Adjusted estimates for time-to-event endpoints

In the analysis of retrospective data or when interpreting results from a single-arm phase II clinical trial relative to historical data, it is often of interest to show plots summarizing time-to-event outcomes comparing treatment groups. If the groups being compared are imbalanced with respect to factors known to influence outcome, these plots can be misleading and seemingly incompatible with results obtained from a regression model that accounts for these imbalances. We consider ways in which covariate information can be used to obtain adjusted curves for time-to-event outcomes. We first review a common model-based method and then suggest another model-based approach that is not as reliant on model assumptions. Finally, an approach that is partially model free is suggested. Each method is applied to an example from hematopoietic cell transplantation.     

Covariate‐adjusted borrowing of historical control data in randomized clinical trials

The borrowing of historical control data can be an efficient way to improve the treatment effect estimate of the current control group in a randomized clinical trial. When the historical and current control data are consistent, the borrowing of historical data can increase power and reduce Type I error rate. However, when these 2 sources of data are inconsistent, it may result in a combination of biased estimates, reduced power, and inflation of Type I error rate. In some situations, inconsistency between historical and current control data may be caused by a systematic variation in the measured baseline prognostic factors, which can be appropriately addressed through statistical modeling. In this paper, we propose a Bayesian hierarchical model that can incorporate patient‐level baseline covariates to enhance the appropriateness of the exchangeability assumption between current and historical control data. The performance of the proposed method is shown through simulation studies, and its application to a clinical trial design for amyotrophic lateral sclerosis is described. The proposed method is developed for scenarios involving multiple imbalanced prognostic factors and thus has meaningful implications for clinical trials evaluating new treatments for heterogeneous diseases such as amyotrophic lateral sclerosis.     

Improving interim decisions in randomized trials by exploiting information on short‐term endpoints and prognostic baseline covariates

Conditional power calculations are frequently used to guide the decision whether or not to stop a trial for futility or to modify planned sample size. These ignore the information in short‐term endpoints and baseline covariates, and thereby do not make fully efficient use of the information in the data. We therefore propose an interim decision procedure based on the conditional power approach which exploits the information contained in baseline covariates and short‐term endpoints. We will realize this by considering the estimation of the treatment effect at the interim analysis as a missing data problem. This problem is addressed by employing specific prediction models for the long‐term endpoint which enable the incorporation of baseline covariates and multiple short‐term endpoints. We show that the proposed procedure leads to an efficiency gain and a reduced sample size, without compromising the Type I error rate of the procedure, even when the adopted prediction models are misspecified. In particular, implementing our proposal in the conditional power approach enables earlier decisions relative to standard approaches, whilst controlling the probability of an incorrect decision. This time gain results in a lower expected number of recruited patients in case of stopping for futility, such that fewer patients receive the futile regimen. We explain how these methods can be used in adaptive designs with unblinded sample size re‐assessment based on the inverse normal P‐value combination method to control Type I error. We support the proposal by Monte Carlo simulations based on data from a real clinical trial.     

An investigation into the two‐stage meta‐analytic copula modelling approach for evaluating time‐to‐event surrogate endpoints which comprise of one or more events of interest

Clinical trials of experimental treatments must be designed with primary endpoints that directly measure clinical benefit for patients. In many disease areas, the recognised gold standard primary endpoint can take many years to mature, leading to challenges in the conduct and quality of clinical studies. There is increasing interest in using shorter‐term surrogate endpoints as substitutes for costly long‐term clinical trial endpoints; such surrogates need to be selected according to biological plausibility, as well as the ability to reliably predict the unobserved treatment effect on the long‐term endpoint. A number of statistical methods to evaluate this prediction have been proposed; this paper uses a simulation study to explore one such method in the context of time‐to‐event surrogates for a time‐to‐event true endpoint. This two‐stage meta‐analytic copula method has been extensively studied for time‐to‐event surrogate endpoints with one event of interest, but thus far has not been explored for the assessment of surrogates which have multiple events of interest, such as those incorporating information directly from the true clinical endpoint. We assess the sensitivity of the method to various factors including strength of association between endpoints, the quantity of data available, and the effect of censoring. In particular, we consider scenarios where there exist very little data on which to assess surrogacy. Results show that the two‐stage meta‐analytic copula method performs well under certain circumstances and could be considered useful in practice, but demonstrates limitations that may prevent universal use.     

