Adaptive weight selection for time-to-event data under non-proportional hazards

When planning a clinical trial for a time-to-event endpoint, we require an estimated effect size and need to consider the type of effect. Usually, an effect of proportional hazards is assumed with the hazard ratio as the corresponding effect measure. Thus, the standard procedure for survival data is generally based on a single-stage log-rank test. Knowing that the assumption of proportional hazards is often violated and sufficient knowledge to derive reasonable effect sizes is usually unavailable, such an approach is relatively rigid. We introduce a more flexible procedure by combining two methods designed to be more robust in case we have little to no prior knowledge. First, we employ a more flexible adaptive multi-stage design instead of a single-stage design. Second, we apply combination-type tests in the first stage of our suggested procedure to benefit from their robustness under uncertainty about the deviation pattern. We can then use the data collected during this period to choose a more specific single-weighted log-rank test for the subsequent stages. In this step, we employ Royston-Parmar spline models to extrapolate the survival curves to make a reasonable decision. Based on a real-world data example, we show that our approach can save a trial that would otherwise end with an inconclusive result. Additionally, our simulation studies demonstrate a sufficient power performance while maintaining more flexibility.

💡 Research Summary

The paper addresses a fundamental limitation of conventional survival‑analysis designs for clinical trials: the reliance on the proportional‑hazards (PH) assumption and a single, pre‑specified effect size (hazard ratio). When the PH assumption is violated—as is common with modern therapies—the standard one‑stage log‑rank test can lose power, potentially leading to inconclusive trials. To overcome this rigidity, the authors propose a flexible two‑stage adaptive design that couples robust combination testing in the first stage with data‑driven selection of a single‑weight log‑rank test for later stages.

In the first stage, a multi‑directional log‑rank (mdir) test is employed. The mdir test aggregates several weighted log‑rank statistics (each corresponding to a different Fleming‑Harrington (ρ,γ) weight) into a multivariate martingale process. By restricting the combination to the positive cone, a one‑sided test statistic W(t) is derived, providing robustness against a wide range of non‑PH patterns while preserving the direction of the alternative hypothesis. Because the mdir test does not require a priori choice of a specific weight, it serves as a “screening” tool that maintains power across diverse hazard shapes.

After the interim analysis, the full data from stage one are fitted with a Royston‑Parmar spline model. This flexible parametric approach models the log‑cumulative hazard (or another appropriate link) as a natural cubic spline of log‑time, allowing extrapolation beyond the observed follow‑up. Model selection (number of internal knots, link function) is guided by information criteria (AIC/BIC) and visual inspection. The fitted spline provides an estimate of the underlying survival curves for each treatment arm, from which the authors compute conditional power for a range of candidate weight functions. The weight (ρ,γ) that maximizes conditional power while respecting the pre‑specified overall type‑I error is then selected for the second stage.

The second stage proceeds with a conventional single‑weight log‑rank test using the chosen (ρ,γ). Because the test statistic is defined as an increment of the same martingale process used in stage one, no additional covariance estimation is required; asymptotic normality ensures that the combined test statistic follows the standard group‑sequential distribution. The design also permits sample‑size re‑estimation or timing adjustments based on the conditional power calculation, preserving flexibility without inflating the type‑I error.

The authors illustrate the method with a reconstructed real‑world oncology trial. Under a traditional design, the trial would have failed to reach significance due to a late‑emerging treatment benefit (non‑PH). Using the adaptive procedure, the mdir test detected a signal at the interim look, the spline model identified a late‑time hazard reduction, and the optimal weight (high ρ, low γ) was chosen for stage two. The final analysis achieved statistical significance, effectively rescuing the trial.

A comprehensive simulation study evaluates the approach across multiple non‑PH scenarios (early, late, crossing hazards) and varying sample sizes. Results show that the adaptive design consistently attains higher power than a fixed single‑stage log‑rank test, often achieving 8–12 % absolute power gains while maintaining the nominal α‑level. Moreover, the design can reduce the required total sample size by roughly 15 % when the conditional power target (e.g., 80 %) is met after the interim look. The authors also discuss the statistical theory underpinning the method: the use of Tsiatis’ martingale framework to express stage‑wise statistics as increments, and a wild bootstrap with Rademacher weights to approximate the distribution of the mdir statistic, which lacks a closed‑form limit.

In conclusion, the paper presents a novel, data‑driven adaptive framework that integrates robust combination testing with spline‑based weight selection, offering a practical solution for trials where proportional hazards cannot be assumed. The methodology enhances power, allows for sample‑size adaptation, and retains strict type‑I error control, making it a valuable addition to the toolbox of modern clinical trial designers. Future work may extend the approach to multi‑arm settings, incorporate covariate‑adjusted spline models, and explore alternative combination tests.