Bayesian design of noninferiority trials for medical devices using historical data

Summary We develop a new Bayesian approach of sample size determination (SSD) for the design of noninferiority clinical trials. We extend the fitting and sampling priors of Wang and Gelfand (2002, Statistical Science 17 , 193–208) to Bayesian SSD with a focus on controlling the type I error and power. Historical data are incorporated via a hierarchical modeling approach as well as the power prior approach of Ibrahim and Chen (2000, Statistical Science 15 , 46–60). Various properties of the proposed Bayesian SSD methodology are examined and a simulation‐based computational algorithm is developed. The proposed methodology is applied to the design of a noninferiority medical device clinical trial with historical data from previous trials.     

Bayesian Design of Non-inferiority Clinical Trials Via the Bayes Factor

We develop a Bayes factor-based approach for the design of non-inferiority clinical trials with a focus on controlling type I error and power. Historical data are incorporated in the Bayesian design via the power prior discussed in Ibrahim and Chen (Stat Sci 15:46–60, 2000). The properties of the proposed method are examined in detail. An efficient simulation-based computational algorithm is developed to calculate the Bayes factor, type I error, and power. The proposed methodology is applied to the design of a non-inferiority medical device clinical trial.     

Elastic priors to dynamically borrow information from historical data in clinical trials

Use of historical data and real-world evidence holds great potential to improve the efficiency of clinical trials. One major challenge is to effectively borrow information from historical data while maintaining a reasonable type I error and minimal bias. We propose the elastic prior approach to address this challenge. Unlike existing approaches, this approach proactively controls the behavior of information borrowing and type I errors by incorporating a well-known concept of clinically significant difference through an elastic function, defined as a monotonic function of a congruence measure between historical data and trial data. The elastic function is constructed to satisfy a set of prespecified criteria such that the resulting prior will strongly borrow information when historical and trial data are congruent, but refrain from information borrowing when historical and trial data are incongruent. The elastic prior approach has a desirable property of being information borrowing consistent, that is, asymptotically controls type I error at the nominal value, no matter that historical data are congruent or not to the trial data. Our simulation study that evaluates the finite sample characteristic confirms that, compared to existing methods, the elastic prior has better type I error control and yields competitive or higher power. The proposed approach is applicable to binary, continuous, and survival endpoints.     

Bayesian design of biosimilars clinical programs involving multiple therapeutic indications

In this paper, we propose a Bayesian design framework for a biosimilars clinical program that entails conducting concurrent trials in multiple therapeutic indications to establish equivalent efficacy for a proposed biologic compared to a reference biologic in each indication to support approval of the proposed biologic as a biosimilar. Our method facilitates information borrowing across indications through the use of a multivariate normal correlated parameter prior (CPP), which is constructed from easily interpretable hyperparameters that represent direct statements about the equivalence hypotheses to be tested. The CPP accommodates different endpoints and data types across indications (eg, binary and continuous) and can, therefore, be used in a wide context of models without having to modify the data (eg, rescaling) to provide reasonable information-borrowing properties. We illustrate how one can evaluate the design using Bayesian versions of the type I error rate and power with the objective of determining the sample size required for each indication such that the design has high power to demonstrate equivalent efficacy in each indication, reasonably high power to demonstrate equivalent efficacy simultaneously in all indications (ie, globally), and reasonable type I error control from a Bayesian perspective. We illustrate the method with several examples, including designing biosimilars trials for follicular lymphoma and rheumatoid arthritis using binary and continuous endpoints, respectively.     

An optimal Bayesian predictive probability design for phase II clinical trials with simple and complicated endpoints

Most existing phase II clinical trial designs focus on conventional chemotherapy with binary tumor response as the endpoint. The advent of novel therapies, such as molecularly targeted agents and immunotherapy, has made the endpoint of phase II trials more complicated, often involving ordinal, nested, and coprimary endpoints. We propose a simple and flexible Bayesian optimal phase II predictive probability (OPP) design that handles binary and complex endpoints in a unified way. The Dirichlet-multinomial model is employed to accommodate different types of endpoints. At each interim, given the observed interim data, we calculate the Bayesian predictive probability of success, should the trial continue to the maximum planned sample size, and use it to make the go/no-go decision. The OPP design controls the type I error rate, maximizes power or minimizes the expected sample size, and is easy to implement, because the go/no-go decision boundaries can be enumerated and included in the protocol before the onset of the trial. Simulation studies show that the OPP design has satisfactory operating characteristics.     

