Utilizing Bayesian predictive power in clinical trial design

The Bayesian paradigm provides an ideal platform to update uncertainties and carry them over into the future in the presence of data. Bayesian predictive power (BPP) reflects our belief in the eventual success of a clinical trial to meet its goals. In this paper we derive mathematical expressions for the most common types of outcomes, to make the BPP accessible to practitioners, facilitate fast computations in adaptive trial design simulations that use interim futility monitoring, and propose an organized BPP-based phase II-to-phase III design framework.     

The perils with the misuse of predictive power

In early drug development, especially when studying new mechanisms of action or in new disease areas, little is known about the targeted or anticipated treatment effect or variability estimates. Adaptive designs that allow for early stopping but also use interim data to adapt the sample size have been proposed as a practical way of dealing with these uncertainties. Predictive power and conditional power are two commonly mentioned techniques that allow predictions of what will happen at the end of the trial based on the interim data. Decisions about stopping or continuing the trial can then be based on these predictions. However, unless the user of these statistics has a deep understanding of their characteristics important pitfalls may be encountered, especially with the use of predictive power. The aim of this paper is to highlight these potential pitfalls. It is critical that statisticians understand the fundamental differences between predictive power and conditional power as they can have dramatic effects on decision making at the interim stage, especially if used to re-evaluate the sample size. The use of predictive power can lead to much larger sample sizes than either conditional power or standard sample size calculations. One crucial difference is that predictive power takes account of all uncertainty, parts of which are ignored by standard sample size calculations and conditional power. By comparing the characteristics of each of these statistics we highlight important characteristics of predictive power that experimenters need to be aware of when using this approach.     

Bayesian predictive power: choice of prior and some recommendations for its use as probability of success in drug development

Bayesian predictive power, the expectation of the power function with respect to a prior distribution for the true underlying effect size, is routinely used in drug development to quantify the probability of success of a clinical trial. Choosing the prior is crucial for the properties and interpretability of Bayesian predictive power. We review recommendations on the choice of prior for Bayesian predictive power and explore its features as a function of the prior. The density of power values induced by a given prior is derived analytically and its shape characterized. We find that for a typical clinical trial scenario, this density has a u‐shape very similar, but not equal, to a β‐distribution. Alternative priors are discussed, and practical recommendations to assess the sensitivity of Bayesian predictive power to its input parameters are provided. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Methods for clarifying criteria for study continuation at interim analysis

In monitoring clinical trials, the question of futility, or whether the data thus far suggest that the results at the final analysis are unlikely to be statistically successful, is regularly of interest over the course of a study. However, the opposite viewpoint of whether the study is sufficiently demonstrating proof of concept (POC) and should continue is a valuable consideration and ultimately should be addressed with high POC power so that a promising study is not prematurely terminated. Conditional power is often used to assess futility, and this article interconnects the ideas of assessing POC for the purpose of study continuation with conditional power, while highlighting the importance of the POC type I error and the POC type II error for study continuation or not at the interim analysis. Methods for analyzing subgroups motivate the interim analyses to maintain high POC power via an adjusted interim POC significance level criterion for study continuation or testing against an inferiority margin. Furthermore, two versions of conditional power based on the assumed effect size or the observed interim effect size are considered. Graphical displays illustrate the relationship of the POC type II error for premature study termination to the POC type I error for study continuation and the associated conditional power criteria.     

Evaluation of program success for programs with multiple trials in binary outcomes

A late‐stage clinical development program typically contains multiple trials. Conventionally, the program's success or failure may not be known until the completion of all trials. Nowadays, interim analyses are often used to allow evaluation for early success and/or futility for each individual study by calculating conditional power, predictive power and other indexes. It presents a good opportunity for us to estimate the probability of program success (POPS) for the entire clinical development earlier. The sponsor may abandon the program early if the estimated POPS is very low and therefore permit resource savings and reallocation to other products. We provide a method to calculate probability of success (POS) at an individual study level and also POPS for clinical programs with multiple trials in binary outcomes. Methods for calculating variation and confidence measures of POS and POPS and timing for interim analysis will be discussed and evaluated through simulations. We also illustrate our approaches on historical data retrospectively from a completed clinical program for depression. Copyright © 2015 John Wiley & Sons, Ltd.     

