Treatment effect quantification for time‐to‐event endpoints–Estimands,analysis strategies,and beyond

A draft addendum to ICH E9 has been released for public consultation in August 2017. The addendum focuses on two topics particularly relevant for randomized confirmatory clinical trials: estimands and sensitivity analyses. The need to amend ICH E9 grew out of the realization of a lack of alignment between the objectives of a clinical trial stated in the protocol and the accompanying quantification of the “treatment effect” reported in a regulatory submission. We embed time‐to‐event endpoints in the estimand framework and discuss how the four estimand attributes described in the addendum apply to time‐to‐event endpoints. We point out that if the proportional hazards assumption is not met, the estimand targeted by the most prevalent methods used to analyze time‐to‐event endpoints, logrank test, and Cox regression depends on the censoring distribution. We discuss for a large randomized clinical trial how the analyses for the primary and secondary endpoints as well as the sensitivity analyses actually performed in the trial can be seen in the context of the addendum. To the best of our knowledge, this is the first attempt to do so for a trial with a time‐to‐event endpoint. Questions that remain open with the addendum for time‐to‐event endpoints and beyond are formulated, and recommendations for planning of future trials are given. We hope that this will provide a contribution to developing a common framework based on the final version of the addendum that can be applied to design, protocols, statistical analysis plans, and clinical study reports in the future.     

Sensitivity to censored‐at‐random assumption in the analysis of time‐to‐event endpoints

Over the past years, significant progress has been made in developing statistically rigorous methods to implement clinically interpretable sensitivity analyses for assumptions about the missingness mechanism in clinical trials for continuous and (to a lesser extent) for binary or categorical endpoints. Studies with time‐to‐event outcomes have received much less attention. However, such studies can be similarly challenged with respect to the robustness and integrity of primary analysis conclusions when a substantial number of subjects withdraw from treatment prematurely prior to experiencing an event of interest. We discuss how the methods that are widely used for primary analyses of time‐to‐event outcomes could be extended in a clinically meaningful and interpretable way to stress‐test the assumption of ignorable censoring. We focus on a ‘tipping point’ approach, the objective of which is to postulate sensitivity parameters with a clear clinical interpretation and to identify a setting of these parameters unfavorable enough towards the experimental treatment to nullify a conclusion that was favorable to that treatment. Robustness of primary analysis results can then be assessed based on clinical plausibility of the scenario represented by the tipping point. We study several approaches for conducting such analyses based on multiple imputation using parametric, semi‐parametric, and non‐parametric imputation models and evaluate their operating characteristics via simulation. We argue that these methods are valuable tools for sensitivity analyses of time‐to‐event data and conclude that the method based on piecewise exponential imputation model of survival has some advantages over other methods studied here. Copyright © 2016 John Wiley & Sons, Ltd.     

Nested combination tests with a time‐to‐event endpoint using a short‐term endpoint for design adaptations

Adaptive trial methodology for multiarmed trials and enrichment designs has been extensively discussed in the past. A general principle to construct test procedures that control the family‐wise Type I error rate in the strong sense is based on combination tests within a closed test. Using survival data, a problem arises when using information of patients for adaptive decision making, which are under risk at interim. With the currently available testing procedures, either no testing of hypotheses in interim analyses is possible or there are restrictions on the interim data that can be used in the adaptation decisions as, essentially, only the interim test statistics of the primary endpoint may be used. We propose a general adaptive testing procedure, covering multiarmed and enrichment designs, which does not have these restrictions. An important application are clinical trials, where short‐term surrogate endpoints are used as basis for trial adaptations, and we illustrate how such trials can be designed. We propose statistical models to assess the impact of effect sizes, the correlation structure between the short‐term and the primary endpoint, the sample size, the timing of interim analyses, and the selection rule on the operating characteristics.     

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.     

