Sample size re-estimation in stochastic curtailment tests with time-to-events outcome in the case of non-proportional hazards utilizing two Weibull distributions with unknown shape parameters

Abstract

Stochastic curtailment tests for phase II two-arm trials with time-to-event endpoints are traditionally performed using the log-rank test. Recent advances in designing time-to-event trials have utilized the Weibull distribution with a known shape parameter estimated from historical studies. As sample size calculations depend on the value of this shape parameter, these methods either cannot be used, orlikely underperform/overperform when the natural variation around the point estimate is ignored. We demonstrate that when the magnitude of the Weibull shape parameters changes, unblinded interim information on the shape of the survival curves can be useful to enrich the final analysis for re-estimation of the sample size. For such scenarios, we propose two Bayesian solutions to estimate the natural variations of the Weibull shape parameter. We implement these approaches under the framework of the newly proposed relative time method which allows non-proportional hazards and non-proportional time. We also demonstrate the sample size re-estimation for the relative time method using three different approaches (Internal Pilot study approach, Conditional Power, and Predictive Power approach) at the interim stage of the trial. We demonstrate our methods using a hypothetical example and provide insights regarding the practical constraints for the proposed methods.

1.1. Introduction

Two-arm randomized clinical trials are considered to be the gold standard and are often carried out in phase II/III clinical trial design. Many therapeutic areas such as oncology, immunology, and rare diseases use time-to-event (TTE) as a primary endpoint rather than continuous or binary outcomes. When designing clinical trials with TTE outcomes, sample size calculations are traditionally conducted assuming proportional hazards (PH) or in some cases naively assuming exponentially distributed survival times. However, if the underlying assumptions are violated, the traditional methods are not likely to perform well. Recent advances in literature in this area have seen the growth of methods proposed to handle the non-proportional hazards (NPH) scenario. Some of these are with: piecewise exponential model ¹, delayed effect ^2–4, RMST design ^5–9, max-combo test^10–12, Yang-Prentice model ^12–14, cure rate model ^3,15–19. Very recently, Phadnis & Mayo²⁰ have developed sample size calculation method for a fixed two-arm trial when the survival time in the two arms (control and treatment) follows two different Weibull distributions utilizing the concept of Relative Time (RT). In the most general setting of their RT framework, both non-constant hazard ratio, as well as non-constant time ratio, are permitted with some restrictions on the crossing of two survival curves. However, Phadnis & Mayo²⁰has assumed that the Weibull shape parameter is known or can be accurately estimated from historical studies. Since their sample size calculations are sensitive to this shape parameter, it is important that this parameter be estimated from previous studies with a reasonable level of accuracy. When such estimation is not reliable (as in the case of estimating the parameter from previous studies with small sample sizes), their proposed method may not be likely to perform well (the current study will be either overpowered or underpowered). In such scenarios, it is therefore important to offer adjustments to design to accommodate the initial lack of accurate knowledge of the Weibull shape parameter.

One solution, therefore, is to conduct an adaptive trial by incorporating a two-stage procedure – {i} make reasonable initial assumptions about an input (or nuisance) parameter and using this assumption calculate the sample size needed to have sufficient power to detect a clinically meaningful treatment effect, and, {ii} use the data collected at the interim to update our knowledge about the input (or nuisance) parameter and then use this updated knowledge to determine if the initially calculated sample size yields good operating characteristics and if not, to re-estimate the sample size to rectify the shortcomings. The second stage is referred to as sample size re-estimation method (SSR) and is one of the key features of a well-planned adaptive trial. A SSR is an adaptive design that allows one to reassess the sample size at any prespecified interim time point of the trial ²¹. Designing a phase II/phase III TTE trial with SSR mitigates the inaccuracy of the sample size calculations conducted at the start of the trial. Furthermore, SSR design allows us to modify the initial assumptions of the design parameter while maintaining statistical rigor of the trial ^22,23. Additionally, an adaptive design improves the effectiveness of the drug development process by better allocating the available resources and increasing the probability of trial success ²⁴. Recent guidelines published by the U.S Food and Drug Administration in 2010 for adaptive clinical trials for drug and biologics ²⁵ also encourage the utilization of adaptive design in confirmatory clinical trials. Analogously, other FDA brunches such as the Center for Drug Evaluation and Research (CDER), Center for Biologics Evaluation and Research (CBER), and the Center for Devices and Radiological Health (CDRH) have also published their guidelines and rules of adaptive trial in their respective area of research. SSR methods are also encouraged by the Data Monitoring Committee (DMC) ^26,27.

To conduct such trials by implementing SSR, several adaptive methods have been proposed in the literature. Traditionally, in literature, sample size re-estimation methods were mainly focused on continuous or binary endpoints. Methods for sample size re-estimation have been developed for continuous response variables with the assumption of normal distribution ^28–32. Sample size modifications based on observed treatment differences have been proposed by several researchers ^33–37. Studies showed that if the sample size adjustment is based on within-group variance, the type I error rate will not be inflated ^28,29,38,39. However, if the SSR is based on observed treatment difference, type I error rate could be inflated and an appropriate adjustment may require controlling the error rate ^23,30,34. Other common method used for SSR is Cui, Hung, and Wang (CHW) method ³⁵. The CHW test statistic is derived from a weighted combination of independent test statistics from each stage of the trial and control the type I error rate at a prespecified level ²¹. One of the major criticisms of the CHW test is the appropriate distribution of the weight since weighting in the late stage is not proportional to the sample size in that particular stage ²¹. Chen et al. ³⁷ showed classical Wald test statistics can be used instead of the CHW method if the conditional power is greater than 50% at interim stage to re-estimate the sample size and it also controls the type I error rate. Some other methods deployed are based on group sequential designs (GSD) or stochastic curtailment (SC) based methods. A SC test is used to calculate the probability of rejecting the null hypothesis at the end of the trial based on the observed data at the interim stage. Such information helps to decide whether the trial should be continued or stopped due to early evidence of futility or overwhelming efficacy ^27,40,41. Some of the most popular SC methods are conditional power (CP) ^27,40–43, predictive power (PP), and Bayesian predictive probability (BPP) ^{40,41,44–46}.

Sharma & Phadnis⁴⁷ extended the work of Phadnis & Mayo²⁰ to the domain of SC tests for phase II trials with TTE endpoints. They have shown how conditional power (CP), predictive power (PP) and Bayesian predictive probability (BPP) calculations can be done using the RT framework. However, similar to Phadnis & Mayo²⁰, the limitation of SC tests proposed by Sharma & Phadnis⁴⁷ is that the sample size calculations require the Weibull shape parameter in the control arm to be known putatively from historical studies. Therefore, in this paper, we aim to continue the statistical framework developed by Phadnis & Mayo²⁰ by considering the scenarios when no historical information is available to estimate of the control arm of the Weibull shape parameter at the design stage. By estimating this nuisance parameter at the interim stage using the data collected thus far, and, by re-adjusting the sample size, we propose to mitigate and resolve the limitations of some of the issues mentioned above. The idea of such two-stage clinical trial design is very common where the first stage is used to estimate the nuisance parameter to adjust in the later stage to maintain the power of a hypothesis test. Wittes & Brittain ²⁸ in their seminal paper called it as an internal pilot study (IPS) design. Under this design, an initial sample size is calculated at the design stage based on the prior estimation of the nuisance parameter and the sample size is recalculated based on the internal pilot study data. We will implement this approach to re-estimate the sample size in our manuscript. Furthermore, we also evaluate the CP and PP approach to recalculate the sample size based on observed treatment effect at interim stage.

The manuscript is organized as follows. After introducing some notation in section 1.2, we briefly review the Relative Time method (allowing both non-proportional hazard and non-proportional time approach) followed by a motivating example. In section 1.3, we propose sample size re-estimation method at a prespecified interim stage under fixed design. We also discuss sample size adjustment under conditional and predictive power scenarios.Section 1.5 is outlined for a simulation study to assess the operation characteristics of the proposed methods. Finally, in section 1.6, we provide a brief discussion of the topics presented in this manuscript.

1.2. Concept of Relative time framework

1.2.1. Motivating Example

In this paper, we consider the examples described by Phadnis & Mayo²⁰ and have shown how we can use our proposed methods to perform the SSR calculations. Although those magnitudes of the treatment effects mentioned in these examples are hypothetical, they represent the real-life scenarios in designing clinical trials at our academic institution. Let us consider a two-arm randomized cancer trial where the researcher expects that the experimental arm will show a clinically meaningful improvement in the median progression-free survival (PFS) over time compared with the control group. In terms of design perspective, the improvement for the 10^th percentile of PFS will be a factor of 1.5 and the improvement for the 90^th percentile of PFS will be by a factor of 2.0 which showed the benefit of the treatment is not an instantaneous rather gradual improvement over time. In the second example, we are considering a hypothetical surgical intervention trial where we are comparing the benefit of the new surgical procedure vs. standard of care. We are assuming that benefit of the surgical intervention is considerably higher at the beginning than the standard of care, but the improvement will wane off over time. This scenario can be reflected as the improvement for the 10^th percentile of PFS will be a factor of 2.0 and the improvement for the 90^th percentile of PFS will be by a factor of 1.5. The effect size can be defined as the improvement in the median PFS. A graphical representation of the above scenarios is presented in figure 1. This figure is adopted from Phadnis & Mayo²⁰to make easier to explain the above scenarios. The left panel of this figure illustrates the Kaplan Meier (KM) curve when $RT\left(p_1=0.10\right)=1.50,RT\left(p_2=0.90\right)=2$ and right panel in this figure depicts the scenarios when $RT\left(p_1=0.10\right)=2,RT\left(p_2=0.90\right)=1.5$.

