Estimation after subpopulation selection in adaptive seamless trials

Adaptive enrichment designs with a continuous biomarker

A popular design for clinical trials assessing targeted therapies is the two-stage adaptive enrichment design with recruitment in stage 2 limited to a biomarker-defined subgroup chosen based on data from stage 1. The data-dependent selection leads to statistical challenges if data from both stages are used to draw inference on treatment effects in the selected subgroup. If subgroups considered are nested, as when defined by a continuous biomarker, treatment effect estimates in different subgroups follow the same distribution as estimates in a group-sequential trial. This result is used to obtain tests controlling the familywise type I error rate (FWER) for six simple subgroup selection rules, one of which also controls the FWER for any selection rule. Two approaches are proposed: one based on multivariate normal distributions suitable if the number of possible subgroups, k, is small, and one based on Brownian motion approximations suitable for large k. The methods, applicable in the wide range of settings with asymptotically normal test statistics, are illustrated using survival data from a breast cancer trial.

Point estimation for adaptive trial designs I: A methodological review

Recent FDA guidance on adaptive clinical trial designs defines bias as "a systematic tendency for the estimate of treatment effect to deviate from its true value," and states that it is desirable to obtain and report estimates of treatment effects that reduce or remove this bias. The conventional end-of-trial point estimates of the treatment effects are prone to bias in many adaptive designs, because they do not take into account the potential and realized trial adaptations. While much of the methodological developments on adaptive designs have tended to focus on control of type I error rates and power considerations, in contrast the question of biased estimation has received relatively less attention. This article is the first in a two-part series that studies the issue of potential bias in point estimation for adaptive trials. Part I provides a comprehensive review of the methods to remove or reduce the potential bias in point estimation of treatment effects for adaptive designs, while part II illustrates how to implement these in practice and proposes a set of guidelines for trial statisticians. The methods reviewed in this article can be broadly classified into unbiased and bias-reduced estimation, and we also provide a classification of estimators by the type of adaptive design. We compare the proposed methods, highlight available software and code, and discuss potential methodological gaps in the literature.

Bias correction based on weighted likelihood for conditional estimation of subgroup effects in randomized clinical trials

Currently, many confirmatory randomized clinical trials (RCTs) with predictive markers have taken the all-comers approach because of the difficulty in developing predictive markers that are biologically compelling enough to apply the enrichment approach to restrict the patient population to a marker-defined subgroup. However, such a RCT with weak marker credentials can conclude that the new treatment is efficacious only in the subgroup, especially when the primary analysis demonstrates some treatment efficacy in the subgroup, but the overall treatment efficacy is not significant under a control of study-wise alpha rate. In this article, we consider conditional estimation of subgroup treatment effects, given the negative result in testing the overall treatment efficacy in the trial. To address the problem of unstable estimation due to the truncation in the distribution of the test statistic on overall treatment efficacy, we propose a new approach based on a weighted likelihood for the truncated distribution. The weighted likelihood can be derived by invoking a randomized test with a smooth critical function for the overall test. Our approach allows for point and interval estimations of the conditional effects consistently based on the standard maximum likelihood inference. Numerical evaluations, including simulations and application to real clinical trials, and guidelines for implementing our methods with R-codes, are provided.

Estimation of treatment effect in 2-in-1 adaptive design and some of its extensions

The 2-in-1 adaptive design allows seamless expansion of an ongoing Phase II trial into a Phase III trial to expedite a drug development program. Since its publication, it has generated a lot of interest. So far, most of the related research focused on type I error control. Similar to most adaptive designs, 2-in-1 design could also pose a great challenge on estimation of treatment effect due to the data-driven adaptation. In addition, the use of intermediate endpoint for interim adaptive decision-making is a less well-studied field. In this paper, we investigate the bias and variances in estimation for 2-in-1 design and some of its extensions, and propose some bias-adjusted estimators for 2-in-1 design. The properties of the proposed estimators are further studied theoretically and/or numerically, so as to provide guidance on how to interpret the estimated treatment effect of 2-in-1 design.