Predicting analysis time in events‐driven clinical trials using accumulating time‐to‐event surrogate information

For clinical trials with time‐to‐event endpoints, predicting the accrual of the events of interest with precision is critical in determining the timing of interim and final analyses. For example, overall survival (OS) is often chosen as the primary efficacy endpoint in oncology studies, with planned interim and final analyses at a pre‐specified number of deaths. Often, correlated surrogate information, such as time‐to‐progression (TTP) and progression‐free survival, are also collected as secondary efficacy endpoints. It would be appealing to borrow strength from the surrogate information to improve the precision of the analysis time prediction. Currently available methods in the literature for predicting analysis timings do not consider utilizing the surrogate information. In this article, using OS and TTP as an example, a general parametric model for OS and TTP is proposed, with the assumption that disease progression could change the course of the overall survival. Progression‐free survival, related both to OS and TTP, will be handled separately, as it can be derived from OS and TTP. The authors seek to develop a prediction procedure using a Bayesian method and provide detailed implementation strategies under certain assumptions. Simulations are performed to evaluate the performance of the proposed method. An application to a real study is also provided. Copyright © 2015 John Wiley & Sons, Ltd.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

Control‐based imputation for sensitivity analyses in informative censoring for recurrent event data

In clinical trials, missing data commonly arise through nonadherence to the randomized treatment or to study procedure. For trials in which recurrent event endpoints are of interests, conventional analyses using the proportional intensity model or the count model assume that the data are missing at random, which cannot be tested using the observed data alone. Thus, sensitivity analyses are recommended. We implement the control‐based multiple imputation as sensitivity analyses for the recurrent event data. We model the recurrent event using a piecewise exponential proportional intensity model with frailty and sample the parameters from the posterior distribution. We impute the number of events after dropped out and correct the variance estimation using a bootstrap procedure. We apply the method to an application of sitagliptin study.     

Investigation of mixed model repeated measures analyses and non‐linear random coefficient models in the context of long‐term efficacy data

The longitudinal data from 2 published clinical trials in adult subjects with upper limb spasticity (a randomized placebo‐controlled study [NCT01313299] and its long‐term open‐label extension [NCT01313312]) were combined. Their study designs involved repeat intramuscular injections of abobotulinumtoxinA (Dysport®), and efficacy endpoints were collected accordingly. With the objective of characterizing the pattern of response across cycles, Mixed Model Repeated Measures analyses and Non‐Linear Random Coefficient (NLRC) analyses were performed and their results compared. The Mixed Model Repeated Measures analyses, commonly used in the context of repeated measures with missing dependent data, did not involve any parametric shape for the curve of changes over time. Based on clinical expectations, the NLRC included a negative exponential function of the number of treatment cycles, with its asymptote and rate included as random coefficients in the model. Our analysis focused on 2 specific efficacy parameters reflecting complementary aspects of efficacy in the study population. A simulation study based on a similar study design was also performed to further assess the performance of each method under different patterns of response over time. This highlighted a gain of precision with the NLRC model, and most importantly the need for its assumptions to be verified to avoid potentially biased estimates. These analyses describe a typical situation and the conditions under which non‐linear mixed modeling can provide additional insights on the behavior of efficacy parameters over time. Indeed, the resulting estimates from the negative exponential NLRC can help determine the expected maximal effect and the treatment duration required to reach it.     

Joint modeling tumor burden and time to event data in oncology trials

The tumor burden (TB) process is postulated to be the primary mechanism through which most anticancer treatments provide benefit. In phase II oncology trials, the biologic effects of a therapeutic agent are often analyzed using conventional endpoints for best response, such as objective response rate and progression‐free survival, both of which causes loss of information. On the other hand, graphical methods including spider plot and waterfall plot lack any statistical inference when there is more than one treatment arm. Therefore, longitudinal analysis of TB data is well recognized as a better approach for treatment evaluation. However, longitudinal TB process suffers from informative missingness because of progression or death. We propose to analyze the treatment effect on tumor growth kinetics using a joint modeling framework accounting for the informative missing mechanism. Our approach is illustrated by multisetting simulation studies and an application to a nonsmall‐cell lung cancer data set. The proposed analyses can be performed in early‐phase clinical trials to better characterize treatment effect and thereby inform decision‐making. Copyright © 2014 John Wiley & Sons, Ltd.     