Figure 1:

Plot of Relative Time method under two scenarios: Reprinted from Phadnis and Mayo (2021), Biometrical Journal 63 (2021), Pg. 1409. Copyright [2021] by John Wiley & Sons, Inc. Reprinted with permission.

1.2.2. Notation and Preliminaries

We consider a two-arm randomized clinical trial comparing the experimental treatment arm with a control arm ($i=0,1$ denote the control and treatment arm). Let us assume that each subject from i^th group, $T_i$ denotes the event time which follows the Weibull distribution with unknown shape and scale parameter. Subjects are accrued over a period of length $t_a$ and the length of follow-up period denoted as $t_f$. For simplicity, we assumed that the entry time is uniformly distributed on $\left[0,t_a\right].$Also, assume administrative censoring incorporated at a prespecified time $\left(\epsilon \right)$ and random loss to follow-up denoted as $\rho$. Therefore, $v_{ij}=\min \left(\mathit{max}\left(0,\epsilon -t_a\right),\rho \right)$ representsthe censoring time due to administrative censoring or random loss to follow-up. Let us assume that during the accrual phase of the trial, $n_j$ subjects of the i^th group are enrolled in the study. If we denote the observed time as $X_{ij}$ then,

$$X_{ij}=min\left(T_{ij},v_{ij}\right)\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\delta }_{ij}=1_{(T_{ij}<v_{ij})},j=1,\dots .,n_j,i=\mathrm{0,1}.$$

Survival status indicator ${\delta }_{ij}=0$ indicates the censored observations and ${\delta }_{ij}=1$ indicate an event. $T_{ij}$ represents the true event time from the Weibull distribution. Some popular parametric TTE models assume that the underlying survival time follows exponential, piecewise exponential and/or Weibull distribution. The exponential distribution assumes constant hazard over time and most commonly used in designing randomized clinical trials. However, in recent years, TTE trial with Weibull distribution became popular due to mathematical tractability and flexibility of Weibull distribution ^20,47–50. We also assume that each subject’s survival time on each arm is an independent and identically distributed (IID) observation and independently follows Weibull distribution and the probability density function of two- parameter Weibull distribution can be written as

$$f\left(t|\theta ,\beta \right)=\frac{\beta }{{\theta }^{\beta }}{t}^{\beta -1}{\mathrm{e}}^{-{\left(\frac{t}{\theta }\right)}^{\beta }};\theta ,\beta >0,t>0$$ #(1.1)

Here, $\theta$ is the scale parameter and $\beta$ is the shape parameter of the Weibull distribution with $\theta ={\theta }_i$ and $\beta ={\beta }_i$ ($i=\mathrm{0,1}$ denote the control and treatment arm). Weibull shape parameter determines the shape of the hazard function i.e., when the shape parameter is positive ($\beta >1$) indicates the hazard increasing over time. Similarly, when the Weibull shape parameter is negative ($\beta <1$) indicates the hazard decreasing over time. In special case, $\beta =1$ (exponential distribution) represents the constant hazard over time.

1.2.3. Sample Size estimation using Relative Time method

The methodological framework for two-arm RCT with TTE outcomes in the case of non-proportional hazards and non-proportional time developed by Phadnis & Mayo²⁰ and is briefly revisited in Appendix A.1–A.3. While the full details are available in their paper, Appendix A.1–A.3 provides a quick overview of the Relative Time concept, the modeling framework setting up the hypothesis, and sample size calculations accounting for administrative censoring and loss to follow-up.

Under this design, for a given allocation ratio r, the required number of events in two arm trial with one-sided hypothesis can be written as $d_0={\left[\frac{\left(Z_{\omega }+Z_{1-\alpha }\right)}{\mathrm{l}\mathrm{n}\left\{RT\left(p_{mid}\right)\right\}}\right]}^2\left(\frac{1}{r{\beta }_1^2}+\frac{1}{{\beta }_0^2}\right)\mathrm{a}\mathrm{n}\mathrm{d}{d}_1=r{d}_0$ where $\mathit{\alpha }$ is the type I error for the one-sided test and $\mathit{\omega }$ is the power. Here, ${\mathit{d}}_{\mathbf{0}}$ and ${\mathit{d}}_{\mathbf{1}}$ are the number of events in the control and treatment arm, respectively. Furthermore, we can readjust the sample size for an administrative censoring $\left(\mathit{v}\right)$ and to account for loss due to drop-out (right-censored observations), we can inflate the sample size calculated after by simply dividing by 1 minus the drop-out rate. Thus, for a drop-out rate $\mathit{\rho }$, the final sample size in the two study arms can be calculated as ${\mathit{n}}_{\mathbf{0}}=\frac{{\mathit{d}}_{\mathbf{0}}}{\left(\mathbf{1}-\mathit{\rho }\right)\mathit{v}}$ and ${\mathit{n}}_{\mathbf{1}}=\mathit{r}{\mathit{n}}_{\mathbf{0}}$ where $\mathit{r}$ is the allocation ratio.

1.3. Methods

1.3.1. Initial sample size determination for the shape parameter of the control arm

It should be noted that the two-arm fixed design calculations of Phadnis & Mayo²⁰ require a reasonably accurate estimate of ${\beta }_0$, the control arm shape parameter of the Weibull distribution. Then, given the values of $p_1$, $p_2$, $RT\left(p_1\right),RT\left(p_2\right)$ and median survival time in the control arm (see Appendix A.1–A.3), the parameters ${\beta }_1$, ${\theta }_0$, and ${\theta }_1$ can be estimated leading to the sample size calculation using the formulas mentioned in Section 1.2.2. Given that we do not have reliable prior information regarding the distribution of the survival time, it is reasonable to assume that survival time in the control arm follows an exponential distribution since the exponential distribution is popularly used to model survival time in many clinical trials. That is, we assume that if a priori we do not have reliable knowledge of the Weibull shape parameter, then a safe choice is to assume the Weibull shape parameter ${\beta }_0=1$ leading to an exponential distribution with constant hazard in the control arm. This point estimate is used to calculate the initial sample size for the study (at the design stage). Then, at a prespecified interim stage, when data has been collected on a fraction of these subjects, we ask ourselves “How can we correct for the initial misspecification of ${\beta }_0$, given that the data observed at the interim comes from a Weibull distribution with a different value for ${\beta }_0$”? This question leads us to the elicitation of a prior distribution for ${\beta }_0$ with the idea that this prior distribution can be combined with the information gained from the data collected at the interim to obtain the posterior distribution of ${\beta }_0$ which in turn can be used for sample size re-estimation.

1.3.2. Prior elicitation of the Weibull shape parameter

In this section, we will discuss the prior elicitation of the Weibull shape and scale parameter of control arm. When a reliable estimate of the Weibull shape parameter is not available from previous clinical trials or historical studies, we may consider an independent prior specification of the Weibull shape and scale parameter ^16,51. Thus, we may choose a gamma prior for ${\beta }_0~G({\alpha }_0,{\delta }_o)$, such that the mean and mode of the Weibull shape parameter centered around 1. Here, $G\left({\beta }_0\right|{\alpha }_0,{\delta }_o)$ denote a gamma distribution with unknown shape parameter ${\alpha }_0$ and rate parameter ${\delta }_0.$ We can choose unknown hyperparameter $({\alpha }_0,{\delta }_o)$ such that the mode of the gamma distribution is approximately centered around 1 and the distribution has some reasonable variance. By reasonable, it is meant that if we want to choose a non-informative prior, we should choose the $({\alpha }_0,{\delta }_o)$ such that the variance is very large. On the other hand, appropriate choice of $({\alpha }_0,{\delta }_o)$ with smaller variance provides informative prior of the control Weibull shape parameter. The smaller the variance, stronger is our prior belief that the survival times in the control arm come from an exponential distribution, and vice versa. We choose $({\alpha }_0,{\delta }_o)$ for informative and weakly informative prior with some trial-and-error method and by plotting the gamma distribution under this parameterization, $\frac{{\alpha }_0-1}{{\delta }_0}={\beta }_{prior}\approx 1$ and $\frac{{\alpha }_0}{{\delta }_0}={\beta }_{prior}\approx 1$ and the prior standard deviation can be calculated by $\frac{\sqrt{{\alpha }_0}}{{\delta }_0}$. We can also check whether the 95% of the highest posterior density (HPD) of the Weibull shape parameter contains 1. We use a vague prior with independent normal distribution for the control Weibull scale parameter $\left({\theta }_0\right)$ with ${\theta }_0~N\left({\mu }_0,{\tau }_0^2\right)$. We can specify the Weibull scale parameter based on median PFS of the control arm. We know that Weibull scale parameter has direct relationship with the median PFS and can be defined as ${\theta }_0=\frac{m}{{\log \left(2\right)}^{1/\beta }}$; here $m$ is the median progression free survival time. Also, the parametrization of the Weibull pdf described in our paper is slightly different from the OpenBugs software ⁵² parameterization, therefore, we use the following transformation, ${\omega }_0=log\left(\frac{1}{{(\theta }_0)^{\beta }_0}\right)$ to define the scale parameter where ${\omega }_0~N\left(\mathrm{0,0.001}\right)$. In Bayesian modeling, OpenBugs uses precision as a parameter to specify the normal distribution instead of variance.