A dynamic power prior for borrowing historical data in noninferiority trials with binary endpoint

Traditionally, noninferiority hypotheses have been tested using a frequentist method with a fixed margin. Given that information for the control group is often available from previous studies, it is interesting to consider a Bayesian approach in which information is “borrowed” for the control group to improve efficiency. However, construction of an appropriate informative prior can be challenging. In this paper, we consider a hybrid Bayesian approach for testing noninferiority hypotheses in studies with a binary endpoint. To account for heterogeneity between the historical information and the current trial for the control group, a dynamic P value–based power prior parameter is proposed to adjust the amount of information borrowed from the historical data. This approach extends the simple test‐then‐pool method to allow a continuous discounting power parameter. An adjusted α level is also proposed to better control the type I error. Simulations are conducted to investigate the performance of the proposed method and to make comparisons with other methods including test‐then‐pool and hierarchical modeling. The methods are illustrated with data from vaccine clinical trials.     

Study design of single‐arm phase II immunotherapy trials with long‐term survivors and random delayed treatment effect

In the traditional study design of a single‐arm phase II cancer clinical trial, the one‐sample log‐rank test has been frequently used. A common practice in sample size calculation is to assume that the event time in the new treatment follows exponential distribution. Such a study design may not be suitable for immunotherapy cancer trials, when both long‐term survivors (or even cured patients from the disease) and delayed treatment effect are present, because exponential distribution is not appropriate to describe such data and consequently could lead to severely underpowered trial. In this research, we proposed a piecewise proportional hazards cure rate model with random delayed treatment effect to design single‐arm phase II immunotherapy cancer trials. To improve test power, we proposed a new weighted one‐sample log‐rank test and provided a sample size calculation formula for designing trials. Our simulation study showed that the proposed log‐rank test performs well and is robust of misspecified weight and the sample size calculation formula also performs well.     

A novel equivalence probability weighted power prior for using historical control data in an adaptive clinical trial design: A comparison to standard methods

A standard two-arm randomised controlled trial usually compares an intervention to a control treatment with equal numbers of patients randomised to each treatment arm and only data from within the current trial are used to assess the treatment effect. Historical data are used when designing new trials and have recently been considered for use in the analysis when the required number of patients under a standard trial design cannot be achieved. Incorporating historical control data could lead to more efficient trials, reducing the number of controls required in the current study when the historical and current control data agree. However, when the data are inconsistent, there is potential for biased treatment effect estimates, inflated type I error and reduced power. We introduce two novel approaches for binary data which discount historical data based on the agreement with the current trial controls, an equivalence approach and an approach based on tail area probabilities. An adaptive design is used where the allocation ratio is adapted at the interim analysis, randomising fewer patients to control when there is agreement. The historical data are down-weighted in the analysis using the power prior approach with a fixed power. We compare operating characteristics of the proposed design to historical data methods in the literature: the modified power prior; commensurate prior; and robust mixture prior. The equivalence probability weight approach is intuitive and the operating characteristics can be calculated exactly. Furthermore, the equivalence bounds can be chosen to control the maximum possible inflation in type I error.     

Treatment effect measures under nonproportional hazards

In a clinical trial with a time-to-event endpoint the treatment effect can be measured in various ways. Under proportional hazards all reasonable measures (such as the hazard ratio and the difference in restricted mean survival time) are consistent in the following sense: Take any control group survival distribution such that the hazard rate remains above zero; if there is no benefit by any measure there is no benefit by all measures, and as the magnitude of treatment benefit increases by any measure it increases by all measures. Under nonproportional hazards, however, survival curves can cross, and the direction of the effect for any pair of measures can be inconsistent. In this paper we critically evaluate a variety of treatment effect measures in common use and identify flaws with them. In particular, we demonstrate that a treatment's benefit has two distinct and independent dimensions which can be measured by the difference in the survival rate at the end of follow-up and the difference in restricted mean survival time, and that commonly used measures do not adequately capture both dimensions. We demonstrate that a generalized hazard difference, which can be estimated by the difference in exposure-adjusted subject incidence rates, captures both dimensions, and that its inverse, the number of patient-years of follow-up that results in one fewer event (the NYNT), is an easily interpretable measure of the magnitude of clinical benefit.     