Sequential parallel comparison design with two coprimary endpoints

A placebo‐controlled randomized clinical trial is required to demonstrate that an experimental treatment is superior to its corresponding placebo on multiple coprimary endpoints. This is particularly true in the field of neurology. In fact, clinical trials for neurological disorders need to show the superiority of an experimental treatment over a placebo in two coprimary endpoints. Unfortunately, these trials often fail to detect a true treatment effect for the experimental treatment versus the placebo owing to an unexpectedly high placebo response rate. Sequential parallel comparison design (SPCD) can be used to address this problem. However, the SPCD has not yet been discussed in relation to clinical trials with coprimary endpoints. In this article, our aim was to develop a hypothesis‐testing method and a method for calculating the corresponding sample size for the SPCD with two coprimary endpoints. In a simulation, we show that the proposed hypothesis‐testing method achieves the nominal type I error rate and power and that the proposed sample size calculation method has adequate power accuracy. In addition, the usefulness of our methods is confirmed by returning to an SPCD trial with a single primary endpoint of Alzheimer disease‐related agitation.     

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.     

Testing treatment‐by‐period interaction in four‐period crossover trials

Statistical analyses of crossover clinical trials have mainly focused on assessing the treatment effect, carryover effect, and period effect. When a treatment‐by‐period interaction is plausible, it is important to test such interaction first before making inferences on differences among individual treatments. Considerably less attention has been paid to the treatment‐by‐period interaction, which has historically been aliased with the carryover effect in two‐period or three‐period designs. In this article, from the data of a newly developed four‐period crossover design, we propose a statistical method to compare the effects of two active drugs with respect to two response variables. We study estimation and hypothesis testing considering the treatment‐by‐period interaction. Constrained least squares is used to estimate the treatment effect, period effect, and treatment‐by‐period interaction. For hypothesis testing, we extend a general multivariate method for analyzing the crossover design with multiple responses. Results from simulation studies have shown that this method performs very well. We also illustrate how to apply our method to the real data problem.     

A temporal correlation function for the association between patient‐reported outcome and clinical endpoints

For clinical trials with multiple endpoints, the primary interest is usually to evaluate the relationship of these endpoints and treatment interventions. Studying the correlation of two clinical trial endpoints can also be of interests. For example, the association between patient‐reported outcome and clinically assessed endpoint could answer important research questions and also generate interesting hypothesis for future research. However, it is not straightforward to quantify such association. In this article, we proposed a multiple event approach to profile such association with a temporal correlation function, visualized by a correlation function plot over time with a confidence band. We developed this approach by extending the existing methodology in recurrent event literature. This approach was shown to be generally unbiased and could be a useful tool for data visualization and inference. We demonstrated the use of this method with data from a real clinical trial. Although this approach was developed to evaluate the association between patient‐reported outcome and adverse events, it can also be used to evaluate the association of any two endpoints that can be translated to time‐to‐event endpoints. Copyright © 2015 John Wiley & Sons, Ltd.     

Hazard ratio inference in stratified clinical trials with time‐to‐event endpoints and limited sample size