1.3.3. Generating the posterior of the Weibull Shape and Scale Parameters

When the Weibull scale (${\theta }_0)$ and shape (${\beta }_0)$ parameter is unknown, there are no suitable joint prior distribution available. For this purpose, let us assume that the independent prior distribution for control scale and shape parameter as ${\theta }_0~N({\mu }_0,{\tau }_o^2)$ and ${\beta }_0~G({\alpha }_0,{\delta }_o)$, then the joint posterior distribution of $\pi \left({\theta }_0,{\beta }_0\right)$ can be expressed as

$$\pi \left({\theta }_0,{\beta }_0|D\right)\propto L\left({\theta }_0,{\beta }_0|D\right)\pi \left({\theta }_0|{\mu }_0,{\tau }_o^2\right)\pi \left({\beta }_0|{\alpha }_0,{\delta }_o\right)$$ #(1.2)

where ${\theta }_0$, ${\beta }_0$ indicates the parameter of interest and D indicates the observed interim data. Since the posterior distribution is complex and no closed form solution exists, we can use the Markov Chain Monte Carlo (MCMC) simulation method to draw posterior samples. In this case, we have used Gibbs sampling algorithm to draw the posterior sample. Gibbs sampling is the simplest and most easily implemented method to conduct MCMC simulations. Therefore, we have used Gibbs sampling implemented in OpenBugs software and integrated it within R environment to deduce the statistical inference regarding the unknown shape parameter. Although there are several R packages available to do such calculations, we have used “R2OpenBUGS” package ⁵³ to tackle this issue. We checked the convergence of the model parameters and discarded some initial samples as burn-in. We have implemented two approaches to calculate the unknown Weibull shape parameter (${\beta }_0$) of the control arm based on posterior MCMC samples.

1.3.3.1. Maximum A posteriori (MAP) estimation

The first approach is to estimate the Maximum A Posteriori (MAP). A Maximum A Posteriori (MAP) estimator is a Bayesian based approach that maximizes the posterior distribution of the model parameter given the observed data. A MAP estimator equal to the mode of the posterior distribution and provides an important summary statistic of the posterior samples. MAP estimator can be represented as

$${\widehat{\beta }}_{MAP}=\arg \underset{\beta }{\max }\pi \left({\theta }_0,{\beta }_0|D\right)=\arg \underset{\beta }{\max }L\left({\theta }_0,{\beta }_0|D\right)\pi \left({\theta }_0|{\mu }_0,{\tau }_o^2\right)\pi \left({\beta }_0|{\alpha }_0,{\delta }_o\right)$$ #(1.3)

This method provides an alternative probabilistic framework to the maximum likelihood-based approach, except instead of maximizing the likelihood, we are maximizing the likelihood by incorporating prior knowledge of our estimate.

1.3.3.2. Utilizing the Full posterior distribution

Second approach is to utilize the full posterior distribution instead of point estimator (MAP) to estimate the control Weibull shape parameter (${\widehat{\beta }}_{full}$ This approach takes into consideration the variability of the posterior samples, however computationally it can turn out to be very time consuming. We used weighted average of the entire posterior samples to calculate the Weibull shape parameter and subsequently used it to recalculate the sample size. Here, the updated shape parameter can be parameterized as ${\widehat{\beta }}_{full}^w=\frac{\sum _{k=1}^n{w}_k{\beta }_k}{\sum _{k=1}^n{w}_k},$ where, ${\beta }_k$ represents the posterior samples of $\pi ({\widehat{\theta }}_0,{\widehat{\beta }}_0|D)$ and $w_k$ represents the frequency of the posterior samples.

1.3.4. Sample size Re-estimation using the IPS Approach

Here, we have used the traditional internal pilot study approach (IPS) to re-estimate the sample size at a prespecified time point. However, internal pilot study approach only allows for upward adjustment to the sample size ^28,54. In survival analysis, this seems to be a reasonable assumption as also suggested by Chiang ⁵⁵. In this paper, we consider conducting only a single preplanned interim analysis for the SSR calculations. We compute the estimated variance of the control and treatment arm, $\widehat{{\sigma }_0^2}$ and $\widehat{{\sigma }_1^2}$ using the updated Weibull shape parameter $\widehat{{\beta }_0}$ and $\widehat{{\beta }_1}$ as discussed previously based upon the interim data accumulated so far. The SSR procedure is described in the following steps:

1. A proper statistical plan should be outlined by mentioning that sample size re-estimation will be conducted at a prespecified time point.

2. Based on the expertise knowledge of clinician/researcher, sample size calculation should be conducted as accurately as possible. In our design, since we did not have any prior knowledge of the shape parameter or the variability of the variance, we initially designed the study under the assumption of exponential distribution and specify the effect size under the relative time framework.

3. Using the sample size estimated in step 2, we will now begin the trial with prespecified accrual and follow-up rate. Also, we need to consider administrative censoring and random loss to follow-up.

4. At interim stage, based on unblinded observed data with the method described above, we can calculate the control arm Weibull shape and scale parameter (${\theta }_0\approx \widehat{{\theta }_0},{\beta }_0\approx \widehat{{\beta }_0}$) and subsequently be able to calculate the treatment arm shape and scale parameter ${\theta }_1\approx \widehat{{\theta }_1},{\beta }_1\approx \widehat{{\beta }_1})$ using the relationship between control and treatment group Weibull parameters as illustrated in the relative time methodology²⁰. Furthermore, we can calculate the pooled variance by estimating the control and treatment arm variance parameter $({\sigma }_0^2\approx \widehat{{\sigma }_0^2},{\sigma }_1^2\approx \widehat{{\sigma }_1^2})$

5. Based on the estimates of the nuisance parameters at the interim stage, the trial is re-evaluated to determine the expected number of events given the fixed planned sample size. If the interim estimates of the sample size are consistent with the current estimates, no additional sample size adjustment will be required.

6. If the interim estimates are different from the initial estimates at the design stage, we would require to re-estimate the sample size with the updated estimates. Then the number of additional events would require, say $d^{\mathrm{*}}$ will be $d^{\mathrm{*}}=\max \left(d,\widetilde{d^{\mathrm{*}}}\right)-d,$ where $\widetilde{d^{\mathrm{*}}}$ is the new event calculated at the interim stage and $d$ be the fixed sample design event number. Therefore, the total required event is $D=d+d^{\mathrm{*}}$.

7. Based on the newly estimated event number, we can recalculate the sample size by accounting for administrative censoring ($v$) and drop-out rate ($\rho$). The final analysis will be performed at the end of the trial by utilizing the data for all the N subjects.

1.3.5. Sample size Re-estimation using the Conditional power (CP) and Predictive power (PP)

Conditional power (CP) was first proposed by Lan et al. ⁴² in their seminal paper. CP is defined as the conditional probability of the final analysis will have a significant result at the end of the trial given the observed interim data ^27,40,41,56.

