}false(tfalse)d{\mathrm{\Lambda }}_{0}false(tfalse),$$ $${\upsilon }_1={\int }_0^{\infty }Gfalse(tfalse)S_{1}false(tfalse)d$$…”

Single‐arm phase II trial design under parametric cure models

“…For the sample size calculation, the exact variance of W under the alternative has been derived by Wu [13]. Let the exact mean and variance of W at the alternative be $E_{H_1}false(Wfalse)=\sqrt{n}\omega$ and Var H 1 ( W ) = σ 2 , respectively, where ω and σ 2 are given below.…”

Weighted log‐rank test for time‐to‐event data in immunotherapy trials with random delayed treatment effect and cure rate

“…When only the cure rate was present, Boag 4 proposed a cure rate model to appropriately model the data. Significant generalizations, 5-10 computation programs, 11,12 and applications to clinical study designs [13][14][15][16][17] of mixture cure rate models have been extensively studied. In recent years, another type of cure rate model, the proportional hazards cure rate model (PHCRM), has been FIGURE 1 Ipilimumab with delay time and cure rate proposed 18 and applied.…”