An adaptive two-stage design with treatment selection using the conditional error function approach

As an approach to combining the phase II dose finding trial and phase III pivotal trials, we propose a two-stage adaptive design that selects the best among several treatments in the first stage and tests significance of the selected treatment in the second stage. The approach controls the type I error defined as the probability of selecting a treatment and claiming its significance when the selected treatment is indifferent from placebo, as considered in Bischoff and Miller (2005). Our approach uses the conditional error function and allows determining the conditional type I error function for the second stage based on information observed at the first stage in a similar way to that for an ordinary adaptive design without treatment selection. We examine properties such as expected sample size and stage-2 power of this design with a given type I error and a maximum stage-2 sample size under different hypothesis configurations. We also propose a method to find the optimal conditional error function of a simple parametric form to improve the performance of the design and have derived optimal designs under some hypothesis configurations. Application of this approach is illustrated by a hypothetical example.     

Self-designing two-stage trials to minimize expected costs

In the design of clinical trials, the sample size for the trial is traditionally calculated from estimates of parameters of interest, such as the mean treatment effect, which can often be inaccurate. However, recalculation of the sample size based on an estimate of the parameter of interest that uses accumulating data from the trial can lead to inflation of the overall Type I error rate of the trial. The self-designing method of Fisher, also known as the variance-spending method, allows the use of all accumulating data in a sequential trial (including the estimated treatment effect) in determining the sample size for the next stage of the trial without inflating the Type I error rate. We propose a self-designing group sequential procedure to minimize the expected total cost of a trial. Cost is an important parameter to consider in the statistical design of clinical trials due to limited financial resources. Using Bayesian decision theory on the accumulating data, the design specifies sequentially the optimal sample size and proportion of the test statistic's variance needed for each stage of a trial to minimize the expected cost of the trial. The optimality is with respect to a prior distribution on the parameter of interest. Results are presented for a simple two-stage trial. This method can extend to nonmonetary costs, such as ethical costs or quality-adjusted life years.     

Power gains by using external information in clinical trials are typically not possible when requiring strict type I error control

In the era of precision medicine, novel designs are developed to deal with flexible clinical trials that incorporate many treatment strategies for multiple diseases in one trial setting. This situation often leads to small sample sizes in disease-treatment combinations and has fostered the discussion about the benefits of borrowing of external or historical information for decision-making in these trials. Several methods have been proposed that dynamically discount the amount of information borrowed from historical data based on the conformity between historical and current data. Specifically, Bayesian methods have been recommended and numerous investigations have been performed to characterize the properties of the various borrowing mechanisms with respect to the gain to be expected in the trials. However, there is common understanding that the risk of type I error inflation exists when information is borrowed and many simulation studies are carried out to quantify this effect. To add transparency to the debate, we show that if prior information is conditioned upon and a uniformly most powerful test exists, strict control of type I error implies that no power gain is possible under any mechanism of incorporation of prior information, including dynamic borrowing. The basis of the argument is to consider the test decision function as a function of the current data even when external information is included. We exemplify this finding in the case of a pediatric arm appended to an adult trial and dichotomous outcome for various methods of dynamic borrowing from adult information to the pediatric arm. In conclusion, if use of relevant external data is desired, the requirement of strict type I error control has to be replaced by more appropriate metrics.     

An alternative metric for evaluating the potential patient benefit of response-adaptive randomization procedures

When planning a two-arm group sequential clinical trial with a binary primary outcome that has severe implications for quality of life (e.g., mortality), investigators may strive to find the design that maximizes in-trial patient benefit. In such cases, Bayesian response-adaptive randomization (BRAR) is often considered because it can alter the allocation ratio throughout the trial in favor of the treatment that is currently performing better. Although previous studies have recommended using fixed randomization over BRAR based on patient benefit metrics calculated from the realized trial sample size, these previous comparisons have been limited by failures to hold type I and II error rates constant across designs or consider the impacts on all individuals directly affected by the design choice. In this paper, we propose a metric for comparing designs with the same type I and II error rates that reflects expected outcomes among individuals who would participate in the trial if enrollment is open when they become eligible. We demonstrate how to use the proposed metric to guide the choice of design in the context of two recent trials in persons suffering out of hospital cardiac arrest. Using computer simulation, we demonstrate that various implementations of group sequential BRAR offer modest improvements with respect to the proposed metric relative to conventional group sequential monitoring alone.     