Detailed derivation of CP and PP utilizing the relative time (RT) method developed by Sharma & Phadnis ⁴⁷. Let, at an interim stage $m,Z_m$ denote the interim test statistic. The CP at interim stage $m$ is defined by $CP\left(\varphi \right)=\mathrm{P}\mathrm{r}\{Z_N\mathrm{w}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{r}\mathrm{e}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}{H}_0\left|Z_m\right\}$. Here, to follow the conventional mathematical notation of stochastic curtailment-based tests and ease of readability, we are defining $\varphi$ as a relative time effect size $RT\left(P_{mid}\right)$ and $Z_m(m=1,2,3,\dots ,N-1)$ plays the role of test statistics $Q\text{'}$ which is defined in the Appendix A.1–A.3. The test continues to final stage N where it rejects $H_0$ if $Z_N>Z_{1-\alpha }$ and accept otherwise. The conditional distribution of $Z_N$ given $Z_m$ is approximately follows normal distribution with $Z_N|Z_m~N(Z_N\sqrt{\frac{I_m}{I_N}}+\left\{\frac{\varphi \left(I_N-I_m\right)}{\sqrt{I_N}}\right\},1-\frac{I_m}{I_N})$. Hence, the CP at stage $m$ for rejecting $H_0$ about a parameter $\varphi$ at the end of the study, given $Z_m$ is

$$\begin{array}{c}C{P}_m\left(\varphi \right)=\mathrm{\Phi }\left(\frac{Z_m\sqrt{I_m}-Z_{1-\alpha }\sqrt{I_N}+\varphi \left(I_N-I_m\right)}{\sqrt{I_N-I_m}}\right),m=1,2,3,\dots ,N-1\#\end{array}$$

Here, $I_m$ = information time at stage $m=\frac{1}{Variance}=\frac{1}{{\sigma }_m^2}$ and $I_N$ = information time at stage $N=\frac{1}{{\sigma }_N^2}$. Under the current trend, utilizing the concept from Brownian motion framework ⁵⁷, the effect size can be written as ${\varphi }_T=\frac{B_t}{t}=\frac{Z_t\surd t}{t}=\frac{Z_t}{\surd t}\approx \frac{Z_m}{\surd f}$. Here, we are utilizing the linear relationship of interim test statistic and the concept of Brownian motion: $B_t=Z_t\surd t$ (assume that the future data derives from the same distribution given observed data so far, CP under the current trend becomes $C{P}_m\left({\varphi }_T\right)=\text{Φ}\left(\frac{Z_m\sqrt{I_m}-Z_{1-\alpha }\sqrt{I_N}+{\varphi }_T\left(I_N-I_m\right)}{\sqrt{I_N-I_m}}\right)$.

On the other hand, Predictive power (PP) was introduced by Spiegelhalter et al. ⁴⁴ to monitor the trial. PP can be calculated as the conditional power function (frequentist component) is averaged over with posterior distribution function (Bayesian component) of $\varphi$ given its estimate $\widehat{Z_m}$ at the interim stage $m$ ^40,41,58.

Let $Z_m$ and $Z_N$ denote the test statistics computed at the interim and final stage. The mathematical formula for the predictive power can be expressed as:

$$PP\left(\varphi \right)=\int C{P}_m\left(\varphi \right)\pi (\varphi \widehat{Z_m})d\varphi \#$$

where $C{P}_m\left(\varphi \right)$ is the CP function described above and $\pi (\varphi \widehat{Z_m})$ is the posterior density of $\varphi$ given its estimate $\widehat{Z_m}$ at the interim stage $m$. Under uniform prior distribution, the above equation can be written as $PP(\varphi )=\mathrm{\Phi }\left\{\frac{z_m-\sqrt{t}{z}_{1-ϵ}}{\sqrt{1-t}}\right\}$. We would like to refer to Sharma & Phadnis⁴⁷ manuscript for detailed derivation for CP and PP based approaches.

The general procedure consists of the following steps and associated decision rules. A similar procedure can be applied to either CP or PP based approach.

1. A proper statistical plan with prespecified upper and lower Conditional and Predictive power should be outlined by mentioning that sample size re-estimation will be conducted at a prespecified time point. Based on the expertise knowledge of clinician/researcher, sample size calculation should be conducted as accurately as possible.

2. Begin the trial and estimate the nuisance parameters (treatment effect, Weibull shape and scale parameters, pooled variance etc.) and test statistics at the interim time (i.e., when $d_0$ and $d_1$ events are reported at the prespecified time in the control and treatment group, respectively).

3. Conduct stochastic curtailment tests (CP or PP) based on the observed data at this interim stage.

4. If the CP/PP is less than a prespecified level, we can utilize numerical optimization algorithm to find the minimum number of events to achieve that prespecified CP/PP level. We can use either equation described in section 1.3.5 (depends on which method will be used) to estimate the events $\tilde{d}$ to achieve $CP\left(\varphi \right)=C{P}_u\left(\varphi \right)$ or $PP\left(\varphi \right)=P{P}_u\left(\varphi \right)$. In our paper, we use binary search algorithms based on the modification of the bisection method to find the minimum events required to achieve the targeted upper $C{P}_u$ or $P{P}_u.$ Here, $C{P}_u$ or $P{P}_u$ is the desirable CP/PP for the trial assuming the treatment trend estimated at the interim stage will continue until the end of the trial.

5. If $\tilde{d}<d_{min}$, reset $\tilde{d}=d_{min}$, where $d_{min}$ is the minimum events required as per with the fixed protocol design. On the other hand, $\tilde{d}>d_{max}$, reset $\tilde{d}=d_{max}$ where $d_{max}$ is the maximum affordable event for the trial.

6. If $CP\left(\varphi \right)<C{P}_{min}\left(\varphi \right)$ or $PP\left(\varphi \right)<P{P}_{min}\left(\varphi \right)$ then it can be considered as an unfavorable zone. This indicates the interim result is disappointing and it is not worth increasing the sample size. Here, $C{P}_{min}\left(\varphi \right)$ is the minimum acceptable CP for continue the trial. If $C{P}_{min}\left(\varphi \right)\le CP\left(\varphi \right)<1-\beta$, then it can be considered as a promising zone and the trial should continue with a newly re-estimated sample size (event) of $d_{min}<\tilde{d}<d_{max}.$ If $CP\left(\varphi \right)\ge 1-\beta$, then the interim result indicates that the trial favors the treatment and there is no need to adaptively increase the sample size. Here, $1-\beta$ is the pre-specified threshold level for the trial usually between 0.8 – 1.00. Similar mechanism can be adopted for PP based approach.

7. Based on the newly estimated event number, we can recalculate the sample size by accounting for administrative censoring ($v$) and drop-out rate ($\rho$). The final analysis will be performed at the end of the trial by utilizing the data for all the N subjects.

1.5. Simulation Study

1.5.1. Study Set-up

We carried out a simulation study to assess the operating characteristics of the sample size re-estimation method (SSR) under the relative time framework as described earlier. Data were generated by simulating two distinct Weibull distributions based on RT framework, each representing the treatment and control arms. Specifically, we set the median PFS of 4 months in the control arm while maintaining a censoring rate of 20%. Scale parameter of Weibull distribution was calculated for control arm is ${\theta }_0=\frac{4}{{\log \left(2\right)}^{\frac{1}{\beta }}}$ represent a median PFS time at 4 months. Treatment arm scale $({\theta }_1$) and shape $({\beta }_1$) parameter were calculated based on the linear relationship of control and treatment arm as specified in the RT method. We used R for statistical software (Version: 1.3.1073) to carry out the simulation and Openbugs software was implemented for Bayesian calculation. The design parameters of the relative time framework are described below. We will consider four different scenarios under the relative time method:

1. $RT\left(p_1=0.10\right)=1.52,RT\left(p_2=0.90\right)=1.98$ implying that at the 10^th percentile the treatment arm has progression-free survival by a factor of 1.52 whereas at the 90^th percentile the PFS in the control arm will be 1.98 which indicates that the gradual improvement of treatment effect from 10^th percentile to 90^th percentile rather than an instantaneous effect.

2. Similarly,$RT\left(p_1=0.10\right)=2,RT\left(p_2=0.90\right)=1.5$ implying a gradual decline in longevity over time in both treatment and control arm

3. Likewise, we also consider the scenarios where improvement of PFS will be at $RT\left(p_1=0.25\right)=1.667,RT\left(p_2=0.75\right)=1.5$ and

4. $RT\left(p_1=0.25\right)=1.50,RT\left(p_2=0.75\right)=1.667$.

Assume that a fixed sample two-arm trial is designed with no prior knowledge of the Weibull shape parameter. Therefore, under this assumption with scenario 1, with a uniform accrual rate of 12 months and follow-up rate of 12 months, with assuming no drop-out rate, we would require 92 subjects to detect 80% power with a type I error rate of 5%. The fixed sample size varies under different scenarios as mentioned in (I-IV). We have considered two design options: (a) Fixed sample size design by allowing for readjustment at interim stage (b) sample size re-estimation using the CP and PP approach based on observed data at an interim stage. The effect of type I error and power was also assessed under the misspecification of the shape parameter. In each case, 10,000 simulations were performed.

