A clinical trial design using the concept of proportional time using the generalized gamma ratio distribution

Group sequential design for time-to-event outcome with non-proportional hazards using the concept of relative time utilizing two different Weibull distributions

A group sequential design allows investigators to sequentially monitor efficacy and safety as part of interim testing in phase III trials. Literature is well developed in the case of continuous and binary outcomes, however, in case of trials with a time-to-event outcome, popular methods of sample size calculation often assume proportional hazards. In situations where the proportional hazards assumption is inappropriate as indicated by historical data, these popular methods are very restrictive. In this paper, a novel simulation-based group sequential design is proposed for a two-arm randomized phase III clinical trial with a survival endpoint for the non-proportional hazards scenario. By assuming that the survival times for each treatment arm follow two different Weibull distributions, the proposed method utilizes the concept of Relative Time to calculate the efficacy and safety boundaries at selected interim testing points. The test statistic used to generate these boundaries is asymptotically normal, allowing p-value calculation at each boundary. Many design features specific to time-to-event data can be incorporated with ease. Additionally, the proposed method allows the flexibility of having the accelerated failure time model and the proportional hazards model as constrained special cases. Real life applications are discussed demonstrating the practicality of the proposed method.

Stochastic curtailment tests for phase II trial with time-to-event outcome using the concept of relative time in the case of non-proportional hazards

As part of the drug development process, interim analysis is frequently used to design efficient phase II clinical trials. A stochastic curtailment framework is often deployed wherein a decision to continue or curtail the trial is taken at each interim look based on the likelihood of observing a positive or negative treatment effect if the trial were to continue to its anticipated end. Thus, curtailment can take place due to evidence of early efficacy or futility. Traditionally, in the case of time-to-event endpoints, interim monitoring is conducted in a two-arm clinical trial using the log-rank test, often with the assumption of proportional hazards. However, when this is violated, the log-rank test may not be appropriate, resulting in loss of power and subsequently inaccurate sample sizes. In this paper, we propose stochastic curtailment methods for two-arm phase II trial with the flexibility to allow non-proportional hazards. The proposed methods are built utilizing the concept of relative time assuming that the survival times in the two treatment arms follow two different Weibull distributions. Three methods - conditional power, predictive power and Bayesian predictive probability - are discussed along with corresponding sample size calculations. The monitoring strategy is discussed with a real-life example.

Sample size calculation for two-arm trials with time-to-event endpoint for nonproportional hazards using the concept of Relative Time when inference is built on comparing Weibull distributions

Sample size calculations for two-arm clinical trials with a time-to-event endpoint have traditionally used the assumption of proportional hazards (PH) or the assumption of exponentially distributed survival times. Available software provides methods for sample size calculation using a nonparametric logrank test, Schoenfeld's formula for Cox PH model, or parametric calculations specific to the exponential distribution. In cases where the PH assumption is not valid, the first-choice method is to compute sample size assuming a piecewise linear survival curve (Lakatos approach) for both the control and treatment arms with judiciously chosen cut-points. Recent advances in literature have used the assumption of Weibull distributed times for single-arm trials, and, newer methods have emerged that allow sample size calculations for two-arm trials using the assumption of proportional time (PT) while considering non-PH. These methods, however, always assume an instantaneous effect of treatment relative to control requiring that the effect size be defined by a single number whose magnitude is preserved throughout the trial duration. Here, we consider the scenarios where the hypothesized benefit of treatment relative to control may not be constant giving rise to the notion of Relative Time (RT). By assuming that survival times for control and treatment arm come from two different Weibull distributions with different location and shape parameters, we develop the methodology for sample size calculation for specific cases of both non-PH and non-PT. Simulations are conducted to assess the operation characteristics of the proposed method and a practical example is discussed.

Impact of a non-constant baseline hazard on detection of time-dependent treatment effects: a simulation study

Background

Non-proportional hazards are common with time-to-event data but the majority of randomised clinical trials (RCTs) are designed and analysed using approaches which assume the treatment effect follows proportional hazards (PH). Recent advances in oncology treatments have identified two forms of non-PH of particular importance - a time lag until treatment becomes effective, and an early effect of treatment that ceases after a period of time. In sample size calculations for treatment effects on time-to-event outcomes where information is based on the number of events rather than the number of participants, there is crucial importance in correct specification of the baseline hazard rate amongst other considerations. Under PH, the shape of the baseline hazard has no effect on the resultant power and magnitude of treatment effects using standard analytical approaches. However, in a non-PH context the appropriateness of analytical approaches can depend on the shape of the underlying hazard.

Methods

A simulation study was undertaken to assess the impact of clinically plausible non-constant baseline hazard rates on the power, magnitude and coverage of commonly utilized regression-based measures of treatment effect and tests of survival curve difference for these two forms of non-PH used in RCTs with time-to-event outcomes.

Results

In the presence of even mild departures from PH, the power, average treatment effect size and coverage were adversely affected. Depending on the nature of the non-proportionality, non-constant event rates could further exacerbate or somewhat ameliorate the losses in power, treatment effect magnitude and coverage observed. No single summary measure of treatment effect was able to adequately describe the full extent of a potentially time-limited treatment benefit whilst maintaining power at nominal levels.