The stratified Cox model is commonly used for stratified clinical trials with time‐to‐event endpoints. The estimated log hazard ratio is approximately a weighted average of corresponding stratum‐specific Cox model estimates using inverse‐variance weights; the latter are optimal only under the (often implausible) assumption of a constant hazard ratio across strata. Focusing on trials with limited sample sizes (50‐200 subjects per treatment), we propose an alternative approach in which stratum‐specific estimates are obtained using a refined generalized logrank (RGLR) approach and then combined using either sample size or minimum risk weights for overall inference. Our proposal extends the work of Mehrotra et al, to incorporate the RGLR statistic, which outperforms the Cox model in the setting of proportional hazards and small samples. This work also entails development of a remarkably accurate plug‐in formula for the variance of RGLR‐based estimated log hazard ratios. We demonstrate using simulations that our proposed two‐step RGLR analysis delivers notably better results through smaller estimation bias and mean squared error and larger power than the stratified Cox model analysis when there is a treatment‐by‐stratum interaction, with similar performance when there is no interaction. Additionally, our method controls the type I error rate while the stratified Cox model does not in small samples. We illustrate our method using data from a clinical trial comparing two treatments for colon cancer.     

Practical approaches for design and analysis of clinical trials of infertility treatments: crossover designs and the Mantel–Haenszel method are recommended

Crossover designs have some advantages over standard clinical trial designs and they are often used in trials evaluating the efficacy of treatments for infertility. However, clinical trials of infertility treatments violate a fundamental condition of crossover designs, because women who become pregnant in the first treatment period are not treated in the second period. In previous research, to deal with this problem, some new designs, such as re‐randomization designs, and analysis methods including the logistic mixture model and the beta‐binomial mixture model were proposed. Although the performance of these designs and methods has previously been evaluated in large‐scale clinical trials with sample sizes of more than 1000 per group, the actual sample sizes of infertility treatment trials are usually around 100 per group. The most appropriate design and analysis for these moderate‐scale clinical trials are currently unclear. In this study, we conducted simulation studies to determine the appropriate design and analysis method of moderate‐scale clinical trials for irreversible endpoints by evaluating the statistical power and bias in the treatment effect estimates. The Mantel–Haenszel method had similar power and bias to the logistic mixture model. The crossover designs had the highest power and the smallest bias. We recommend using a combination of the crossover design and the Mantel–Haenszel method for two‐period, two‐treatment clinical trials with irreversible endpoints. Copyright © 2015 John Wiley & Sons, Ltd.     

Some statistical considerations in the clinical development of cancer immunotherapies

Immuno‐oncology has emerged as an exciting new approach to cancer treatment. Common immunotherapy approaches include cancer vaccine, effector cell therapy, and T‐cell–stimulating antibody. Checkpoint inhibitors such as cytotoxic T lymphocyte–associated antigen 4 and programmed death‐1/L1 antagonists have shown promising results in multiple indications in solid tumors and hematology. However, the mechanisms of action of these novel drugs pose unique statistical challenges in the accurate evaluation of clinical safety and efficacy, including late‐onset toxicity, dose optimization, evaluation of combination agents, pseudoprogression, and delayed and lasting clinical activity. Traditional statistical methods may not be the most accurate or efficient. It is highly desirable to develop the most suitable statistical methodologies and tools to efficiently investigate cancer immunotherapies. In this paper, we summarize these issues and discuss alternative methods to meet the challenges in the clinical development of these novel agents. For safety evaluation and dose‐finding trials, we recommend the use of a time‐to‐event model‐based design to handle late toxicities, a simple 3‐step procedure for dose optimization, and flexible rule‐based or model‐based designs for combination agents. For efficacy evaluation, we discuss alternative endpoints/designs/tests including the time‐specific probability endpoint, the restricted mean survival time, the generalized pairwise comparison method, the immune‐related response criteria, and the weighted log‐rank or weighted Kaplan‐Meier test. The benefits and limitations of these methods are discussed, and some recommendations are provided for applied researchers to implement these methods in clinical practice.     