Optimizing subgroup selection in two-stage adaptive enrichment and umbrella designs

We design two‐stage confirmatory clinical trials that use adaptation to find the subgroup of patients who will benefit from a new treatment, testing for a treatment effect in each of two disjoint subgroups. Our proposal allows aspects of the trial, such as recruitment probabilities of each group, to be altered at an interim analysis. We use the conditional error rate approach to implement these adaptations with protection of overall error rates. Applying a Bayesian decision‐theoretic framework, we optimize design parameters by maximizing a utility function that takes the population prevalence of the subgroups into account. We show results for traditional trials with familywise error rate control (using a closed testing procedure) as well as for umbrella trials in which only the per‐comparison type 1 error rate is controlled. We present numerical examples to illustrate the optimization process and the effectiveness of the proposed designs.

A Bayesian Decision-Theoretic Design for Simultaneous Biomarker-Based Subgroup Selection and Efficacy Evaluation

The success of drug development of targeted therapy often hinges on an appropriate selection of the sensitive patient population, mostly based on patients' biomarker levels. At the planning stage of a phase II study, although a potential biomarker may have been identified, a threshold value for defining sensitive patient population is often unavailable for adopting many existing biomarker-guided designs. To address this issue, we propose a two-stage design that allows for simultaneous biomarker threshold selection and efficacy evaluation while accommodating situations where the drug is efficacious in the entire patient population. The design uses a Bayesian decision-theoretic approach and incorporates the benefit and cost considerations of the study into a utility function. The operating characteristics of the proposed design under different scenarios are investigated via simulations. We also provide a discussion on the choice of the benefit and cost parameters in practice.

Bayesian group sequential enrichment designs based on adaptive regression of response and survival time on baseline biomarkers

Precision medicine relies on the idea that, for a particular targeted agent, only a subpopulation of patients is sensitive to it and thus may benefit from it therapeutically. In practice, it is often assumed based on preclinical data that a treatment-sensitive subpopulation is known, and moreover that the agent is substantively efficacious in that subpopulation. Due to important differences between preclinical settings and human biology, however, data from patients treated with a new targeted agent often show that one or both of these assumptions are false. This paper provides a Bayesian randomized group sequential enrichment design that compares an experimental treatment to a control based on survival time and uses early response as an ancillary outcome to assist with adaptive variable selection and enrichment. Initially, the design enrolls patients under broad eligibility criteria. At each interim decision, submodels for regression of response and survival time on a baseline covariate vector and treatment are fit; variable selection is used to identify a covariate subvector that characterizes treatment-sensitive patients and determines a personalized benefit index, and comparative superiority and futility decisions are made. Enrollment of each cohort is restricted to the most recent adaptively identified treatment-sensitive patients. Group sequential decision cutoffs are calibrated to control overall type I error and account for the adaptive enrollment restriction. The design provides a basis for precision medicine by identifying a treatment-sensitive subpopulation, if it exists, and determining whether the experimental treatment is superior to the control in that subpopulation. A simulation study shows that the proposed design reliably identifies a sensitive subpopulation, yields much higher generalized power compared to several existing enrichment designs and a conventional all-comers group sequential design, and is robust.

Point and interval estimation in two-stage adaptive designs with time to event data and biomarker-driven subpopulation selection

In personalized medicine, it is often desired to determine if all patients or only a subset of them benefit from a treatment. We consider estimation in two-stage adaptive designs that in stage 1 recruit patients from the full population. In stage 2, patient recruitment is restricted to the part of the population, which, based on stage 1 data, benefits from the experimental treatment. Existing estimators, which adjust for using stage 1 data for selecting the part of the population from which stage 2 patients are recruited, as well as for the confirmatory analysis after stage 2, do not consider time to event patient outcomes. In this work, for time to event data, we have derived a new asymptotically unbiased estimator for the log hazard ratio and a new interval estimator with good coverage probabilities and probabilities that the upper bounds are below the true values. The estimators are appropriate for several selection rules that are based on a single or multiple biomarkers, which can be categorical or continuous.

Adaptive seamless clinical trials using early outcomes for treatment or subgroup selection: Methods, simulation model and their implementation in R

Adaptive seamless designs combine confirmatory testing, a domain of phase III trials, with features such as treatment or subgroup selection, typically associated with phase II trials. They promise to increase the efficiency of development programmes of new drugs, for example, in terms of sample size and/or development time. It is well acknowledged that adaptive designs are more involved from a logistical perspective and require more upfront planning, often in the form of extensive simulation studies, than conventional approaches. Here, we present a framework for adaptive treatment and subgroup selection using the same notation, which links the somewhat disparate literature on treatment selection on one side and on subgroup selection on the other. Furthermore, we introduce a flexible and efficient simulation model that serves both designs. As primary endpoints often take a long time to observe, interim analyses are frequently informed by early outcomes. Therefore, all methods presented accommodate interim analyses informed by either the primary outcome or an early outcome. The R package asd, previously developed to simulate designs with treatment selection, was extended to include subgroup selection (so‐called adaptive enrichment designs). Here, we describe the functionality of the R package asd and use it to present some worked‐up examples motivated by clinical trials in chronic obstructive pulmonary disease and oncology. The examples both illustrate various features of the R package and provide insights into the operating characteristics of adaptive seamless studies.

