Design and analysis of group-sequential clinical trials based on a modestly weighted log-rank test in anticipation of a delayed separation of survival curves: A practical guidance

Visualizing hypothesis tests in survival analysis under anticipated delayed effects

What can be considered an appropriate statistical method for the primary analysis of a randomized clinical trial (RCT) with a time-to-event endpoint when we anticipate non-proportional hazards owing to a delayed effect? This question has been the subject of much recent debate. The standard approach is a log-rank test and/or a Cox proportional hazards model. Alternative methods have been explored in the statistical literature, such as weighted log-rank tests and tests based on the Restricted Mean Survival Time (RMST). While weighted log-rank tests can achieve high power compared to the standard log-rank test, some choices of weights may lead to type-I error inflation under particular conditions. In addition, they are not linked to a mathematically unambiguous summary measure. Test statistics based on the RMST, on the other hand, allow one to investigate the average difference between two survival curves up to a pre-specified time pointτ $$ \tau $$ -a mathematically unambiguous summary measure. However, by emphasizing differences prior toτ $$ \tau $$ , such test statistics may not fully capture the benefit of a new treatment in terms of long-term survival. In this article, we introduce a graphical approach for direct comparison of weighted log-rank tests and tests based on the RMST. This new perspective allows a more informed choice of the analysis method, going beyond power and type I error comparison.

Simultaneous inference procedures for the comparison of multiple characteristics of two survival functions

Survival time is the primary endpoint of many randomized controlled trials, and a treatment effect is typically quantified by the hazard ratio under the assumption of proportional hazards. Awareness is increasing that in many settings this assumption is a priori violated, for example, due to delayed onset of drug effect. In these cases, interpretation of the hazard ratio estimate is ambiguous and statistical inference for alternative parameters to quantify a treatment effect is warranted. We consider differences or ratios of milestone survival probabilities or quantiles, differences in restricted mean survival times, and an average hazard ratio to be of interest. Typically, more than one such parameter needs to be reported to assess possible treatment benefits, and in confirmatory trials, the according inferential procedures need to be adjusted for multiplicity. A simple Bonferroni adjustment may be too conservative because the different parameters of interest typically show considerable correlation. Hence simultaneous inference procedures that take into account the correlation are warranted. By using the counting process representation of the mentioned parameters, we show that their estimates are asymptotically multivariate normal and we provide an estimate for their covariance matrix. We propose according to the parametric multiple testing procedures and simultaneous confidence intervals. Also, the logrank test may be included in the framework. Finite sample type I error rate and power are studied by simulation. The methods are illustrated with an example from oncology. A software implementation is provided in the R package nph.

Group sequential designs for cancer immunotherapy trial with delayed treatment effect

Cancer immunotherapy trials are frequently characterized by delayed treatment effects such that the proportional hazards assumption is violated and the log-rank test suffers a substantial loss of statistical power. To increase the efficacy of the trial design, a variety of weighted log-rank tests have been proposed for fixed sample and group sequential trial designs. However, in such a group sequential design, it is often not recommended for futility interim monitoring due to possible delayed treatment effect which could result a high false-negative rate. To resolve this problem, we propose a group sequential design using a piecewise weighted log-rank test which provides an event-driven approach based on number of events after the delayed time. That is, the interim looks will not be conducted until the planned number of events observed after the delay time. Thus, it avoids the possibility of false-negative rate due to the delayed treatment effect. Furthermore, with an event-driven approach, the proposed group sequential design is robust against the underlying survival, accrual and censoring distributions. The group sequential designs using Fleming-Harrington-(ρ , γ ) weighted log-rank test and a new weighted log-rank test are also discussed.

Sequential monitoring of cancer immunotherapy trial with random delayed treatment effect

Cancer immunotherapy trials are frequently characterized by a delayed treatment effect that violates the proportional hazards assumption. The log-rank test (LRT) suffers a substantial loss of statistical power under the nonproportional hazards model. Various group sequential designs using weighted LRTs (WLRTs) have been proposed under the fixed delayed treatment effect model. However, patients enrolled in immunotherapy trials are often heterogeneous, and the duration of the delayed treatment effect is a random variable. Therefore, we propose group sequential designs under the random delayed effect model using the random delayed distribution WLRT. The proposed group sequential designs are developed for monitoring the efficacy of the trial using the method of Lan-DeMets alpha-spending function with O'Brien-Fleming stopping boundaries or a gamma family alpha-spending function. The maximum sample size for the group sequential design is obtained by multiplying an inflation factor with the sample size for the fixed sample design. Simulations are conducted to study the operating characteristics of the proposed group sequential designs. The robustness of the proposed group sequential designs for misspecifying random delay time distribution and domain is studied via simulations.

The Inverse Log-Rank Test: A Versatile Procedure for Late Separating Survival Curves

Often in the planning phase of a clinical trial, a researcher will need to choose between a standard versus weighted log-rank test (LRT) for investigating right-censored survival data. While a standard LRT is optimal for analyzing evenly distributed but distinct survival events (proportional hazards), an appropriately weighted LRT test may be better suited for handling non-proportional, delayed treatment effects. The "a priori" misspecification of this alternative may result in a substantial loss of power when determining the effectiveness of an experimental drug. In this paper, the standard unweighted and inverse log-rank tests (iLRTs) are compared with the multiple weight, default Max-Combo procedure for analyzing differential late survival outcomes. Unlike combination LRTs that depend on the arbitrary selection of weights, the iLRT by definition is a single weight test and does not require implicit multiplicity correction. Empirically, both weighted methods have reasonable flexibility for assessing continuous survival curve differences from the onset of a study. However, the iLRT may be preferable for accommodating delayed separating survival curves, especially when one arm finishes first. Using standard large-sample methods, the power and sample size for the iLRT are easily estimated without resorting to complex and timely simulations.

Stratified modestly weighted log-rank tests in settings with an anticipated delayed separation of survival curves

Delayed separation of survival curves is a common occurrence in confirmatory studies in immuno-oncology. Many novel statistical methods that aim to efficiently capture potential long-term survival improvements have been proposed in recent years. However, the vast majority do not consider stratification, which is a major limitation considering that most large confirmatory studies currently employ a stratified primary analysis. In this article, we combine recently proposed weighted log-rank tests that have been designed to work well under a delayed separation of survival curves, with stratification by a baseline variable. The aim is to increase the efficiency of the test when the stratifying variable is highly prognostic for survival. As there are many potential ways to combine the two techniques, we compare several possibilities in an extensive simulation study. We also apply the techniques retrospectively to two recent randomized clinical trials.