1.5.2. Results

Let us assume that a fixed sample two-arm design is designed using the concept of RT framework discussed in Appendix A.1–A.3 to test the hypothesis of $H_0:R{T}_{mid}<1$ vs. $H_a:R{T}_{mid}>1$ with type I error rate was set to 5%, and the empirical power is 80%. Simulations were conducted under the initial assumption of exponential distributed survival time (as discussed in Section 1.3.1) to assess the empirical power and type I error. We also evaluated the impact of different priors for Weibull shape parameters. For informative priors, we chose the Weibull shape parameter as $G({\alpha }_0=101,{\delta }_0=100)$ which has mode centered around 1 and transformed scale parameter as $N({\mu }_0=0,{\tau }_0=0.001)$. Likewise, for weakly informative priors, we chose the Weibull shape parameter as $G({\alpha }_0=41,{\delta }_0=40)$ and transformed scale parameter as $N({\mu }_0=0,{\tau }_0=0.001)$. Table 1 displays the sample size re-estimation calculations using the IPS approach outlined in section 1.3.3. Here, the maximum a posteriori (MAP) approach was utilized with informative and weakly informative prior to calculating the Weibull shape parameter of the control arm. Let us consider the example described in scenario 1 with $RT\left(p_1=0.10\right)=1.52,RT\left(p_2=0.90\right)=1.98$ and the investigators have anticipated a uniform accrual rate. Assuming exponentially distributed survival times, we require 82 events with a sample size of 92 to achieve desired power (80%). Suppose an interim analysis was conducted based on the IPS approach with the data monitoring committee (DMC) recommendation at a prespecified calendar time (18 months). For simplicity, we assume that all the participants have been accrued (12 months) and followed up for 6 months. We generated a sample size of 92 with various shape parameters, as shown in Table 1. We want to observe the required sample size (events) in case of the prior assumption that the Weibull shape parameter is grossly mis-specified at the interim stage. Suppose the true Weibull shape parameter was 0.25 (decreasing hazard), in that case, we can see from Table 1 that with the MAP approach (Informative prior was used), the estimated interim shape parameter was estimated as 0.462, and the expected event size is almost four-fold that of the original design (IPS approach was used). In this context, we would require an additional 306 events to achieve the desired power. Such an increase in sample size/event would put a very substantial practical burden to continue the study. If we choose weakly informative prior, we will require an additional 591 events to achieve the desired power with the re-adjusted shape parameter of 0.355. As we can see, if the shape parameter is close to the protocol specified values (Weibull shape parameter $\beta =1$), the expected event/sample size is slightly greater than the original fixed design. However, this can be easily attainable from a practical standpoint. As expected, if the Weibull shape parameter is greater than 1 (increasing hazard with relatively shorter survival time), we observed a minimal increase in desired event size/sample size than the original hypothesized sample. In this case, it would be reasonable to take a conservative approach to continue the trial if ethically permissible rather than terminate it due to efficacy. We can also see from Table 2 that the empirical type I error rate is still preserved at the nominal level even if the Weibull shape parameter is mis-specified. On the other hand, power is slightly decreased compared to that of the original fixed sample design and is affected by the misspecification of the shape parameter. Table 2 also displays the absolute relative bias ($ARB$) and coverage probability ($Cov$) under misspecification of Weibull shape parameter. We have observed similar trends of the empirical type I error rate under stochastic curtailment (CP and PP) based sample size re-estimation approaches, although in some cases slightly inflated as the IPS approach. Table 3–5 displays the expected estimated event size under CP and PP approaches. Under scenario 1 mentioned above with ${\beta }_0=0.25$, the $C{P}_T$ approach with informative prior provides the re-estimated number of events is 213, which is lower than the IPS approach (under current trend). However, if we choose $CP(H_a)$, we will require 401 more events to achieve 80% CP. Likewise, if we choose the weakly informative prior, we will require 291 additional events to achieve the desired 80% $C{P}_T$ given the current trend continues at the end of the study. Similar strategy can be adopted for PP-based approach and the results are displayed in Table 5. Moreover, one might utilize the full posterior samples instead of the MAP approach as shown in Table 6–7. The estimated Weibull shape parameter by using the full posterior samples is similar to the MAP-based approach; however, it is computationally intensive to calculate and would require a substantial amount of time. Although the interim shape parameter computed using these two approaches provides slightly different values, the conclusion drawn based on these methods is quite similar. Therefore, it seems that for most scenarios, one can easily adopt the MAP approach to estimate the unknown Weibull shape parameter ($\beta$) rather than using the full posterior sampling-based method. We also considered the above scenarios with unequal allocation ratio for IPS, CP, PP methods as shown in supplementary table 1 and 2 in appendix A.4. As expected, RT method with unequal allocation ratio would require a higher sample size than the equal allocation given similar design criteria.

Table 1:

Sample size re-estimation for the relative time method using the IPS approach with r=1 (equal allocation ratio), $\alpha$ = 0.05, one-sided test, power 80%, accrual time 12 months and follow-up time 6 months respectively and control median PFS = 4 months. Unknown Weibull control shape parameter was calculated using the MAP approach with informative and weakly informative prior information of the shape parameter.

							Informative Prior			Weakly Informative Prior
RT	$p_0$	$p_1$	$D_f$	$N_f$	$D_m$ $(C_m/T_m)$	${\beta }_0$	${\widehat{\beta }}_{0(MI)}$	$\widehat{{\beta }_1}$	$\tilde{D}(D_{\mathrm{*}})$	${\widehat{\beta }}_{0(MW)}$	$\widehat{{\beta }_1}$	$\tilde{D}(D_{\mathrm{*}})$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	82	92	53(28/25)	0.25	0.462	0.444	306(359)	0.355	0.333	591(644)
	0.10	0.90	82	92	59(32/27)	0.50	0.699	0.659	101(160)	0.606	0.576	152(211)
	0.10	0.90	82	92	65(36/29)	0.75	0.861	0.802	42(107)	0.809	0.757	56(121)
	0.10	0.90	82	92	70(40/30)	1.00	0.989	0.911	12(82)	0.987	0.910	13(83)
	0.10	0.90	82	92	74(42/32)	1.25	1.099	1.004	---	1.148	1.045	---
	0.10	0.90	82	92	77(44/33)	1.50	1.192	1.081	---	1.296	1.167	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	86	94	54(28/26)	0.25	0.465	0.486	356(410)	0.359	0.371	640(694)
	0.10	0.90	86	94	61(33/28)	0.50	0.711	0.762	110(171)	0.622	0.661	165(226)
	0.10	0.90	86	94	68(37/31)	0.75	0.885	0.965	41(109)	0.843	0.915	53(121)
	0.10	0.90	86	94	74(40/34)	1.00	1.029	1.138	6(80)	1.043	1.155	4(78)
	0.10	0.90	86	94	79(43/36)	1.25	1.155	1.294	---	1.234	1.394	---
	0.10	0.90	86	94	82(45/37)	1.50	1.267	1.437	---	1.413	1.627	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	124	138	80(42/38)	0.25	0.403	0.392	646(726)	0.323	0.316	1043(1123)
	0.25	0.75	124	138	90 (48/42)	0.50	0.649	0.622	195(285)	0.575	0.554	271(361)
	0.25	0.75	124	138	99(54/45)	0.75	0.834	0.789	76(175)	0.793	0.753	94(193)
	0.25	0.75	124	138	107(59/48)	1.00	0.988	0.927	19(126)	0.985	0.924	20(127)
	0.25	0.75	124	138	114(63/51)	1.25	1.123	1.044	---	1.170	1.085	---
	0.25	0.75	124	138	119(65/54)	1.50	1.243	1.147	---	1.341	1.230	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.25	0.75	114	126	73 (38/35)	0.25	0.422	0.434	588(661)	0.334	0.342	989(1062)
	0.25	0.75	114	126	83 (44/39)	0.50	0.675	0.707	171(254)	0.597	0.622	244(327)
	0.25	0.75	114	126	92 (49/43)	0.75	0.865	0.918	61(153)	0.825	0.873	77(169)
	0.25	0.75	114	126	100(54/46)	1.00	1.025	1.100	8(108)	1.035	1.078	12(112)
	0.25	0.75	114	126	107(57/50)	1.25	1.169	1.268	---	1.236	1.347	---
	0.25	0.75	114	126	111(59/52)	1.50	1.300	1.424	---	1.431	1.583	---

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter = 1. $D_f,N_f$ indicate the fixed sample event and sample size as per the design protocol. $D_m(C_m/T_m)$ represent the interim event (Control event/ Treatment event). ${\beta }_0$ showed the survival data generated with the specific control shape parameter. ${\widehat{\beta }}_{0(MI)}$ and ${\widehat{\beta }}_{0(MW)}$ represents the estimated Weibull shape parameter using the MAP approach with informative prior and weakly informative prior, respectively. $\widehat{{\beta }_1}$ is the estimated Weibull control scale parameter. Similarly, $\tilde{D}(D_{\mathrm{*}})$ represent the event required after the interim analysis (total event).

Table 2:

Operating characteristics of the study design were calculated under the relative time design when the sample size readjustment was considered using the using the IPS with the MAP approach and using the informative and weakly informative prior information of the control shape parameter.