Adjusting overall survival for treatment switches: commonly used methods and practical application

In parallel group trials, long‐term efficacy endpoints may be affected if some patients switch or cross over to the alternative treatment arm prior to the event. In oncology trials, switch to the experimental treatment can occur in the control arm following disease progression and potentially impact overall survival. It may be a clinically relevant question to estimate the efficacy that would have been observed if no patients had switched, for example, to estimate ‘real‐life’ clinical effectiveness for a health technology assessment. Several commonly used statistical methods are available that try to adjust time‐to‐event data to account for treatment switching, ranging from naive exclusion and censoring approaches to more complex inverse probability of censoring weighting and rank‐preserving structural failure time models. These are described, along with their key assumptions, strengths, and limitations. Best practice guidance is provided for both trial design and analysis when switching is anticipated. Available statistical software is summarized, and examples are provided of the application of these methods in health technology assessments of oncology trials. Key considerations include having a clearly articulated rationale and research question and a well‐designed trial with sufficient good quality data collection to enable robust statistical analysis. No analysis method is universally suitable in all situations, and each makes strong untestable assumptions. There is a need for further research into new or improved techniques. This information should aid statisticians and their colleagues to improve the design and analysis of clinical trials where treatment switch is anticipated. Copyright © 2013 John Wiley & Sons, Ltd.     

Incorporating historical two‐arm data in clinical trials with binary outcome: A practical approach

The feasibility of a new clinical trial may be increased by incorporating historical data of previous trials. In the particular case where only data from a single historical trial are available, there exists no clear recommendation in the literature regarding the most favorable approach. A main problem of the incorporation of historical data is the possible inflation of the type I error rate. A way to control this type of error is the so‐called power prior approach. This Bayesian method does not “borrow” the full historical information but uses a parameter 0 ≤ δ ≤ 1 to determine the amount of borrowed data. Based on the methodology of the power prior, we propose a frequentist framework that allows incorporation of historical data from both arms of two‐armed trials with binary outcome, while simultaneously controlling the type I error rate. It is shown that for any specific trial scenario a value δ > 0 can be determined such that the type I error rate falls below the prespecified significance level. The magnitude of this value of δ depends on the characteristics of the data observed in the historical trial. Conditionally on these characteristics, an increase in power as compared to a trial without borrowing may result. Similarly, we propose methods how the required sample size can be reduced. The results are discussed and compared to those obtained in a Bayesian framework. Application is illustrated by a clinical trial example.     

The win ratio: Impact of censoring and follow‐up time and use with nonproportional hazards

The win ratio has been studied methodologically and applied in data analysis and in designing clinical trials. Researchers have pointed out that the results depend on follow‐up time and censoring time, which are sometimes used interchangeably. In this article, we distinguish between follow‐up time and censoring time, show theoretically the impact of censoring on the win ratio, and illustrate the impact of follow‐up time. We then point out that, if the treatment has long‐term benefit from a more important but less frequent endpoint (eg, death), the win ratio can show that benefit by following patients longer, avoiding masking by more frequent but less important outcomes, which occurs in conventional time‐to‐first‐event analyses. For the situation of nonproportional hazards, we demonstrate that the win ratio can be a good alternative to methods such as landmark survival rate, restricted mean survival time, and weighted log‐rank tests.     

An evaluation of the trimmed mean approach in clinical trials with dropout

The trimmed mean is a method of dealing with patient dropout in clinical trials that considers early discontinuation of treatment a bad outcome rather than leading to missing data. The present investigation is the first comprehensive assessment of the approach across a broad set of simulated clinical trial scenarios. In the trimmed mean approach, all patients who discontinue treatment prior to the primary endpoint are excluded from analysis by trimming an equal percentage of bad outcomes from each treatment arm. The untrimmed values are used to calculated means or mean changes. An explicit intent of trimming is to favor the group with lower dropout because having more completers is a beneficial effect of the drug, or conversely, higher dropout is a bad effect. In the simulation study, difference between treatments estimated from trimmed means was greater than the corresponding effects estimated from untrimmed means when dropout favored the experimental group, and vice versa. The trimmed mean estimates a unique estimand. Therefore, comparisons with other methods are difficult to interpret and the utility of the trimmed mean hinges on the reasonableness of its assumptions: dropout is an equally bad outcome in all patients, and adherence decisions in the trial are sufficiently similar to clinical practice in order to generalize the results. Trimming might be applicable to other inter‐current events such as switching to or adding rescue medicine. Given the well‐known biases in some methods that estimate effectiveness, such as baseline observation carried forward and non‐responder imputation, the trimmed mean may be a useful alternative when its assumptions are justifiable.     