Type I error and power in trials with one interim futility analysis

Futility analysis reduces the opportunity to commit Type I error. For a superiority study testing a two‐sided hypothesis, an interim futility analysis can substantially reduce the overall Type I error while keeping the overall power relatively intact. In this paper, we quantify the extent of the reduction for both one‐sided and two‐sided futility analysis. We argue that, because of the reduction, we should be allowed to set the significance level for the final analysis at a level higher than the allowable Type I error rate for the study. We propose a method to find the significance level for the final analysis. We illustrate the proposed methodology and show that a design employing a futility analysis can reduce the sample size, and therefore reduce the exposure of patients to unnecessary risk and lower the cost of a clinical trial. Copyright © 2004 John Wiley & Sons, Ltd.     

Decision‐making in a phase II clinical trial: a new approach combining Bayesian and frequentist concepts

The aim of a phase II clinical trial is to decide whether or not to develop an experimental therapy further through phase III clinical evaluation. In this paper, we present a Bayesian approach to the phase II trial, although we assume that subsequent phase III clinical trials will have standard frequentist analyses. The decision whether to conduct the phase III trial is based on the posterior predictive probability of a significant result being obtained. This fusion of Bayesian and frequentist techniques accepts the current paradigm for expressing objective evidence of therapeutic value, while optimizing the form of the phase II investigation that leads to it. By using prior information, we can assess whether a phase II study is needed at all, and how much or what sort of evidence is required. The proposed approach is illustrated by the design of a phase II clinical trial of a multi‐drug resistance modulator used in combination with standard chemotherapy in the treatment of metastatic breast cancer. Copyright © 2005 John Wiley & Sons, Ltd     

Integrating phase 2 into phase 3 based on an intermediate endpoint while accounting for a cure proportion—With an application to the design of a clinical trial in acute myeloid leukemia

For a trial with primary endpoint overall survival for a molecule with curative potential, statistical methods that rely on the proportional hazards assumption may underestimate the power and the time to final analysis. We show how a cure proportion model can be used to get the necessary number of events and appropriate timing via simulation. If phase 1 results for the new drug are exceptional and/or the medical need in the target population is high, a phase 3 trial might be initiated after phase 1. Building in a futility interim analysis into such a pivotal trial may mitigate the uncertainty of moving directly to phase 3. However, if cure is possible, overall survival might not be mature enough at the interim to support a futility decision. We propose to base this decision on an intermediate endpoint that is sufficiently associated with survival. Planning for such an interim can be interpreted as making a randomized phase 2 trial a part of the pivotal trial: If stopped at the interim, the trial data would be analyzed, and a decision on a subsequent phase 3 trial would be made. If the trial continues at the interim, then the phase 3 trial is already underway. To select a futility boundary, a mechanistic simulation model that connects the intermediate endpoint and survival is proposed. We illustrate how this approach was used to design a pivotal randomized trial in acute myeloid leukemia and discuss historical data that informed the simulation model and operational challenges when implementing it.     

An exact test for comparing two predictive values in small‐size clinical trials

Positive and negative predictive values describe the performance of a diagnostic test. There are several methods to test the equality of predictive values in paired designs. However, these methods were premised on large sample theory, and they may not be suitable for small‐size clinical trials because of inflation of the type 1 error rate. In this study, we propose an exact test to control the type 1 error rate strictly for conducting a small‐size clinical trial that investigates the equality of predictive values in paired designs. In addition, we execute simulation studies to evaluate the performance of the proposed exact test and existing methods in small‐size clinical trials. The proposed test can calculate the exact P value, and as a result of simulations, the empirical type 1 error rate for the proposed test did not exceed the significance level regardless of the setting, and the empirical power for the proposed test is not much different from the other methods based on large‐sample theory. Therefore, it is considered that the proposed exact test is useful when the type 1 error rate needs to be controlled strictly.     