Blinded sample size recalculation in multicentre trials with normally distributed outcome

The internal pilot study design enables to estimate nuisance parameters required for sample size calculation on the basis of data accumulated in an ongoing trial. By this, misspecifications made when determining the sample size in the planning phase can be corrected employing updated knowledge. According to regulatory guidelines, blindness of all personnel involved in the trial has to be preserved and the specified type I error rate has to be controlled when the internal pilot study design is applied. Especially in the late phase of drug development, most clinical studies are run in more than one centre. In these multicentre trials, one may have to deal with an unequal distribution of the patient numbers among the centres. Depending on the type of the analysis (weighted or unweighted), unequal centre sample sizes may lead to a substantial loss of power. Like the variance, the magnitude of imbalance is difficult to predict in the planning phase. We propose a blinded sample size recalculation procedure for the internal pilot study design in multicentre trials with normally distributed outcome and two balanced treatment groups that are analysed applying the weighted or the unweighted approach. The method addresses both uncertainty with respect to the variance of the endpoint and the extent of disparity of the centre sample sizes. The actual type I error rate as well as the expected power and sample size of the procedure is investigated in simulation studies. For the weighted analysis as well as for the unweighted analysis, the maximal type I error rate was not or only minimally exceeded. Furthermore, application of the proposed procedure led to an expected power that achieves the specified value in many cases and is throughout very close to it.     

Exact sample size determination for a single Poisson random sample

Classical power analysis for sample size determination is typically performed in clinical trials. A “hybrid” classical Bayesian or a “fully Bayesian” approach can be alternatively used in order to add flexibility to the design assumptions needed at the planning stage of the study and to explicitly incorporate prior information in the procedure. In this paper, we exploit and compare these approaches to obtain the optimal sample size of a single-arm trial based on Poisson data. We adopt exact methods to establish the rejection of the null hypothesis within a frequentist or a Bayesian perspective and suggest the use of a conservative criterion for sample size determination that accounts for the not strictly monotonic behavior of the power function in the presence of discrete data. A Shiny web app in R has been developed to provide a user-friendly interface to easily compute the optimal sample size according to the proposed criteria and to assure the reproducibility of the results.     

A parallel phase I/II clinical trial design for combination therapies

The use of multiple drugs in a single clinical trial or as a therapeutic strategy has become common, particularly in the treatment of cancer. Because traditional trials are designed to evaluate one agent at a time, the evaluation of therapies in combination requires specialized trial designs. In place of the traditional separate phase I and II trials, we propose using a parallel phase I/II clinical trial to evaluate simultaneously the safety and efficacy of combination dose levels, and select the optimal combination dose. The trial is started with an initial period of dose escalation, then patients are randomly assigned to admissible dose levels. These dose levels are compared with each other. Bayesian posterior probabilities are used in the randomization to adaptively assign more patients to doses with higher efficacy levels. Combination doses with lower efficacy are temporarily closed and those with intolerable toxicity are eliminated from the trial. The trial is stopped if the posterior probability for safety, efficacy, or futility crosses a prespecified boundary. For illustration, we apply the design to a combination chemotherapy trial for leukemia. We use simulation studies to assess the operating characteristics of the parallel phase I/II trial design, and compare it to a conventional design for a standard phase I and phase II trial. The simulations show that the proposed design saves sample size, has better power, and efficiently assigns more patients to doses with higher efficacy levels.     

Hierarchical commensurate and power prior models for adaptive incorporation of historical information in clinical trials

Bayesian clinical trial designs offer the possibility of a substantially reduced sample size, increased statistical power, and reductions in cost and ethical hazard. However when prior and current information conflict, Bayesian methods can lead to higher than expected type I error, as well as the possibility of a costlier and lengthier trial. This motivates an investigation of the feasibility of hierarchical Bayesian methods for incorporating historical data that are adaptively robust to prior information that reveals itself to be inconsistent with the accumulating experimental data. In this article, we present several models that allow for the commensurability of the information in the historical and current data to determine how much historical information is used. A primary tool is elaborating the traditional power prior approach based upon a measure of commensurability for Gaussian data. We compare the frequentist performance of several methods using simulations, and close with an example of a colon cancer trial that illustrates a linear models extension of our adaptive borrowing approach. Our proposed methods produce more precise estimates of the model parameters, in particular, conferring statistical significance to the observed reduction in tumor size for the experimental regimen as compared to the control regimen.     

Sample size reestimation by Bayesian prediction

We review a Bayesian predictive approach for interim data monitoring and propose its application to interim sample size reestimation for clinical trials. Based on interim data, this approach predicts how the sample size of a clinical trial needs to be adjusted so as to claim a success at the conclusion of the trial with an expected probability. The method is compared with predictive power and conditional power approaches using clinical trial data. Advantages of this approach over the others are discussed.     