The shrinking or disappearing observed treatment effect

It is frequently noted that an initial clinical trial finding was not reproduced in a later trial. This is often met with some surprise. Yet, there is a relatively straightforward reason partially responsible for this observation. In this article, we examine this reason by first reviewing some findings in a recent publication in the Journal of the American Medical Association. To help explain the non‐negligible chance of failing to reproduce a previous positive finding, we compare a series of trials to successive diagnostic tests used for identifying a condition. To help explain the suspicion that the treatment effect, when observed in a subsequent trial, seems to have decreased in magnitude, we draw a conceptual analogy between phases II–III development stages and interim analyses of a trial with a group sequential design. Both analogies remind us that what we observed in an early trial could be a false positive or a random high. We discuss statistical sources for these occurrences and discuss why it is important for statisticians to take these into consideration when designing and interpreting trial results. Copyright © 2014 John Wiley & Sons, Ltd.     

The Interpretation of Clinical Trials of Immediate Versus Delayed Therapy

Prospective, randomised clinical comparisons with a control group are the ideal way to evaluate the effectiveness of a new therapy. However if the new therapy is already available, then it may be unethical to refuse patients this treatment indefinitely. In some trials, patients randomised to the control group receive placebo until their condition deteriorates, and are then switched to the new therapy. Patients randomised to the experimental group receive the new treatment immediately. An analysis following the intention-to-treat principle will be a valid comparison of the two treatment policies actually used. However, such an analysis will underestimate the effect of immediate therapy relative to completely untreated controls, and in particular a negative conclusion should not be interpreted to mean that the therapy is ineffective.In this paper we introduce a parametric approach, which models the dependence between the survival time and the time of switching to the new treatment. This is used first to illustrate the lack of power of the intention-to-treat analysis for evaluating the therapy relative to a pure control. More speculatively, an alternative method of analysis based on our model is presented. We illustrate the issues with data from a prospective randomised study, in which one group of HIV-positive patients received zidovudine immediately after randomisation while control patients were switched to zidovudine only when their condition deteriorated.     

Critical appraisal of Bayesian dynamic borrowing from an imperfectly commensurate historical control

Bayesian dynamic borrowing designs facilitate borrowing information from historical studies. Historical data, when perfectly commensurate with current data, have been shown to reduce the trial duration and the sample size, while inflation in the type I error and reduction in the power have been reported, when imperfectly commensurate. These results, however, were obtained without considering that Bayesian designs are calibrated to meet regulatory requirements in practice and even no‐borrowing designs may use information from historical data in the calibration. The implicit borrowing of historical data suggests that imperfectly commensurate historical data may similarly impact no‐borrowing designs negatively. We will provide a fair appraiser of Bayesian dynamic borrowing and no‐borrowing designs. We used a published selective adaptive randomization design and real clinical trial setting and conducted simulation studies under varying degrees of imperfectly commensurate historical control scenarios. The type I error was inflated under the null scenario of no intervention effect, while larger inflation was noted with borrowing. The larger inflation in type I error under the null setting can be offset by the greater probability to stop early correctly under the alternative. Response rates were estimated more precisely and the average sample size was smaller with borrowing. The expected increase in bias with borrowing was noted, but was negligible. Using Bayesian dynamic borrowing designs may improve trial efficiency by stopping trials early correctly and reducing trial length at the small cost of inflated type I error.     

Use of a historical control group in a noninferiority trial assessing a new antibacterial treatment: A case study and discussion of practical implementation aspects

When recruitment into a clinical trial is limited due to rarity of the disease of interest, or when recruitment to the control arm is limited due to ethical reasons (eg, pediatric studies or important unmet medical need), exploiting historical controls to augment the prospectively collected database can be an attractive option. Statistical methods for combining historical data with randomized data, while accounting for the incompatibility between the two, have been recently proposed and remain an active field of research. The current literature is lacking a rigorous comparison between methods but also guidelines about their use in practice. In this paper, we compare the existing methods based on a confirmatory phase III study design exercise done for a new antibacterial therapy with a binary endpoint and a single historical dataset. A procedure to assess the relative performance of the different methods for borrowing information from historical control data is proposed, and practical questions related to the selection and implementation of methods are discussed. Based on our examination, we found that the methods have a comparable performance, but we recommend the robust mixture prior for its ease of implementation.     