Informing the selection of futility stopping thresholds: case study from a late-phase clinical trial

In an environment where (i) potential risks to subjects participating in clinical studies need to be managed carefully, (ii) trial costs are increasing, and (iii) there are limited research resources available, it is necessary to prioritize research projects and sometimes re-prioritize if early indications suggest that a trial has low probability of success. Futility designs allow this re-prioritization to take place. This paper reviews a number of possible futility methods available and presents a case study from a late-phase study of an HIV therapeutic, which utilized conditional power-based stopping thresholds. The two most challenging aspects of incorporating a futility interim analysis into a trial design are the selection of optimal stopping thresholds and the timing of the analysis, both of which require the balancing of various risks. The paper outlines a number of graphical aids that proved useful in explaining the statistical risks involved to the study team. Further, the paper outlines a decision analysis undertaken which combined expectations of drug performance with conditional power calculations in order to produce probabilities of different interim and final outcomes, and which ultimately led to the selection of the final stopping thresholds.     

A new futility test approach in clinical interim data monitoring

Sequential monitoring of efficacy and safety data has become a vital component of modern clinical trials. It affords companies the opportunity to stop studies early in cases when it appears as if the primary objective will not be achieved or when there is clear evidence that the primary objective has already been met. This paper introduces a new concept of the backward conditional hypothesis test (BCHT) to evaluate clinical trial success. Unlike the regular conditional power approach that relies on the probability that the final study result will be statistically significant based on the current interim look, the BCHT was constructed based on the hypothesis test framework. The framework comprises a significant test level as opposed to the arbitrary fixed futility index utilized in the conditional power method. Additionally, the BCHT has proven to be a uniformly most powerful test. Noteworthy features of the BCHT method compared with the conditional power method will be presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Joint probability of statistical success of multiple phase III trials

In drug development, after completion of phase II proof‐of‐concept trials, the sponsor needs to make a go/no‐go decision to start expensive phase III trials. The probability of statistical success (PoSS) of the phase III trials based on data from earlier studies is an important factor in that decision‐making process. Instead of statistical power, the predictive power of a phase III trial, which takes into account the uncertainty in the estimation of treatment effect from earlier studies, has been proposed to evaluate the PoSS of a single trial. However, regulatory authorities generally require statistical significance in two (or more) trials for marketing licensure. We show that the predictive statistics of two future trials are statistically correlated through use of the common observed data from earlier studies. Thus, the joint predictive power should not be evaluated as a simplistic product of the predictive powers of the individual trials. We develop the relevant formulae for the appropriate evaluation of the joint predictive power and provide numerical examples. Our methodology is further extended to the more complex phase III development scenario comprising more than two (K > 2) trials, that is, the evaluation of the PoSS of at least k₀ () trials from a program of K total trials. Copyright © 2013 John Wiley & Sons, Ltd.     

An adaptive biomarker basket design in phase II oncology trials

The phase II basket trial in oncology is a novel design that enables the simultaneous assessment of treatment effects of one anti-cancer targeted agent in multiple cancer types. Biomarkers could potentially associate with the clinical outcomes and re-define clinically meaningful treatment effects. It is therefore natural to develop a biomarker-based basket design to allow the prospective enrichment of the trials with the adaptive selection of the biomarker-positive (BM+) subjects who are most sensitive to the experimental treatment. We propose a two-stage phase II adaptive biomarker basket (ABB) design based on a potential predictive biomarker measured on a continuous scale. At Stage 1, the design incorporates a biomarker cutoff estimation procedure via a hierarchical Bayesian model with biomarker as a covariate (HBMbc). At Stage 2, the design enrolls only BM+ subjects, defined as those with the biomarker values exceeding the biomarker cutoff within each cancer type, and subsequently assesses the early efficacy and/or futility stopping through the pre-defined interim analyses. At the end of the trial, the response rate of all BM+ subjects for each cancer type can guide drug development, while the data from all subjects can be used to further model the relationship between the biomarker value and the clinical outcome for potential future research. The extensive simulation studies show that the ABB design could produce a good estimate of the biomarker cutoff to select BM+ subjects with high accuracy and could outperform the existing phase II basket biomarker cutoff design under various scenarios.     