				Informative Prior					Weakly Informative Prior
RT	$p_0$	$p_1$	${\beta }_0$	${\widehat{\beta }}_{0(MI)}$	$\alpha$	Power	$ARB$ $(H_a)$	$Cov$ $(H_a)$	${\widehat{\beta }}_{0(MW)}$	$\alpha$	Power	$ARB$ $(H_a)$	$Cov$ $(H_a)$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	0.25	0.462	0.040	0.812	0.031	0.893	0.355	0.043	0.807	0.031	0.895
	0.10	0.90	0.50	0.699	0.039	0.813	0.030	0.887	0.606	0.039	0.824	0.032	0.889
	0.10	0.90	0.75	0.861	0.041	0.821	0.033	0.881	0.809	0.039	0.817	0.032	0.889
	0.10	0.90	1.00	0.989	0.036	0.816	0.033	0.878	0.987	0.037	0.815	0.033	0.877
	0.10	0.90	1.25	1.099	0.036	0.828	0.037	0.879	1.148	0.035	0.820	0.037	0.876
	0.10	0.90	1.50	1.192	0.037	0.829	0.037	0.876	1.296	0.040	0.830	0.038	0.868
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	0.25	0.465	0.047	0.777	0.021	0.896	0.359	0.053	0.786	0.027	0.897
	0.10	0.90	0.50	0.711	0.050	0.780	0.026	0.889	0.622	0.049	0.779	0.023	0.889
	0.10	0.90	0.75	0.885	0.056	0.767	0.024	0.879	0.843	0.055	0.776	0.026	0.881
	0.10	0.90	1.00	1.029	0.062	0.774	0.028	0.876	1.043	0.059	0.773	0.025	0.876
	0.10	0.90	1.25	1.155	0.059	0.771	0.027	0.877	1.234	0.068	0.773	0.030	0.871
	0.10	0.90	1.50	1.267	0.067	0.777	0.029	0.866	1.413	0.069	0.778	0.033	0.867
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	0.25	0.403	0.042	0.810	0.020	0.898	0.323	0.042	0.807	0.018	0.900
	0.25	0.75	0.50	0.649	0.041	0.818	0.019	0.894	0.575	0.042	0.811	0.017	0.892
	0.25	0.75	0.75	0.834	0.044	0.814	0.021	0.879	0.793	0.041	0.822	0.023	0.887
	0.25	0.75	1.00	0.988	0.042	0.816	0.019	0.878	0.985	0.043	0.815	0.020	0.876
	0.25	0.75	1.25	1.123	0.042	0.820	0.024	0.881	1.170	0.047	0.813	0.024	0.873
	0.25	0.75	1.50	1.243	0.045	0.818	0.022	0.873	1.341	0.043	0.813	0.021	0.874
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}2}=\mathbf{1.50}$	0.25	0.75	0.25	0.422	0.046	0.785	0.016	0.893	0.334	0.056	0.788	0.016	0.901
	0.25	0.75	0.50	0.675	0.049	0.786	0.020	0.893	0.597	0.054	0.784	0.018	0.895
	0.25	0.75	0.75	0.865	0.054	0.787	0.021	0.885	0.825	0.067	0.784	0.020	0.884
	0.25	0.75	1.00	1.025	0.059	0.772	0.018	0.874	1.035	0.056	0.780	0.022	0.879
	0.25	0.75	1.25	1.169	0.060	0.778	0.019	0.874	1.236	0.079	0.776	0.021	0.874
	0.25	0.75	1.50	1.300	0.064	0.772	0.022	0.871	1.431	0.085	0.781	0.022	0.857

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter is 1). ${\beta }_0$ showed the survival data generated with the specific control shape parameter. ${\widehat{\beta }}_{0(MI)}$ and ${\widehat{\beta }}_{0(MW)}$ represents the estimated Weibull shape parameter using the MAP approach with informative prior and weakly informative prior, respectively.

Table 3:

Sample size re-estimation for the relative time method using the Conditional power approach (fixed and current trend effect size) with r=1 (equal allocation ratio), $\alpha$ = 0.05, one-sided test, power 80%, accrual time 12 months and follow-up time 6 months respectively and control median PFS = 4 months. Unknown Weibull control shape parameter was calculated using the MAP approach with informative and weakly informative prior information of the shape parameter.

				Informative Prior					Weakly Informative Prior
RT	$p_0$	$p_1$	$D_m$ $(C_m/T_m)$	${\widehat{\beta }}_{0(MI)}$	$CP$ $(H_a)$	$\tilde{D}(D_{\mathrm{*}})$	$C{P}_T$	$\tilde{D}(D_{\mathrm{*}})$	${\widehat{\beta }}_{0(MW)}$	$CP$ $(H_a)$	$\tilde{D}(D_{\mathrm{*}})$	$C{P}_T$	$\tilde{D}(D_{\mathrm{*}})$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	53(28/25)	0.462	0.269	401(454)	0.396	213(266)	0.355	0.190	695(748)	0.268	291(344)
	0.10	0.90	59(32/27)	0.699	0.437	151(210)	0.593	107(166)	0.606	0.361	215(274)	0.524	137(196)
	0.10	0.90	65(36/29)	0.861	0.713	79(144)	0.825	---	0.809	0.625	97(162)	0.765	73(138)
	0.10	0.90	70(40/30)	0.989	0.983	---	0.991	---	0.987	0.979	---	0.989	---
	0.10	0.90	74(42/32)	1.099	---	---	---	---	1.148	---	---	---	---
	0.10	0.90	77(44/33)	1.192	---	---	---	---	1.296	---	---	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	54(28/26)	0.465	0.251	416(470)	0.366	228(282)	0.359	0.219	714(768)	0.332	310(364)
	0.10	0.90	61(33/28)	0.711	0.427	153(214)	0.581	111(172)	0.622	0.352	213(274)	0.513	141(202)
	0.10	0.90	68(37/31)	0.885	0.748	76(144)	0.849	---	0.843	0.673	90(158)	0.799	---
	0.10	0.90	74(40/34)	1.029	0.999	---	0.999	---	1.043	0.999	---	0.999	---
	0.10	0.90	79(43/36)	1.155	---	---	---	---	1.234	---	---	---	---
	0.10	0.90	82(45/37)	1.267	---	---	---	---	1.413	---	---	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	80(42/38)	0.403	0.112	888(968)	0.219	468(548)	0.323	0.09	1484(1564)	0.197	630(710)
	0.25	0.75	90 (48/42)	0.649	0.232	304(394)	0.411	216(306)	0.575	0.185	400(490)	0.357	266(356)
	0.25	0.75	99(54/45)	0.834	0.532	149(248)	0.713	119(218)	0.793	0.445	175(274)	0.646	135(234)
	0.25	0.75	107(59/48)	0.988	0.908	---	0.953	---	0.985	0.975	---	0.989	---
	0.25	0.75	114(63/51)	1.123	---	---	---	---	1.170	---	---	---	---
	0.25	0.75	119(65/54)	1.243	---	---	---	---	1.341	---	---	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.25	0.75	73 (38/35)	0.422	0.141	721(794)	0.253	387(460)	0.334	0.131	901(974)	0.240	443(516)
	0.25	0.75	83 (44/39)	0.675	0.289	245(328)	0.471	177(260)	0.597	0.267	271(354)	0.448	191(274)
	0.25	0.75	92 (49/43)	0.865	0.651	118(210)	0.801	---	0.825	0.597	128(220)	0.759	102(194)
	0.25	0.75	100(54/46)	1.025	0.999	60(152)	0.999	---	1.035	0.997	---	0.998	---
	0.25	0.75	107(57/50)	1.169	---	7(107)	---	---	1.236	---	---	---	---
	0.25	0.75	111(59/52)	1.300	---	---	---	---	1.431	---	---	---	---

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter is 1). $D_m(C_m/T_m)$ represent the interim event (Control event/ Treatment event). ${\beta }_0$ showed the survival data generated with the specific control shape parameter.${\widehat{\beta }}_{0(MI)}$ and ${\widehat{\beta }}_{0(MW)}$ represents the estimated Weibull shape parameter using the MAP approach with informative and weakly informative prior. Similarly, $\tilde{D}(D_{\mathrm{*}})$ represent the event required after the interim analysis (total event).

Table 5:

Sample size re-estimation for the relative time method using the Predictive power approach with r=1 (equal allocation ratio), $\alpha$ = 0.05, one-sided test, power 80%, accrual time 12 months and follow-up time 6 months respectively and control median PFS = 4 months. Unknown Weibull control shape parameter was calculated using the MAP approach with informative and weakly informative prior information of the shape parameter.

