## 1 Introduction

Sample size determination is an important part of clinical trial design and conventionally involves power calculations. However, the power of a trial does not necessarily give the probability of the trial demonstrating a treatment effect, as the true treatment effect may be different to that assumed in the power calculation. Several authors have proposed a hybrid classical-Bayesian approach for assessing the probability of a successful trial, given the sample size only, which can then be used to inform sample size decisions.

The hybrid method was first considered by Spiegelhalter and Freedman [1]. They constructed an unconditional probability of having a desired outcome and called this unconditional probability the average power. O'Hagan and Stevens [2] used this method for choosing sample sizes for clinical trials of cost-effectiveness. They referred to the unconditional probability of a successful trial as the ‘assurance’ of the trial, and we use this term here. O'Hagan *et al.* [3] extended assurance methods to two-sided testing and equivalence trials, covering the use of non-conjugate prior distributions for uncertain parameters. Chuang-Stein [4] discussed the difference between traditional power calculations and assurance calculations to determine sample sizes, giving an example of planning the next trial based on the results of an early trial. Chuang-Stein and Yang [5] reviewed the concept of assurance and illustrated its use when planning phase III trials. They also applied assurance to study designs when re-estimating a sample size based on an interim analysis.

An assurance calculation requires a prior distribution for the treatment effect, but does not necessarily involve a Bayesian analysis of the trial data. The method of analysis, and in particular the criteria for which the trial is deemed a ‘success’, are determined externally, for example, by a regulator. Once the criteria have been specified, a prior distribution is used to assess the probability that these criteria will be met. Typically, the prior distribution will only be used in the design stage and not the analysis. At the design stage, the risk of trial failure is primarily the trial sponsor's, and so it should be uncontroversial for a trial sponsor to use all their prior knowledge in assessing such a risk.

We consider clinical trials in which the endpoint of interest is a survival time. For time-to-event outcome measures, power and sample size calculations have been well studied under various model assumptions. For example, Schoenfeld and Richter [6] developed a power function with a limited recruitment period and a pre-specified follow-up period under the assumption that the survival times in each treatment group follow exponential distributions and patients enter the trial uniformly. Gross and Clark [7] provided a method of calculating sample size by assuming that the sample mean survival time is approximately normally distributed under Weibull models for the survival times. Freedman [8] and Schoenfeld [9] derived sample size formulae under the assumption of proportional hazards based on asymptotic properties of the logrank statistic.

Little has been done in calculating assurance for survival endpoints. Assuming proportional hazards, Spiegelhalter *et al.* [10] derived an assurance formula in the case of equal allocation and follow-up. The only uncertain variable considered was the log hazard ratio, and a normal prior was assumed. In this paper, we extend assurance calculations to accommodate both parametric and proportional hazards models. Under proportional hazards models, we derive an assurance formula assuming uniform patient entry over a limited recruitment period. We consider uncertainty in both the log hazard ratio and the baseline survivor function.

In Section 2, we review how assurance is calculated to determine the unconditional probability of having a desired outcome. In Section 3, we derive assurance calculations for exponential and Weibull survival models and describe the elicitation methods for the required prior distributions. In Section 4, we extend assurance calculations to accommodate proportional hazards models, considering uncertainty in both treatment effect and baseline survivor function. We also describe the procedure of generating the baseline survivor function. Examples are given in Section 5.