Power and sample size for randomized phase III survival trials under the Weibull model

Statistical power and sample size calculations for time-to-event analysis

OBJECTIVE

To provide thoracic and cardiovascular surgeons with the necessary tools for performing sample size and power calculations for studies with time-to-event outcomes.

METHODS

Power and sample size calculations enhance the overall quality of research studies by providing readers with assurance and insight into the number of patients included in the study. A 5-step approach is presented for performing sample size calculations in comparing groups on time-to-event endpoints. The steps are as follows: (1) identify the primary outcome of interest, (2) define size of the effect and desired power, (3) determine the appropriate statistical test, (4) perform calculations of the required sample size, and (5) write formal power and sample size statement. This approach is demonstrated with 5 clinical examples for time-to-event studies in cardiovascular surgery, featuring Cox regression, 2-sample log-rank test, 1-sample log-rank test, and competing risks analysis.

CONCLUSIONS

Statistical power is an essential element for designing studies to ensure sufficient sample sizes for detecting treatment effects or group differences in time-to-event patient outcomes. Power and sample size justification not only adds statistical rigor and credibility to research manuscripts, but also provides the reader with assurance that the findings and conclusions are valid and based on a sufficient number of patients.

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.

Parameter estimation of Weibull distribution for the number of days between drug administration and early onset adverse event

The Weibull distribution is applied to the number of days between the start date of drug administration and the date of occurrence of an adverse event. The tendency of occurrence of adverse events can be clarified by estimating the two- or three-parameter Weibull distribution, using the data regarding the number of days. Our purpose is to estimate the parameters of the Weibull distribution with high accuracy, even in low-reported adverse events, such as new drugs, polypharmacy and small clinical trials. Furthermore, the two-sample Kolmogorov - Smirnov test (two-sided) is used to examine whether the tendency of occurrence of adverse events is different for two Weibull distributions estimated from two drugs with similar efficacy. We used discrete data derived from FDA Adverse Event Reporting System (FAERS), as the FAERS data are presented in years, months and days without hours and minutes. Because this study focuses on early onset adverse events, data may be contained 0 days. The discreteness of the data and the fact that it may include zero make this distribution different from the general Weibull distribution, which is defined for continuous data greater than zero. We search for the optimal parameter estimation method for the Weibull distribution under these two conditions, and verify its effectiveness using Monte Carlo simulations and FAERS data. Because the results obtained from FAERS data may differ depending on data handling, we describe the of data handling technique and the sample code that can reproduce the results.

Cancer patient survival can be parametrized to improve trial precision and reveal time-dependent therapeutic effects

Individual participant data (IPD) from oncology clinical trials is invaluable for identifying factors that influence trial success and failure, improving trial design and interpretation, and comparing pre-clinical studies to clinical outcomes. However, the IPD used to generate published survival curves are not generally publicly available. We impute survival IPD from ~500 arms of Phase 3 oncology trials (representing ~220,000 events) and find that they are well fit by a two-parameter Weibull distribution. Use of Weibull functions with overall survival significantly increases the precision of small arms typical of early phase trials: analysis of a 50-patient trial arm using parametric forms is as precise as traditional, non-parametric analysis of a 90-patient arm. We also show that frequent deviations from the Cox proportional hazards assumption, particularly in trials of immune checkpoint inhibitors, arise from time-dependent therapeutic effects. Trial duration therefore has an underappreciated impact on the likelihood of success.

Analysis of more than 150 Phase 3 oncology clinical trials supports parametric statistical analysis, significantly increasing the precision of small early-phase trials and relating deviations from the Cox proportional hazards model to trial duration.

Complex survival trial design by the product integration method

Nonproportional hazards (NPHs) are often observed in survival trials such as the immunotherapy cancer trials. Under NPH, the classical log-rank test can be inefficient, and the estimated hazards ratio from the Cox model is difficult to interpret. The weighted log-rank test, and the tests for comparing the restricted mean survival time or the milestone survival become increasingly popular in handling NPH. The sample size calculation for these tests may require high-dimensional numerical integration. We present a sample size determination method for survival trials via product integration on the basis of a continuous-time multistate Markov model. The main challenge of the method lies in the design of the multistate model under a complex NPH pattern, and this is illustrated for NPH induced by delayed effect with individual heterogeneity in the lag duration, cure fractions, and treatment switching due to disease progression or noncompliance. Numerical examples are presented to demonstrate the accuracy of the proposed method. We obtain the following findings. The powers of the tests for milestone survival and RMST depend on both the trial duration and milestone timepoint, and may not increase as the milestone timepoint increases. If the milestone timepoint is appropriately chosen, the RMST test can be more powerful than the conventional log-rank test in the presence of diminishing treatment effect or in the proportional hazards cure model. In general, the RMST test yields lower power than a proper Fleming-Harrington weighted log-rank test.

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.

Spontaneous and Immune Checkpoint Inhibitor-Induced Autoimmune Diseases: Analysis of Temporal Information by Using the Japanese Adverse Drug Event Report Database

BACKGROUND AND OBJECTIVE

Immune checkpoint inhibitors (ICIs) such as programmed cell death protein 1 (PD-1) and cytotoxic T-lymphocyte antigen 4 (CTLA-4) inhibitors have greatly improved cancer treatment. However, they are associated with immune-related adverse events, including autoimmune diseases (ADs) owing to their immune enhancement effect. As there are few comprehensive studies of ADs by ICIs, it is necessary to analyze the period information of drug-induced ADs. We also assumed that the temporal information may be useful to estimate the similarity of the pathogenic mechanism between spontaneous and ICI-induced ADs.