Bayesian adaptive patient enrollment restriction to identify a sensitive subpopulation using a continuous biomarker in a randomized phase 2 trial

With the development of molecular targeted drugs, predictive biomarkers have played an increasingly important role in identifying patients who are likely to receive clinically meaningful benefits from experimental drugs (i.e., sensitive subpopulation) even in early clinical trials. For continuous biomarkers, such as mRNA levels, it is challenging to determine cutoff value for the sensitive subpopulation, and widely accepted study designs and statistical approaches are not currently available. In this paper, we propose the Bayesian adaptive patient enrollment restriction (BAPER) approach to identify the sensitive subpopulation while restricting enrollment of patients from the insensitive subpopulation based on the results of interim analyses, in a randomized phase 2 trial with time‐to‐endpoint outcome and a single biomarker. Applying a four‐parameter change‐point model to the relationship between the biomarker and hazard ratio, we calculate the posterior distribution of the cutoff value that exhibits the target hazard ratio and use it for the restriction of the enrollment and the identification of the sensitive subpopulation. We also consider interim monitoring rules for termination because of futility or efficacy. Extensive simulations demonstrated that our proposed approach reduced the number of enrolled patients from the insensitive subpopulation, relative to an approach with no enrollment restriction, without reducing the likelihood of a correct decision for next trial (no‐go, go with entire population, or go with sensitive subpopulation) or correct identification of the sensitive subpopulation. Additionally, the four‐parameter change‐point model had a better performance over a wide range of simulation scenarios than a commonly used dichotomization approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal inference for Simon's two-stage design with over or under enrollment at the second stage

Simon's two-stage designs are widely used in clinical trials to assess the activity of a new treatment. In practice, it is often the case that the second stage sample size is different from the planned one. For this reason, the critical value for the second stage is no longer valid for statistical inference. Existing approaches for making statistical inference are either based on asymptotic methods or not optimal. We propose an approach to maximize the power of a study while maintaining the type I error rate, where the type I error rate and power are calculated exactly from binomial distributions. The critical values of the proposed approach are numerically searched by an intelligent algorithm over the complete parameter space. It is guaranteed that the proposed approach is at least as powerful as the conditional power approach which is a valid but non-optimal approach. The power gain of the proposed approach can be substantial as compared to the conditional power approach. We apply the proposed approach to a real Phase II clinical trial.     

Assurance in clinical trial design

Conventional clinical trial design involves considerations of power, and sample size is typically chosen to achieve a desired power conditional on a specified treatment effect. In practice, there is considerable uncertainty about what the true underlying treatment effect may be, and so power does not give a good indication of the probability that the trial will demonstrate a positive outcome. Assurance is the unconditional probability that the trial will yield a ‘positive outcome’. A positive outcome usually means a statistically significant result, according to some standard frequentist significance test. The assurance is then the prior expectation of the power, averaged over the prior distribution for the unknown true treatment effect. We argue that assurance is an important measure of the practical utility of a proposed trial, and indeed that it will often be appropriate to choose the size of the sample (and perhaps other aspects of the design) to achieve a desired assurance, rather than to achieve a desired power conditional on an assumed treatment effect. We extend the theory of assurance to two‐sided testing and equivalence trials. We also show that assurance is straightforward to compute in some simple problems of normal, binary and gamma distributed data, and that the method is not restricted to simple conjugate prior distributions for parameters. Several illustrations are given. Copyright © 2005 John Wiley & Sons, Ltd.     

Early stopping by using stochastic curtailment in a three-arm sequential trial

Summary. Interim analysis is important in a large clinical trial for ethical and cost considerations. Sometimes, an interim analysis needs to be performed at an earlier than planned time point. In that case, methods using stochastic curtailment are useful in examining the data for early stopping while controlling the inflation of type I and type II errors. We consider a three-arm randomized study of treatments to reduce perioperative blood loss following major surgery. Owing to slow accrual, an unplanned interim analysis was required by the study team to determine whether the study should be continued. We distinguish two different cases: when all treatments are under direct comparison and when one of the treatments is a control. We used simulations to study the operating characteristics of five different stochastic curtailment methods. We also considered the influence of timing of the interim analyses on the type I error and power of the test. We found that the type I error and power between the different methods can be quite different. The analysis for the perioperative blood loss trial was carried out at approximately a quarter of the planned sample size. We found that there is little evidence that the active treatments are better than a placebo and recommended closure of the trial.     

Planning for an adaptive design: a case study in COPD

We discuss the practical and clinical considerations encountered when planning a Phase IIa trial in chronic obstructive pulmonary disease (COPD). Various adaptive strategies for reducing the cost of the trial and the statistical implications of these are explored. Use of the EAST software to evaluate the properties of the study designs with one or more interim analyses for futility, efficacy or either is described. We emphasize the rationale for choosing between alternative designs and the relationship between the clinical and statistical considerations.