Prediction Intervals for Interim Events in Randomized Clinical Trials with Time-to-Event Endpoints

TL;DR

The paper tackles interim monitoring in randomized trials with time-to-event endpoints by enabling prediction intervals for the future number of events, a key need when information accrues mainly through events. It extends a reliability-engineering DP-MC framework to clinical trials, modeling each patient as a unit with covariates, staggered entry, and potential dependence between entry and loss to follow-up, and uses a parametric bootstrap to obtain a conditional CMF for the future event count $Y$. The future count is represented as a Poisson-binomial distribution, allowing heterogeneous per-patient probabilities $\pi_j$ while accounting for censoring and covariates. Through simulations and a pediatric ALL case study, the authors show that the proposed PI method achieves near-nominal coverage under correct specifications, with horizon, follow-up maturity, and censoring patterns strongly guiding performance; model-choice considerations and extrapolative checks are important for reliable interval interpretation in practice. The framework thereby provides a flexible, interpretable tool for real-time decision-making in interim analyses and can support sensitivity analyses across competing survival-model specifications.

Abstract

Time-to-event endpoints are central to evaluate treatment efficacy across many disease areas. Many trial protocols include interim analyses within group-sequential designs that control type I error via spending functions or boundary methods, with operating characteristics determined by the number of looks and the information accrued. Planning interim analyses with time-to-event endpoints is challenging because statistical information depends on the number of observed events, so adequate follow-up to accrue the required events is critical and interim prediction of information at scheduled looks and at the final analysis becomes essential. While several methods have been developed to predict the calendar time required to reach a target number of events, to the best of our knowledge there is no established framework that addresses the prediction of the number of events at a future date with corresponding prediction intervals. Starting from prediction interval approach originally developed in reliability engineering for the number of future component failures, we reformulated and extended it to the context of interim monitoring in clinical trials. This adaptation yields a general framework for event-count prediction intervals in the clinical setting, taking the patient as the unit of analysis and accommodating a range of parametric survival models, patient-level covariates, stagged entry and possible dependence between entry dates and loss to follow-up. Prediction intervals are obtained in a frequentist framework from a bootstrap estimator of the conditional distribution of future events. The performance of the proposed approach is investigated via simulation studies and illustrated by analyzing a real-world phase III trial in childhood acute lymphoblastic leukaemia.

Figures (18)

• Figure 1: Study $\mathcal{S}_1$. Performance metrics of the 95% two-sided prediction intervals under different factors. $\Delta t$ denotes the prediction horizon, $t_c$ the time of interim from accrual end, $\rho$ the correlation between loss to follow-up times and entry dates, $HR$ is the hazard ratio and $k$ the ratio between the baseline scale parameter and the loss to follow-up rate.
• Figure 2: Study $\mathcal{S}_2$. Performance metrics of the 95% two-sided prediction intervals under different models and administrative censoring only at $t_c$. $\Delta t$ denotes the prediction horizon, $p(t_c)$ denotes the proportion of interim censored patients, $t_c$ the time of interim from accrual end, $HR$ is the hazard ratio and $k$ the ratio between the baseline event rate and the loss to follow-up rate.
• Figure 3: Different survival function transformations vs. log-time by treatment arm, 1 = Active treatment, 0 = standard of care. Panel A shows the log cumulative hazard of EFS $\mathrm{log}(-\mathrm{log}(S(t)))$; Panel B shows the cumulative probit of EFS $-\Phi^{-1}(S(t))$, where $\Phi^{-1}$ is CDF of the standard normal distribution; Panel C shows the cumulative logit of EFS $\mathrm{log}(1- S(t) / S(t))$.
• Figure 4: Two-sided 95% prediction intervals for the additional number of EFS events from June 1$^{\mathrm{st}}$ 2017 to June 1$^{\mathrm{st}}$ 2021 . Black diamonds indicate the observed counts. Panel A shows intervals under the Weibull model and its flexible specifications; Panel B shows under the log-normal model and its flexible specifications; Panel C shows under the log-logistic model and its flexible specifications. RP = Royston--Parmar; $k$ denotes the number of knots.
• Figure 5: Standard errors of the coverage.