Conclusions

Our results show the increased importance of considering plausible potentially non-constant event rates when non-proportionality of treatment effects could be anticipated. In planning clinical trials with the potential for non-PH, even modest departures from an assumed constant baseline hazard could appreciably impact the power to detect treatment effects depending on the nature of the non-PH. Comprehensive analysis plans may be required to accommodate the description of time-dependent treatment effects.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12874-021-01372-0.

Assessing accuracy of Weibull shape parameter estimate from historical studies for subsequent sample size calculation in clinical trials with time-to-event outcome

Background

Recent developments in literature on sample size calculations for time-to-event outcomes involve assumption of Weibull distributed times. These methods require a point estimate of the Weibull shape parameter obtained from historical studies. However, very limited guidance exists in published literature to assess how reliable this point estimate is when it is obtained from published results of a historical study.

Methods

We conduct simulations to assess how accurate and reliable the point estimate of the Weibull shape parameter is when it is estimated from published results of median survival time and/or corresponding interquartile range. Accuracy of this estimate is assessed using the criteria of average relative bias, root mean square error, and coefficient of variation for various combinations of sample sizes and censoring rates. Sensitivity of these calculations is assessed first, by increasing the number of survival quantiles used to calculate accuracy, and second, by using the full Kaplan Meier (KM) curve from the historical study.

Results

Our simulations suggest that point estimate of the shape parameter is reasonably accurate when estimated from historical studies with sample size ≥ 50 with censoring rate approximately 20%. Knowledge of the median and inter-quartile range seems to be adequate for this purpose. For historical studies with small sample sizes or higher censoring rates, more information needs to be abstracted from the published KM curves to improve accuracy.

Conclusions

We conclude that assessing the accuracy of Weibull shape parameter estimate is important before it can be used to conduct sample size calculations for a subsequent trial.

Group sequential design for time-to-event data using the concept of proportional time

Sequential monitoring of efficacy and safety is an important part of clinical trials. A Group Sequential design allows researchers to perform interim monitoring after groups of patients have completed the study. Statistical literature is well developed for continuous and binary outcomes and relies on asymptotic normality of the test statistic. However, in the case of time-to-event data, existing methods of sample size calculation are done either assuming proportional hazards or assuming exponentially distributed lifetimes. In scenarios where these assumptions are not true, as evidenced from historical data, these traditional methods are restrictive and cannot always be used. As interim monitoring is driven by ethical, financial, and administrative considerations, it is imperative that sample size calculations be done in an efficient manner keeping in mind the specific needs of a clinical trial with a time-to-event outcome. To address these issues, a novel group sequential design is proposed using the concept of Proportional Time. This method utilizes the generalized gamma ratio distribution to calculate the efficacy and safety boundaries and can be used for all distributions that are members of the generalized gamma family using an error spending approach. The design incorporates features specific to survival data such as loss to follow-up, administrative censoring, varying accrual times and patterns, binding or non-binding futility rules with or without skips, and flexible alpha and beta spending mechanisms. Three practical examples are discussed, followed by discussion of the important aspects of the proposed design.

Are non-constant rates and non-proportional treatment effects accounted for in the design and analysis of randomised controlled trials? A review of current practice

Background

Most clinical trials with time-to-event primary outcomes are designed assuming constant event rates and proportional hazards over time. Non-constant event rates and non-proportional hazards are seen increasingly frequently in trials. The objectives of this review were firstly to identify whether non-constant event rates and time-dependent treatment effects were allowed for in sample size calculations of trials, and secondly to assess the methods used for the analysis and reporting of time-to-event outcomes including how researchers accounted for non-proportional treatment effects.

Methods

We reviewed all original reports published between January and June 2017 in four high impact medical journals for trials for which the primary outcome involved time-to-event analysis. We recorded the methods used to analyse and present the main outcomes of the trial and assessed the reporting of assumptions underlying these methods. The sample size calculation was reviewed to see if the effect of either non-constant hazard rates or anticipated non-proportionality of the treatment effect was allowed for during the trial design.

Results

From 446 original reports we identified 66 trials with a time-to-event primary outcome encompassing trial start dates from July 1995 to November 2014. The majority of these trials (73%) had sample size calculations that used standard formulae with a minority of trials (11%) using simulation for anticipated changing event rates and/or non-proportional hazards. Well-established analytical methods, Kaplan-Meier curves (98%), the log rank test (88%) and the Cox proportional hazards model (97%), were used almost exclusively for the main outcome. Parametric regression models were considered in 11% of the reports. Of the trials reporting inference from the Cox model, only 11% reported any results of testing the assumption of proportional hazards.

Conclusions

Our review confirmed that when designing trials with time-to-event primary outcomes, methodologies assuming constant event rates and proportional hazards were predominantly used despite potential efficiencies in sample size needed or power achieved using alternative methods. The Cox proportional hazards model was used almost exclusively to present inferential results, yet testing and reporting of the pivotal assumption underpinning this estimation method was lacking.

Electronic supplementary material

The online version of this article (10.1186/s12874-019-0749-1) contains supplementary material, which is available to authorized users.