METHODS

A period analysis including the Weibull analysis was performed on ICI-induced ADs using the Japanese Adverse Drug Event Report (JADER) database. For evaluating the similarity of spontaneous and ICI-induced ADs, a hierarchical cluster analysis was conducted to compare the different onset-time ranges.

RESULTS

Type 1 diabetes mellitus, autoimmune colitis, and pemphigoid occurred earlier with CTLA-4 inhibitors (median: 46, 29.5 and 28 days, respectively) than with PD-1 inhibitors (> 130 days). Myasthenia gravis had a median time to onset of approximately 1 month, and the risk of onset would increase over time in ipilimumab combination therapy. This result reveals ADs that require attention. Using cluster analysis, we estimated six clusters with different patterns of onset times. Based on these results and a detailed previous research survey, the possible pathogenesis of drug-induced ADs was also discussed.

CONCLUSIONS

This paper describes risk profiles with temporal information of ICI-induced ADs and proposes certain indicators for deciphering the mechanism of AD onset.

A unified approach to power and sample size determination for log-rank tests under proportional and nonproportional hazards

Log-rank tests have been widely used to compare two survival curves in biomedical research. We describe a unified approach to power and sample size calculation for the unweighted and weighted log-rank tests in superiority, noninferiority and equivalence trials. It is suitable for both time-driven and event-driven trials. A numerical algorithm is suggested. It allows flexible specification of the patient accrual distribution, baseline hazards, and proportional or nonproportional hazards patterns, and enables efficient sample size calculation when there are a range of choices for the patient accrual pattern and trial duration. A confidence interval method is proposed for the trial duration of an event-driven trial. We point out potential issues with several popular sample size formulae. Under proportional hazards, the power of a survival trial is commonly believed to be determined by the number of observed events. The belief is roughly valid for noninferiority and equivalence trials with similar survival and censoring distributions between two groups, and for superiority trials with balanced group sizes. In unbalanced superiority trials, the power depends also on other factors such as data maturity. Surprisingly, the log-rank test usually yields slightly higher power than the Wald test from the Cox model under proportional hazards in simulations. We consider various nonproportional hazards patterns induced by delayed effects, cure fractions, and/or treatment switching. Explicit power formulae are derived for the combination test that takes the maximum of two or more weighted log-rank tests to handle uncertain nonproportional hazards patterns. Numerical examples are presented for illustration.

Critical design considerations for time-to-event endpoints in amyotrophic lateral sclerosis clinical trials

BACKGROUND

Funding and resources for low prevalent neurodegenerative disorders such as amyotrophic lateral sclerosis (ALS) are limited, and optimising their use is vital for efficient drug development. In this study, we review the design assumptions for pivotal ALS clinical trials with time-to-event endpoints and provide optimised settings for future trials.

METHODS

We extracted design settings from 13 completed placebo-controlled trials. Optimal assumptions were estimated using parametric survival models in individual participant data (n=4991). Designs were compared in terms of sample size, trial duration, drug use and costs.

RESULTS

Previous trials overestimated the hazard rate by 18.9% (95% CI 3.4% to 34.5%, p=0.021). The median expected HR was 0.56 (range 0.33-0.66). Additionally, we found evidence for an increasing mean hazard rate over time (Weibull shape parameter of 2.03, 95% CI 1.93 to 2.15, p<0.001), which affects the design and planning of future clinical trials. Incorporating accrual time and assuming an increasing hazard rate at the design stage reduced sample size by 33.2% (95% CI 27.9 to 39.4), trial duration by 17.4% (95% CI 11.6 to 23.3), drug use by 14.3% (95% CI 9.6 to 19.0) and follow-up costs by 21.2% (95% CI 15.6 to 26.8).

CONCLUSIONS

Implementing distributional knowledge and incorporating accrual at the design stage could achieve large gains in the efficiency of ALS clinical trials with time-to-event endpoints. We provide an open-source platform that helps investigators to make more accurate sample size calculations and optimise the use of their available resources.

Challenges and shifting paradigms in clinical trials in oncology: the case for immunological and targeted therapies

The advent of immunotherapy has undoubtedly changed the current standard for cancer treatment. Immunotherapy offers the possibility of achieving excellent results—a new alternative for patients with advanced-stage or relapsed disease. Nowadays, the progress made in tumour biology has led to multiple advances in clinical and translational cancer research. Many oncogenic pathways responsible for tumour growth and metastases have been described and, consequently, multiple new cancer therapeutic agents have been developed and are under current investigation. Due to this rapid increase in knowledge and pharmaceutical development, traditional clinical trials designs have encountered major limitations. The pharmacological differences (in toxicity profiles and effectiveness patterns) between immunotherapy and chemotherapy have caused traditional clinical trials to evolve in order to meet this emerging need. This review focuses on the different options pertaining to clinical trial design that have arisen in the field of immuno-oncology, as well as the challenges of accurately interpreting traditional survival analyses within this novel area of cancer medicine.

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.

Group Sequential Design for Randomized Phase III Trials under the Weibull Model

In this article, a parametric sequential test is proposed under the Weibull model. The proposed test is asymptotically normal with an independent increment structure. The sample size for a fixed sample test is derived for the purpose of group sequential trial design. In addition, a multi-stage group sequential procedure is given under the Weibull model by applying the Brownian motion property of the test statistic and sequential conditional probability ratio test methodology.