				Informative Prior				Weakly Informative Prior
RT	$p_0$	$p_1$	$D_m$ $(C_m/T_m)$	${\widehat{\beta }}_{0(MI)}$	$PP$	$\alpha$	$\tilde{D}(D_{\mathrm{*}})$	${\widehat{\beta }}_{0(MW)}$	$PP$	$\alpha$	$\tilde{D}(D_{\mathrm{*}})$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	53(28/25)	0.462	0.681	0.043	293(346)	0.355	0.627	0.051	695(748)
	0.10	0.90	59(32/27)	0.699	0.811	0.044	111(170)	0.606	0.764	0.047	157(216)
	0.10	0.90	65(36/29)	0.861	0.925	---	---	0.809	0.899	---	---
	0.10	0.90	70(40/30)	0.989	0.997	---	---	0.987	0.996	---	---
	0.10	0.90	74(42/32)	1.099	---	---	---	1.148	---	---	---
	0.10	0.90	77(44/33)	1.192	---	---	---	1.296	---	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	54(28/25)	0.465	0.679	0.049	302(356)	0.359	0.637	0.052	510(564)
	0.10	0.90	61(33/28)	0.711	0.815	0.041	111(172)	0.622	0.771	0.044	153(214)
	0.10	0.90	68(37/31)	0.885	0.939	---	---	0.843	0.917	---	---
	0.10	0.90	74(40/34)	1.029	0.999	---	---	1.043	0.999	---	---
	0.10	0.90	79(43/36)	1.155	---	---	---	1.234	---	---	---
	0.10	0.90	82(45/37)	1.267	---	---	---	1.413	---	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	80(42/38)	0.403	0.658	0.055	580(660)	0.323	0.622	0.049	960(1040)
	0.25	0.75	90 (48/42)	0.649	0.789	0.044	198(288)	0.575	0.753	0.042	262(352)
	0.25	0.75	99(54/45)	0.834	0.914	---	---	0.793	0.889	---	---
	0.25	0.75	107(59/48)	0.988	0.988	---	---	0.985	0.997	---	---
	0.25	0.75	114(63/51)	1.123	---	---	---	1.170	---	---	---
	0.25	0.75	119(65/54)	1.243	---	---	---	1.341	---	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.25	0.75	73 (38/35)	0.422	0.665	0.062	481(554)	0.334	0.648	0.045	599(672)
	0.25	0.75	83 (44/39)	0.675	0.804	0.051	163(246)	0.597	0.791	0.033	179(262)
	0.25	0.75	92 (49/43)	0.865	0.936	---	---	0.825	0.923	---	---
	0.25	0.75	100(54/46)	1.025	0.999	---	---	1.035	0.999	---	---
	0.25	0.75	107(57/50)	1.169	---	---	---	1.236	---	---	---
	0.25	0.75	111(59/52)	1.300	---	---	---	1.431	---	---	---

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter is 1). $D_m(C_m/T_m)$ represent the interim event (Control event/ Treatment event). ${\beta }_0$ showed the survival data generated with the specific control shape parameter. ${\widehat{\beta }}_{0(MI)}$ and ${\widehat{\beta }}_{0(MW)}$ represents the estimated Weibull shape parameter using the MAP approach with informative and weakly informative prior. Similarly, $\tilde{D}(D_{\mathrm{*}})$ represent the event required after the interim analysis (total event).

Table 6:

Sample size re-estimation for the relative time method using the IPS approach with r=1 (equal allocation ratio), $\alpha$ = 0.05, one-sided test, power 80%, accrual time 12 months and follow-up time 6 months respectively and control median PFS = 4 months. Unknown Weibull control shape parameter was calculated using the full posterior samples with informative and weakly informative prior information of the shape parameter.

							Informative Prior			Weakly Informative Prior
RT	$p_0$	$p_1$	$D_f$	$N_f$	$D_m$ $(C_m/T_m)$	${\beta }_0$	${\widehat{\beta }}_{0(FI)}$	$\widehat{{\beta }_1}$	$\tilde{D}(D_{\mathrm{*}})$	${\widehat{\beta }}_{0(FW)}$	$\widehat{{\beta }_1}$	$\tilde{D}(D_{\mathrm{*}})$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	82	92	53(28/25)	0.25	0.473	0.451	270(323)	0.356	0.345	455(508)
	0.10	0.90	82	92	59(32/27)	0.50	0.698	0.659	89(148)	0.607	0.576	91(156)
	0.10	0.90	82	92	65(36/29)	0.75	0.863	0.803	36(101)	0.811	0.757	53(105)
	0.10	0.90	82	92	70(40/30)	1.00	0.991	0.913	9(79)	0.988	0.909	2(72)
	0.10	0.90	82	92	74(42/32)	1.25	1.100	1.005	---	1.149	1.045	---
	0.10	0.90	82	92	77(44/33)	1.50	1.194	1.082	---	1.297	1.166	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	86	94	54(28/25)	0.25	0.468	0.489	314(368)	0.362	0.375	478(532)
	0.10	0.90	86	94	61(33/28)	0.50	0.715	0.766	98(159)	0.627	0.667	113(174)
	0.10	0.90	86	94	68(37/31)	0.75	0.889	0.971	40(108)	0.849	0.923	66(134)
	0.10	0.90	86	94	74(40/34)	1.00	1.033	1.145	4(78)	1.049	1.166	9(83)
	0.10	0.90	86	94	79(43/36)	1.25	1.160	1.302	---	1.240	1.405	---
	0.10	0.90	86	94	82(45/37)	1.50	1.272	1.445	---	1.422	1.642	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	124	138	80(42/38)	0.25	0.405	0.394	573(653)	0.323	0.316	605(685)
	0.25	0.75	124	138	90 (48/42)	0.50	0.652	0.624	168(258)	0.576	0.554	175(265)
	0.25	0.75	124	138	99(54/45)	0.75	0.837	0.793	67(166)	0.794	0.753	71(170)
	0.25	0.75	124	138	107(59/48)	1.00	0.991	0.929	19(126)	0.989	0.927	21(128)
	0.25	0.75	124	138	114(63/51)	1.25	1.126	1.047	---	1.171	1.085	---
	0.25	0.75	124	138	119(65/54)	1.50	1.246	1.149	---	1.342	1.231	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.25	0.75	114	126	73 (38/35)	0.25	0.426	0.439	526(599)	0.338	0.347	653(726)
	0.25	0.75	114	126	83 (44/39)	0.50	0.680	0.713	143(226)	0.603	0.629	150(233)
	0.25	0.75	114	126	92 (49/43)	0.75	0.871	0.926	63(155)	0.833	0.883	68(160)
	0.25	0.75	114	126	100(54/46)	1.00	1.031	1.109	13(113)	1.044	1.123	---
	0.25	0.75	114	126	107(57/50)	1.25	1.176	1.278	---	1.246	1.361	---
	0.25	0.75	114	126	111(59/52)	1.50	1.272	1.445	---	1.443	1.599	---

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter is 1). $D_f,N_f$ indicate the fixed sample event and sample size at per the design protocol. $D_m(C_m/T_m)$ represent the interim event (Control event/ Treatment event). ${\beta }_0$ showed the survival data generated with the specific control shape parameter. ${\widehat{\beta }}_{0(FI)}$ and ${\widehat{\beta }}_{0(FW)}$ represents the estimated Weibull shape parameter of the full Posterior sampling using informative and weakly informative prior information and $\widehat{{\beta }_1}$ is the Weibull control scale parameter. Similarly, $\tilde{D}(D_{\mathrm{*}})$ represent the event required after the interim analysis (total event).

Table 7:

Operating characteristics of the study design were calculated under the relative time design when the sample size readjustment was done using the using the IPS with full posterior samples with informative and weakly informative prior information of the control shape parameter.

				Informative Prior					Weakly Informative Prior
RT	$p_0$	$p_1$	${\beta }_0$	${\widehat{\beta }}_{0(FI)}$	$\alpha$	Power	$ARB$ $(H_a)$	$Cov$ $(H_a)$	${\widehat{\beta }}_{0(FW)}$	$\alpha$	Power	$ARB$ $(H_a)$	$Cov$ $(H_a)$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	0.25	0.473	0.045	0.835	0.042	0.931	0.356	0.051	0.815	0.038	0.897
	0.10	0.90	0.50	0.698	0.051	0.886	0.032	0.881	0.607	0.043	0.816	0.031	0.889
	0.10	0.90	0.75	0.863	0.044	0.877	0.035	0.872	0.811	0.046	0.807	0.030	0.886
	0.10	0.90	1.00	0.991	0.051	0.855	0.044	0.901	0.988	0.044	0.805	0.037	0.889
	0.10	0.90	1.25	1.100	0.049	0.897	0.034	0.872	1.149	0.040	0.817	0.035	0.903
	0.10	0.90	1.50	1.194	0.066	0.801	0.039	0.887	1.297	0.051	0.801	0.034	0.892
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	0.25	0.468	0.044	0.815	0.026	0.842	0.362	0.053	0.795	0.028	0.886
	0.10	0.90	0.50	0.715	0.057	0.785	0.038	0.891	0.627	0.042	0.785	0.026	0.881
	0.10	0.90	0.75	0.889	0.054	0.799	0.037	0.866	0.849	0.055	0.796	0.033	0.907
	0.10	0.90	1.00	1.033	0.055	0.788	0.048	0.879	1.049	0.061	0.778	0.029	0.893
	0.10	0.90	1.25	1.160	0.040	0.789	0.039	0.874	1.240	0.041	0.779	0.027	0.865
	0.10	0.90	1.50	1.272	0.041	0.788	0.043	0.891	1.422	0.047	0.788	0.033	0.852
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	0.25	0.405	0.054	0.823	0.039	0.811	0.323	0.044	0.813	0.016	0.897
	0.25	0.75	0.50	0.652	0.057	0.804	0.037	0.826	0.576	0.046	0.804	0.021	0.895
	0.25	0.75	0.75	0.837	0.041	0.814	0.038	0.863	0.794	0.047	0.834	0.025	0.884
	0.25	0.75	1.00	0.991	0.039	0.828	0.051	0.872	0.989	0.051	0.818	0.027	0.846
	0.25	0.75	1.25	1.126	0.054	0.798	0.033	0.844	1.171	0.054	0.799	0.026	0.867
	0.25	0.75	1.50	1.246	0.044	0.807	0.039	0.865	1.342	0.057	0.807	0.030	0.897
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.25	0.75	0.25	0.426	0.053	0.787	0.039	0.833	0.338	0.055	0.797	0.019	0.887
	0.25	0.75	0.50	0.680	0.070	0.789	0.042	0.818	0.603	0.053	0.779	0.034	0.874
	0.25	0.75	0.75	0.871	0.063	0.799	0.032	0.901	0.833	0.061	0.796	0.047	0.894
	0.25	0.75	1.00	1.031	0.060	0.796	0.045	0.903	1.044	0.051	0.795	0.058	0.871
	0.25	0.75	1.25	1.176	0.042	0.779	0.037	0.881	1.246	0.051	0.778	0.027	0.854
	0.25	0.75	1.50	1.272	0.056	0.789	0.034	0.882	1.443	0.075	0.779	0.025	0.852