Bayesian Adaptive Randomization and Trial Monitoring with Predictive Probability for Time-to-Event Endpoint

There has been much development in Bayesian adaptive designs in clinical trials. In the Bayesian paradigm, the posterior predictive distribution characterizes the future possible outcomes given the currently observed data. Based on the interim time-to-event data, we develop a new phase II trial design by combining the strength of both Bayesian adaptive randomization and the predictive probability. By comparing the mean survival times between patients assigned to two treatment arms, more patients are assigned to the better treatment on the basis of adaptive randomization. We continuously monitor the trial using the predictive probability for early termination in the case of superiority or futility. We conduct extensive simulation studies to examine the operating characteristics of four designs: the proposed predictive probability adaptive randomization design, the predictive probability equal randomization design, the posterior probability adaptive randomization design, and the group sequential design. Adaptive randomization designs using predictive probability and posterior probability yield a longer overall median survival time than the group sequential design, but at the cost of a slightly larger sample size. The average sample size using the predictive probability method is generally smaller than that of the posterior probability design.     

Optimal conditional error functions for the control of conditional power

Ethical considerations and the competitive environment of clinical trials usually require that any given trial have sufficient power to detect a treatment advance. If at an interim analysis the available data are used to decide whether the trial is promising enough to be continued, investigators and sponsors often wish to have a high conditional power, which is the probability to reject the null hypothesis given the interim data and the alternative of interest. Under this requirement a design with interim sample size recalculation, which keeps the overall and conditional power at a prespecified value and preserves the overall type I error rate, is a reasonable alternative to a classical group sequential design, in which the conditional power is often too small. In this article two-stage designs with control of overall and conditional power are constructed that minimize the expected sample size, either for a simple point alternative or for a random mixture of alternatives given by a prior density for the efficacy parameter. The presented optimality result applies to trials with and without an interim hypothesis test; in addition, one can account for constraints such as a minimal sample size for the second stage. The optimal designs will be illustrated with an example, and will be compared to the frequently considered method of using the conditional type I error level of a group sequential design.     

Sample size calculation for studies comparing binary outcomes using historical controls

In historical control trials (HCTs), the experimental therapy is compared with a control therapy that has been evaluated in a previously conducted trial. Makuch and Simon developed a sample size formula where the observations from the HC group were considered not subject to sampling variability. Many researchers have pointed out that the Makuch–Simon sample size formula does not preserve the nominal power and type I error. We develop a sample size calculation approach that properly accounts for the uncertainty in the true response rate of the HC group. We demonstrate that the empirical power and type I error, obtained over the simulated HC data, have extremely skewed distributions. We then derive a closed‐form sample size formula that enables researchers to control percentiles, instead of means, of the power and type I error accounting for the skewness of the distributions. A simulation study demonstrates that this approach preserves the operational characteristics in a more realistic scenario where the true response rate of the HC group is unknown. We also show that the controlling percentiles can be used to describe the joint behavior of the power and type I error. It provides a new perspective on the assessment of HCTs.     

Quantification of prior impact in terms of effective current sample size

Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual “loss of information” induced by the prior in case of prior-data conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.     

Falsification of a single species hypothesis using the coefficient of variation: a simulation approach.

Dental variation remains an important criterion for assessing whether a morphologically homogeneous fossil primate sample includes more than one species. The Coefficient of Variation (CV) has commonly been used to compare variation in a fossil sample of unknown taxonomic composition with that of extant single-species samples, in order to determine whether more than one species might be present. However, statistical tests for differences between fossil and single species reference sample CVs often lack power, because fossil samples are usually small and confidence limits of the CV are consequently large. The present study presents a new methodology for using the CV to test the hypothesis that a sample represents only one species. Simulated sampling distributions of single-species and pooled-species CVs are generated based on variation observed in dental samples of extant Cercopithecus species. These simulated distributions are used to test a single-species hypothesis for 13 different combinations of two or three sympatric Cercopithecus species across four dental characteristics at different sample sizes. Two different ways to generate the reference value of the CV are used. Results show the proposed methodology has substantially greater power than previous methods for detecting multiple-species composition, while maintaining an acceptable Type I error rate. Results are also presented concerning the dependence of power on sample size and on the average difference between means in a pooled-species combination.     

Statistical inference for self-designing clinical trials with a one-sided hypothesis

In the process of monitoring clinical trials, it seems appealing to use the interim findings to determine whether the sample size originally planned will provide adequate power when the alternative hypothesis is true, and to adjust the sample size if necessary. In the present paper, we propose a flexible sequential monitoring method following the work of Fisher (1998), in which the maximum sample size does not have to be specified in advance. The final test statistic is constructed based on a weighted average of the sequentially collected data, where the weight function at each stage is determined by the observed data prior to that stage. Such a weight function is used to maintain the integrity of the variance of the final test statistic so that the overall type I error rate is preserved. Moreover, the weight function plays an implicit role in termination of a trial when a treatment difference exists. Finally, the design allows the trial to be stopped early when the efficacy result is sufficiently negative. Simulation studies confirm the performance of the method.