A conditional error function approach for adaptive enrichment designs with continuous endpoints

Adaptive enrichment designs offer an efficient and flexible way to demonstrate the efficacy of a treatment in a clinically defined full population or in, eg, biomarker-defined subpopulations while controlling the family-wise Type I error rate in the strong sense. Frequently used testing strategies in designs with two or more stages include the combination test and the conditional error function approach. Here, we focus on the latter and present some extensions. In contrast to previous work, we allow for multiple subgroups rather than one subgroup only. For nested as well as nonoverlapping subgroups with normally distributed endpoints, we explore the effect of estimating the variances in the subpopulations. Instead of using a normal approximation, we derive new t-distribution-based methods for two different scenarios. First, in the case of equal variances across the subpopulations, we present exact results using a multivariate t-distribution. Second, in the case of potentially varying variances across subgroups, we provide some improved approximations compared to the normal approximation. The performance of the proposed conditional error function approaches is assessed and compared to the combination test in a simulation study. The proposed methods are motivated by an example in pulmonary arterial hypertension.

Using simulation studies to evaluate statistical methods

Simulation studies are computer experiments that involve creating data by pseudo-random sampling. A key strength of simulation studies is the ability to understand the behavior of statistical methods because some "truth" (usually some parameter/s of interest) is known from the process of generating the data. This allows us to consider properties of methods, such as bias. While widely used, simulation studies are often poorly designed, analyzed, and reported. This tutorial outlines the rationale for using simulation studies and offers guidance for design, execution, analysis, reporting, and presentation. In particular, this tutorial provides a structured approach for planning and reporting simulation studies, which involves defining aims, data-generating mechanisms, estimands, methods, and performance measures ("ADEMP"); coherent terminology for simulation studies; guidance on coding simulation studies; a critical discussion of key performance measures and their estimation; guidance on structuring tabular and graphical presentation of results; and new graphical presentations. With a view to describing recent practice, we review 100 articles taken from Volume 34 of Statistics in Medicine, which included at least one simulation study and identify areas for improvement.

Point estimation following two-stage adaptive threshold enrichment clinical trials

Recently, several study designs incorporating treatment effect assessment in biomarker-based subpopulations have been proposed. Most statistical methodologies for such designs focus on the control of type I error rate and power. In this paper, we have developed point estimators for clinical trials that use the two-stage adaptive enrichment threshold design. The design consists of two stages, where in stage 1, patients are recruited in the full population. Stage 1 outcome data are then used to perform interim analysis to decide whether the trial continues to stage 2 with the full population or a subpopulation. The subpopulation is defined based on one of the candidate threshold values of a numerical predictive biomarker. To estimate treatment effect in the selected subpopulation, we have derived unbiased estimators, shrinkage estimators, and estimators that estimate bias and subtract it from the naive estimate. We have recommended one of the unbiased estimators. However, since none of the estimators dominated in all simulation scenarios based on both bias and mean squared error, an alternative strategy would be to use a hybrid estimator where the estimator used depends on the subpopulation selected. This would require a simulation study of plausible scenarios before the trial.

Design and estimation in clinical trials with subpopulation selection

Population heterogeneity is frequently observed among patients' treatment responses in clinical trials because of various factors such as clinical background, environmental, and genetic factors. Different subpopulations defined by those baseline factors can lead to differences in the benefit or safety profile of a therapeutic intervention. Ignoring heterogeneity between subpopulations can substantially impact on medical practice. One approach to address heterogeneity necessitates designs and analysis of clinical trials with subpopulation selection. Several types of designs have been proposed for different circumstances. In this work, we discuss a class of designs that allow selection of a predefined subgroup. Using the selection based on the maximum test statistics as the worst‐case scenario, we then investigate the precision and accuracy of the maximum likelihood estimator at the end of the study via simulations. We find that the required sample size is chiefly determined by the subgroup prevalence and show in simulations that the maximum likelihood estimator for these designs can be substantially biased.