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter is 1). ${\beta }_0$ showed the survival data generated with the specific control shape parameter. ${\widehat{\beta }}_{0(FI)}$ and ${\widehat{\beta }}_{0(FW)}$ represents the estimated Weibull shape parameter using the full posterior approach with informative and weakly informative prior information.

1.6. Discussion and Conclusions

In our work, we have extended the newly proposed relative time method of Phadnis and Mayo²⁰ for sample size re-estimation in the context of SC tests. This method allows us to design two arm-trials allowing both non-constant hazard ratio and non-constant time-ratio by using two different Weibull distributions. One advantage of RT method is, it allows us to define the effect size in the metric of time scale which is more practically intuitive. Another attractive feature of this method is when the PH assumption holds, the sample size calculations match to those obtained by the popular Schoenfeld formula for Cox PH models.

However, there are situations when an initial estimation of the Weibull shape parameter may not be readily available to implement such as in rare diseases across different therapeutic areas. In such cases, we provide a solution how to estimate the Weibull shape parameter using data at the interim despite initial misspecification of the same. Given that no prior information is available, we could simply design a trial based on this method assuming exponentially distributed survival time and later adjust the protocol specified sample size with the newly re-estimated sample size using the data collected at the interim stage. In this paper, we have used the traditional IPS approach for sample size re-estimation based on interim data. Furthermore, we also assessed sample size re-estimation by CP and PP approach. We showed that in these scenarios MAP approach or weighted average of the full posterior distribution can be a viable option to estimate the shape parameter. Also, prior specification of the shape parameter can be influenced by the estimation process. It is worth noting that the MAP approach may lead to a similar estimate of the shape parameter as the full posterior distribution. However, the computational cost is relatively high when calculating shape parameter using the full posterior distribution. There may still be some situations when the MAP approach may not be appropriate, such as when the posterior samples have heavier tails or more spread out. In that case, full posterior distribution can be utilized to calculate the Weibull shape parameter. It is shown that when the true Weibull shape parameter is different from the protocol mis-specified parameter at the start of the study, the sample size is heavily influenced and results in practical constraints on the study design. Another limitation of this design based on the simulation results is that the type I error rate can be slightly inflated under some design parameters. Although the IPS approach and CP and PP approach tends to differ the result qualitatively, they may lead to a similar conclusion based on the interim observation. We have used gamma prior for calculating the Weibull shape parameter. Also, conducting sensitivity analysis under different priors can help to validate the findings. The calculation reported using the unblinded ‘Internal Pilot Study’ approach protects the type I error and provides adequate power.

Sample size adjustment should not be a replacement for proper planning and execution of a clinical trial. Also, SSR may not be advisable when the follow-up period is longer than the recruitment time ³⁹. Trial design should reflect the best knowledge of the variability of the nuisance parameter since extending an ongoing trial may potentially raise the economic and administrative complexity of the trial. Furthermore, in TTE setting, there may arise a few unexpected complexities such as what should be done in the scenario of lower accrual or event rate than the anticipated. Although the adaptive trial allows for making necessary adjustments, such a decision should be made with proper caution with the meeting of Data safety monitoring board (DSMB) and followed by the standard guideline for clinical practices. It is also suggested for small phase II trial, we should not conduct interim sample size adjustment more than once ³⁹. Statisticians should work closely with a clinician to address those issues and utilize any information available based on expert knowledge.

Although an interim evaluation could suggest that sample size should be increased, inthe fact it ordinarily will not be worthwhile in practice to expand the trial unless there is significant benefit observed in the treatment arm. Considering all the uncertainties involvedat various stages of the calculations, small changes to the sample size are unlikely to affectthe power practically. Likewise, as a purely practical matter, it is inadvisable to carry out a trial where the SSR calculation requires many multiples of the original sample size. Rather, careful introspection of the factors operating in the trial should be undertaken to determine why the initial sample size was inadequate. Finally, since Bayesian predictive probability is also a part of stochastic curtailment test and have attractive feature to continuously monitor the trail, therefore, as a part of future work, we are planning to utilize Bayesian predictive probability for sample size re-estimation purposes using the RT method.

Table 4:

Operating characteristics of the study design were calculated under the relative time design using Conditional power approach with the MAP approach including informative and weakly informative prior information of the control shape parameter.

				Informative Prior			Weakly Informative Prior
RT	$p_0$	$p_1$	$D_m$ $(C_m/T_m)$	${\widehat{\beta }}_{0(MI)}$	${\alpha }_{CP(H_a)}$	${\alpha }_{C{P}_T}$	${\widehat{\beta }}_{0(MW)}$	${\alpha }_{CP(H_a)}$	${\alpha }_{C{P}_T}$
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.52}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.98}$	0.10	0.90	53(28/25)	0.462	0.044	0.049	0.355	0.049	0.045
	0.10	0.90	59(32/27)	0.699	0.045	0.037	0.606	0.047	0.041
	0.10	0.90	65(36/29)	0.861	0.042	0.044	0.809	0.046	0.047
	0.10	0.90	70(40/30)	0.989	0.052	0.041	0.987	0.055	0.041
	0.10	0.90	74(42/32)	1.099	---	---	1.148	---	---
	0.10	0.90	77(44/33)	1.192	---	---	1.296	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{2.00}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.10	0.90	54(28/26)	0.465	0.055	0.052	0.359	0.052	0.053
	0.10	0.90	61(33/28)	0.711	0.050	0.051	0.622	0.059	0.051
	0.10	0.90	68(37/31)	0.885	0.057	0.055	0.843	0.053	0.051
	0.10	0.90	74(40/34)	1.029	0.061	0.067	1.043	0.061	0.067
	0.10	0.90	79(43/36)	1.155	---	---	1.234	---	---
	0.10	0.90	82(45/37)	1.267	---	---	1.413	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.50}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.667}$	0.25	0.75	80(42/38)	0.403	0.044	0.040	0.323	0.055	0.045
	0.25	0.75	90 (48/42)	0.649	0.042	0.041	0.575	0.041	0.040
	0.25	0.75	99(54/45)	0.834	0.035	0.049	0.793	0.051	0.041
	0.25	0.75	107(59/48)	0.988	0.051	0.052	0.985	0.059	0.051
	0.25	0.75	114(63/51)	1.123	---	---	1.170	---	---
	0.25	0.75	119(65/54)	1.243	---	---	1.341	---	---
$\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{1}}=\mathbf{1.667}$ $\mathit{R}{\mathit{T}}_{\mathit{p}\mathbf{2}}=\mathbf{1.50}$	0.25	0.75	73 (38/35)	0.422	0.056	0.056	0.334	0.055	0.051
	0.25	0.75	83 (44/39)	0.675	0.054	0.054	0.597	0.056	0.054
	0.25	0.75	92 (49/43)	0.865	0.067	0.067	0.825	0.067	0.068
	0.25	0.75	100(54/46)	1.025	0.056	0.059	1.035	0.056	0.058
	0.25	0.75	107(57/50)	1.169	---	---	1.236	---	---
	0.25	0.75	111(59/52)	1.300	---	---	1.431	---	---

Note: Study was initially designed assuming the exponential distributed survival time in both treatment and control arm (in this case Weibull shape parameter is 1). $D_m(C_m/T_m)$ represent the interim event (Control event/ Treatment event). ${\widehat{\beta }}_{0(MI)}$ and ${\widehat{\beta }}_{0(MW)}$ represents the estimated Weibull shape parameter using the MAP approach with informative and weakly informative prior.