Weighted Mean Survival Test Statistics: a Class of Distance Tests for Censored Survival Data

A class of test statistics is introduced which is sensitive against the alternative of stochastic ordering in the two-sample censored data problem. The test statistics for evaluating a cumulative weighted difference in survival distributions are developed while taking into account the imbalances in base-line covariates between two groups. This procedure can be used to test the null hypothesis of no treatment effect, especially when base-line hazards cross and prognostic covariates need to be adjusted. The statistics are semiparametric, not rank based, and can be written as integrated weighted differences in estimated survival functions, where these survival estimates are adjusted for covariate imbalances. The asymptotic distribution theory of the tests is developed, yielding test procedures that are shown to be consistent under a fixed alternative. The choice of weight function is discussed and relies on stability and interpretability considerations. An example taken from a clinical trial for acquired immune deficiency syndrome is presented.     

Sample size re‐estimation incorporating prior information on a nuisance parameter

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter–based sample size re‐estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta‐analytic‐predictive approach. To incorporate external information into the sample size re‐estimation, we propose to update the meta‐analytic‐predictive prior based on the results of the internal pilot study and to re‐estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re‐estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior‐data conflict is present, incorporating external information into the sample size re‐estimation improves the operating characteristics compared to the traditional approach. In the case of a prior‐data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re‐estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re‐estimation, the potential gains should be balanced against the risks.     

A meta-analytic framework to adjust for bias in external control studies

While randomized controlled trials (RCTs) are the gold standard for estimating treatment effects in medical research, there is increasing use of and interest in using real-world data for drug development. One such use case is the construction of external control arms for evaluation of efficacy in single-arm trials, particularly in cases where randomization is either infeasible or unethical. However, it is well known that treated patients in non-randomized studies may not be comparable to control patients—on either measured or unmeasured variables—and that the underlying population differences between the two groups may result in biased treatment effect estimates as well as increased variability in estimation. To address these challenges for analyses of time-to-event outcomes, we developed a meta-analytic framework that uses historical reference studies to adjust a log hazard ratio estimate in a new external control study for its additional bias and variability. The set of historical studies is formed by constructing external control arms for historical RCTs, and a meta-analysis compares the trial controls to the external control arms. Importantly, a prospective external control study can be performed independently of the meta-analysis using standard causal inference techniques for observational data. We illustrate our approach with a simulation study and an empirical example based on reference studies for advanced non-small cell lung cancer. In our empirical analysis, external control patients had lower survival than trial controls (hazard ratio: 0.907), but our methodology is able to correct for this bias. An implementation of our approach is available in the R package ecmeta .     

Landmark estimation of survival and treatment effects in observational studies

Clinical studies aimed at identifying effective treatments to reduce the risk of disease or death often require long term follow-up of participants in order to observe a sufficient number of events to precisely estimate the treatment effect. In such studies, observing the outcome of interest during follow-up may be difficult and high rates of censoring may be observed which often leads to reduced power when applying straightforward statistical methods developed for time-to-event data. Alternative methods have been proposed to take advantage of auxiliary information that may potentially improve efficiency when estimating marginal survival and improve power when testing for a treatment effect. Recently, Parast et al. (J Am Stat Assoc 109(505):384–394, 2014) proposed a landmark estimation procedure for the estimation of survival and treatment effects in a randomized clinical trial setting and demonstrated that significant gains in efficiency and power could be obtained by incorporating intermediate event information as well as baseline covariates. However, the procedure requires the assumption that the potential outcomes for each individual under treatment and control are independent of treatment group assignment which is unlikely to hold in an observational study setting. In this paper we develop the landmark estimation procedure for use in an observational setting. In particular, we incorporate inverse probability of treatment weights (IPTW) in the landmark estimation procedure to account for selection bias on observed baseline (pretreatment) covariates. We demonstrate that consistent estimates of survival and treatment effects can be obtained by using IPTW and that there is improved efficiency by using auxiliary intermediate event and baseline information. We compare our proposed estimates to those obtained using the Kaplan–Meier estimator, the original landmark estimation procedure, and the IPTW Kaplan–Meier estimator. We illustrate our resulting reduction in bias and gains in efficiency through a simulation study and apply our procedure to an AIDS dataset to examine the effect of previous antiretroviral therapy on survival.