Event-Driven Power for Time-To-Event Endpoints

Why Event Targets Require Simulation

For a time-to-event endpoint, the information available for the treatment comparison depends on how many events occur, when they occur, and how much follow-up is censored. An event target therefore does not translate directly into a fixed sample size or a fixed amount of calendar time. Conditional Bayesian power answers a specific design question: if the treatment effect has a specified value, how often would this design meet its Bayesian success rule?

The calculation below uses a deliberately explicit exponential model. It is a useful setting for understanding the simulation steps, not a claim that every clinical endpoint has a constant hazard. In the worked design, all participants enter at time zero, allocation is equal, follow-up is capped at a fixed duration, and the trial aims to analyse at a prespecified number of events. A simulation can accommodate staggered entry by adjusting each participant’s time at risk.

Model, Analysis, and Success Rule

Let \(\lambda_C\) and \(\lambda_T\) denote the constant event hazards in the control and treatment arms, measured per person-year. Define the hazard ratio as \(\theta=\lambda_T/\lambda_C\). The data-generating scenario specifies a fixed value of \(\theta\), with \(\theta<1\) representing a lower event hazard under treatment. For this tutorial’s analysis, assume the control hazard \(\lambda_C\) is known and fixed, for example from a reliable external source. The treatment hazard is unknown.

For each participant, the event time follows an exponential distribution with the hazard of their assigned arm. A separate censoring time is generated independently of the event time, conditional on treatment arm. Administrative censoring occurs at the maximum follow-up time \(\tau\). The independence assumption means that censoring does not depend on the participant’s unobserved event time after conditioning on the model; if it fails, this analysis can be biased.

For the treatment arm, let \(d_T\) be the number of observed events and \(Y_T\) the total observed person-time at risk. With independent, noninformative right censoring, the event-time likelihood for \(\lambda_T\), ignoring factors not involving it, is \(\lambda_T^{d_T}\exp(-\lambda_TY_T)\). Put a Gamma prior on \(\lambda_T\) with shape \(a_0\) and rate \(b_0\). The rate parameter \(b_0\) is in person-years when \(\lambda_T\) is measured per year.

For a concrete decision rule, call the trial a success when the posterior probability that the treatment hazard is below the known control hazard exceeds 0.95:

This rule is a design choice, not a universal standard. It depends on the model, prior, control-hazard assumption, and threshold. The posterior probability is evaluated from the Gamma posterior distribution, usually with a numerical Gamma cumulative distribution function.

Simulating the Event-Driven Trial

Suppose there are \(n\) participants per arm, all enrolled at time zero, a maximum follow-up of \(\tau\) years, and a target of \(D\) total events. For each participant, generate an event time and an independent censoring time. The observed time is the earliest of the event time, censoring time, and \(\tau\). If the \(D\)th event occurs before \(\tau\), stop follow-up at that event time; otherwise, follow to \(\tau\). Person-time is accumulated only up to the applicable stopping time.

For a simple illustration, assume censoring times follow an exponential distribution with rate \(\gamma\), in addition to administrative censoring. This assumption is not needed for the posterior update itself; it is one way to generate censoring under a specified design scenario. In an actual design, the censoring distribution should reflect expected loss to follow-up and administrative follow-up.

Why the Posterior Update Has This Form

For observed event times \(t_i\) and censoring indicators \(\delta_i\) in the treatment arm, an exponential event-time contribution is proportional to \(\lambda_T^{\delta_i}\exp(-\lambda_Tt_i)\). Multiplying these contributions gives the likelihood in terms of the total event count and total observed exposure. Multiplication by the Gamma prior kernel then yields the posterior kernel:

The last expression is the kernel of a Gamma distribution with shape \(a_0+d_T\) and rate \(b_0+Y_T\). Independent, noninformative censoring allows the censoring contributions to be omitted from this likelihood for the event hazard. If censoring depends on the event risk in a way not captured by the model, that omission is not justified.

Estimating Conditional Power

For \(M\) simulated trials, let \(I_j=1\) if trial \(j\) reaches the event target and meets the posterior success rule, and \(I_j=0\) otherwise. The estimated conditional power is \(\hat{q}=\sum_{j=1}^{M} I_j/M\). Because the indicators are Bernoulli outcomes, an approximate Monte Carlo standard error is \(\sqrt{\hat{q}(1-\hat{q})/M}\). This describes simulation precision, not uncertainty about whether the assumed event rates or design scenario are correct.

Varying the Event Target

A larger event target will often provide more information about the treatment effect, but it can also take longer to reach and may not be reached by the maximum follow-up time. The power calculation should therefore repeat the entire stopping and analysis process for each candidate target. Keep the rest of the design and data-generating scenario fixed when the aim is to compare targets.

Diagnostics, Alternatives, and Limitations

Check the implementation before interpreting the estimate. Confirm that simulated event and censoring times are nonnegative, person-time is accumulated only while a participant remains at risk, and the total event count stops at the target when it is reached. Compare event rates and event-target attainment with rough expectations under the assumed hazards. Inspect whether treatment and control event counts, follow-up, and censoring patterns look plausible. Re-run the simulation with more replicates if Monte Carlo error is large relative to differences between candidate designs.

A common mistake is to treat the target number of events as if it must occur. If a design has a maximum follow-up, some simulated trials may not reach the target. Dropping those trials from the denominator estimates success conditional on attaining the target, not the overall design success probability defined here. Keep them as failures when the operational rule requires the target by the cap, or report the conditional-on-attainment quantity separately and label it precisely.

Another important limitation is the assumption that the control hazard is known. In clinical trials it is usually estimated, and uncertainty in that estimate can affect the posterior treatment comparison. The model also assumes constant hazards and independent censoring; nonproportional hazards, delayed treatment effects, competing events, or informative dropout may call for a different data-generating and analysis model. A Bayesian Cox model or a model with uncertain baseline hazards may be more appropriate, but requires corresponding posterior computation in each simulated trial.

A frequentist log-rank or Cox-model power calculation is a common alternative, often using an approximation based primarily on the number of events. Such calculations can be efficient for planning under their assumptions, but they do not directly estimate the probability of crossing this particular posterior decision threshold under the stated simulation model. Simulation is more flexible about censoring, follow-up caps, and decision rules, at the cost of computational effort and dependence on correctly specified assumptions. Neither approach repairs a mismatch between the assumed and plausible event-time process.