Interval and point estimation in adaptive Phase II trials with binary endpoint

Phase II clinical trials are concerned with making decision of whether a treatment is sufficiently efficacious to be worth further investigations in late large scale Phase III trials. In oncology Phase II trials, frequentist single-arm two-stage group-sequential designs with a binary endpoint are commonly used. To allow for more flexibility, adaptive versions of these designs have been proposed. In this paper, we propose point and interval estimation for adaptive designs in which the second stage sample size is a pre-specified function of first stage's number of responses. Our approach is based on sample space orderings, from which we derive p-values, and point and interval estimates. Simulation studies show that our proposed methods perform better, in terms of bias and root mean square error, than the fixed-sample maximum likelihood estimator.

Adaptive designs in clinical trials: why use them, and how to run and report them

Adaptive designs can make clinical trials more flexible by utilising results accumulating in the trial to modify the trial's course in accordance with pre-specified rules. Trials with an adaptive design are often more efficient, informative and ethical than trials with a traditional fixed design since they often make better use of resources such as time and money, and might require fewer participants. Adaptive designs can be applied across all phases of clinical research, from early-phase dose escalation to confirmatory trials. The pace of the uptake of adaptive designs in clinical research, however, has remained well behind that of the statistical literature introducing new methods and highlighting their potential advantages. We speculate that one factor contributing to this is that the full range of adaptations available to trial designs, as well as their goals, advantages and limitations, remains unfamiliar to many parts of the clinical community. Additionally, the term adaptive design has been misleadingly used as an all-encompassing label to refer to certain methods that could be deemed controversial or that have been inadequately implemented.We believe that even if the planning and analysis of a trial is undertaken by an expert statistician, it is essential that the investigators understand the implications of using an adaptive design, for example, what the practical challenges are, what can (and cannot) be inferred from the results of such a trial, and how to report and communicate the results. This tutorial paper provides guidance on key aspects of adaptive designs that are relevant to clinical triallists. We explain the basic rationale behind adaptive designs, clarify ambiguous terminology and summarise the utility and pitfalls of adaptive designs. We discuss practical aspects around funding, ethical approval, treatment supply and communication with stakeholders and trial participants. Our focus, however, is on the interpretation and reporting of results from adaptive design trials, which we consider vital for anyone involved in medical research. We emphasise the general principles of transparency and reproducibility and suggest how best to put them into practice.

Estimation in multi-arm two-stage trials with treatment selection and time-to-event endpoint

We consider estimation of treatment effects in two‐stage adaptive multi‐arm trials with a common control. The best treatment is selected at interim, and the primary endpoint is modeled via a Cox proportional hazards model. The maximum partial‐likelihood estimator of the log hazard ratio of the selected treatment will overestimate the true treatment effect in this case. Several methods for reducing the selection bias have been proposed for normal endpoints, including an iterative method based on the estimated conditional selection biases and a shrinkage approach based on empirical Bayes theory. We adapt these methods to time‐to‐event data and compare the bias and mean squared error of all methods in an extensive simulation study and apply the proposed methods to reconstructed data from the FOCUS trial. We find that all methods tend to overcorrect the bias, and only the shrinkage methods can reduce the mean squared error. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.

Estimation after subpopulation selection in adaptive seamless trials

During the development of new therapies, it is not uncommon to test whether a new treatment works better than the existing treatment for all patients who suffer from a condition (full population) or for a subset of the full population (subpopulation). One approach that may be used for this objective is to have two separate trials, where in the first trial, data are collected to determine if the new treatment benefits the full population or the subpopulation. The second trial is a confirmatory trial to test the new treatment in the population selected in the first trial. In this paper, we consider the more efficient two-stage adaptive seamless designs (ASDs), where in stage 1, data are collected to select the population to test in stage 2. In stage 2, additional data are collected to perform confirmatory analysis for the selected population. Unlike the approach that uses two separate trials, for ASDs, stage 1 data are also used in the confirmatory analysis. Although ASDs are efficient, using stage 1 data both for selection and confirmatory analysis introduces selection bias and consequently statistical challenges in making inference. We will focus on point estimation for such trials. In this paper, we describe the extent of bias for estimators that ignore multiple hypotheses and selecting the population that is most likely to give positive trial results based on observed stage 1 data. We then derive conditionally unbiased estimators and examine their mean squared errors for different scenarios.