Weighted log‐rank test for time‐to‐event data in immunotherapy trials with random delayed treatment effect and cure rate

A cancer clinical trial with an immunotherapy often has 2 special features, which are patients being potentially cured from the cancer and the immunotherapy starting to take clinical effect after a certain delay time. Existing testing methods may be inadequate for immunotherapy clinical trials, because they do not appropriately take the 2 features into consideration at the same time, hence have low power to detect the true treatment effect. In this paper, we proposed a piece‐wise proportional hazards cure rate model with a random delay time to fit data, and a new weighted log‐rank test to detect the treatment effect of an immunotherapy over a chemotherapy control. We showed that the proposed weight was nearly optimal under mild conditions. Our simulation study showed a substantial gain of power in the proposed test over the existing tests and robustness of the test with misspecified weight. We also introduced a sample size calculation formula to design the immunotherapy clinical trials using the proposed weighted log‐rank test.     

A simulation study of methods for selecting subgroup‐specific doses in phase 1 trials

Patient heterogeneity may complicate dose‐finding in phase 1 clinical trials if the dose‐toxicity curves differ between subgroups. Conducting separate trials within subgroups may lead to infeasibly small sample sizes in subgroups having low prevalence. Alternatively,it is not obvious how to conduct a single trial while accounting for heterogeneity. To address this problem,we consider a generalization of the continual reassessment method on the basis of a hierarchical Bayesian dose‐toxicity model that borrows strength between subgroups under the assumption that the subgroups are exchangeable. We evaluate a design using this model that includes subgroup‐specific dose selection and safety rules. A simulation study is presented that includes comparison of this method to 3 alternative approaches,on the basis of nonhierarchical models,that make different types of assumptions about within‐subgroup dose‐toxicity curves. The simulations show that the hierarchical model‐based method is recommended in settings where the dose‐toxicity curves are exchangeable between subgroups. We present practical guidelines for application and provide computer programs for trial simulation and conduct.     

Non‐inferiority and networks: inferring efficacy from a web of data

In the absence of placebo‐controlled trials, the efficacy of a test treatment can be alternatively examined by showing its non‐inferiority to an active control; that is, the test treatment is not worse than the active control by a pre‐specified margin. The margin is based on the effect of the active control over placebo in historical studies. In other words, the non‐inferiority setup involves a network of direct and indirect comparisons between test treatment, active controls, and placebo. Given this framework, we consider a Bayesian network meta‐analysis that models the uncertainty and heterogeneity of the historical trials into the non‐inferiority trial in a data‐driven manner through the use of the Dirichlet process and power priors. Depending on whether placebo was present in the historical trials, two cases of non‐inferiority testing are discussed that are analogs of the synthesis and fixed‐margin approach. In each of these cases, the model provides a more reliable estimate of the control given its effect in other trials in the network, and, in the case where placebo was only present in the historical trials, the model can predict the effect of the test treatment over placebo as if placebo had been present in the non‐inferiority trial. It can further answer other questions of interest, such as comparative effectiveness of the test treatment among its comparators. More importantly, the model provides an opportunity for disproportionate randomization or the use of small sample sizes by allowing borrowing of information from a network of trials to draw explicit conclusions on non‐inferiority. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian design and analysis of composite endpoints in clinical trials with multiple dependent binary outcomes

The author considers studies with multiple dependent primary endpoints. Testing hypotheses with multiple primary endpoints may require unmanageably large populations. Composite endpoints consisting of several binary events may be used to reduce a trial to a manageable size. The primary difficulties with composite endpoints are that different endpoints may have different clinical importance and that higher‐frequency variables may overwhelm effects of smaller, but equally important, primary outcomes. To compensate for these inconsistencies, we weight each type of event, and the total number of weighted events is counted. To reflect the mutual dependency of primary endpoints and to make the weighting method effective in small clinical trials, we use the Bayesian approach. We assume a multinomial distribution of multiple endpoints with Dirichlet priors and apply the Bayesian test of noninferiority to the calculation of weighting parameters. We use composite endpoints to test hypotheses of superiority in single‐arm and two‐arm clinical trials. The composite endpoints have a beta distribution. We illustrate this technique with an example. The results provide a statistical procedure for creating composite endpoints. Published 2013. This article is a U.S. Government work and is in the public domain in the USA.