A Bayesian sequential design with binary outcome

Several researchers have proposed solutions to control type I error rate in sequential designs. The use of Bayesian sequential design becomes more common; however, these designs are subject to inflation of the type I error rate. We propose a Bayesian sequential design for binary outcome using an alpha‐spending function to control the overall type I error rate. Algorithms are presented for calculating critical values and power for the proposed designs. We also propose a new stopping rule for futility. Sensitivity analysis is implemented for assessing the effects of varying the parameters of the prior distribution and maximum total sample size on critical values. Alpha‐spending functions are compared using power and actual sample size through simulations. Further simulations show that, when total sample size is fixed, the proposed design has greater power than the traditional Bayesian sequential design, which sets equal stopping bounds at all interim analyses. We also find that the proposed design with the new stopping for futility rule results in greater power and can stop earlier with a smaller actual sample size, compared with the traditional stopping rule for futility when all other conditions are held constant. Finally, we apply the proposed method to a real data set and compare the results with traditional designs.     

Repeated Confidence Intervals Under Fractional Brownian Motion in Long-Term Clinical Trials

Repeated confidence interval (RCI) is an important tool for design and monitoring of group sequential trials according to which we do not need to stop the trial with planned statistical stopping rules. In this article, we derive RCIs when data from each stage of the trial are not independent thus it is no longer a Brownian motion (BM) process. Under this assumption, a larger class of stochastic processes fractional Brownian motion (FBM) is considered. Comparisons of RCI width and sample size requirement are made to those under Brownian motion for different analysis times, Type I error rates and number of interim analysis. Power family spending functions including Pocock, O'Brien-Fleming design types are considered for these simulations. Interim data from BHAT and oncology trials is used to illustrate how to derive RCIs under FBM for efficacy and futility monitoring.     

Three-stage designs for monitoring clinical trials

Optimal three-stage designs with equal sample sizes at each stage are presented and compared to fixed sample designs, fully sequential designs, designs restricted to use the fixed sample critical value at the final stage, and to modifications of other group sequential designs previously proposed in the literature. Typically, the greatest savings realized with interim analyses are obtained by the first interim look. More than 50% of the savings possible with a fully sequential design can be realized with a simple two-stage design. Three-stage designs can realize as much as 75% of the possible savings. Without much loss in efficiency, the designs can be modified so that the critical value at the final stage equals the usual fixed sample value while maintaining the overall level of significance, alleviating some potential confusion should a final stage be necessary. Some common group sequential designs, modified to allow early acceptance of the null hypothesis, are shown to be nearly optimal in some settings while performing poorly in others. An example is given to illustrate the use of several three-stage plans in the design of clinical trials.     

A flexible multi-stage design for phase II oncology trials

Phase II trials evaluate whether a new drug or a new therapy is worth further pursuing or certain treatments are feasible or not. A typical phase II is a single arm (open label) trial with a binary clinical endpoint (response to therapy). Although many oncology Phase II clinical trials are designed with a two-stage procedure, multi-stage design for phase II cancer clinical trials are now feasible due to increased capability of data capture. Such design adjusts for multiple analyses and variations in analysis time, and provides greater flexibility such as minimizing the number of patients treated on an ineffective therapy and identifying the minimum number of patients needed to evaluate whether the trial would warrant further development. In most of the NIH sponsored studies, the early stopping rule is determined so that the number of patients treated on an ineffective therapy is minimized. In pharmaceutical trials, it is also of importance to know as early as possible if the trial is highly promising and what is the likelihood the early conclusion can sustain. Although various methods are available to address these issues, practitioners often use disparate methods for addressing different issues and do not realize a single unified method exists. This article shows how to utilize a unified approach via a fully sequential procedure, the sequential conditional probability ratio test, to address the multiple needs of a phase II trial. We show the fully sequential program can be used to derive an optimized efficient multi-stage design for either a low activity or a high activity, to identify the minimum number of patients required to assess whether a new drug warrants further study and to adjust for unplanned interim analyses. In addition, we calculate a probability of discordance that the statistical test will conclude otherwise should the trial continue to the planned end that is usually at the sample size of a fixed sample design. This probability can be used to aid in decision making in a drug development program. All computations are based on exact binomial distribution.     

Operating Characteristics for Group Sequential Trials Monitored Under Fractional Brownian Motion

Brownian motion has been used to derive stopping boundaries for group sequential trials, however, when we observe dependent increment in the data, fractional Brownian motion is an alternative to be considered to model such data. In this article we compared expected sample sizes and stopping times for different stopping boundaries based on the power family alpha spending function under various values of Hurst coefficient. Results showed that the expected sample sizes and stopping times will decrease and power increases when the Hurst coefficient increases. With same Hurst coefficient, the closer the boundaries are to that of O'Brien-Fleming, the higher the expected sample sizes and stopping times are; however, power has a decreasing trend for values start from H = 0.6 (early analysis), 0.7 (equal space), 0.8 (late analysis). We also illustrate study design changes using results from the BHAT study.     

A sequential conditional probability ratio test procedure for comparing diagnostic tests

In this paper, we derive sequential conditional probability ratio tests to compare diagnostic tests without distributional assumptions on test results. The test statistics in our method are nonparametric weighted areas under the receiver-operating characteristic curves. By using the new method, the decision of stopping the diagnostic trial early is unlikely to be reversed should the trials continue to the planned end. The conservatism reflected in this approach to have more conservative stopping boundaries during the course of the trial is especially appealing for diagnostic trials since the end point is not death. In addition, the maximum sample size of our method is not greater than a fixed sample test with similar power functions. Simulation studies are performed to evaluate the properties of the proposed sequential procedure. We illustrate the method using data from a thoracic aorta imaging study.     

On the efficiency of adaptive designs for flexible interim decisions in clinical trials

It is shown that the optimal group sequential designs considered in Tsiatis and Mehta [2003. On the inefficiency of the adaptive design for monitoring clinical trials. Biometrika 90, 367–378] are special cases of the more general flexible designs which allow for a valid inference after adapting a predetermined way to spend the rejection and acceptance probabilities. An unforeseen safety issue in a clinical trial, for example, could make a change of the preplanned number of interim analyses and their sample sizes appropriate. We derive flexible designs which have equivalent rejection and acceptance regions if no adaptation is performed, but at the same time allow for an adaptation of the spending functions, and have a conditional optimality property.     

Combining an Internal Pilot with an Interim Analysis for Single Degree of Freedom Tests

An internal pilot with interim analysis (IPIA) design combines interim power analysis (an internal pilot) with interim data analysis (two stage group sequential). We provide IPIA methods for single df hypotheses within the Gaussian general linear model, including one and two group t tests. The design allows early stopping for efficacy and futility while also re-estimating sample size based on an interim variance estimate. Study planning in small samples requires the exact and computable forms reported here. The formulation gives fast and accurate calculations of power, type I error rate, and expected sample size.     

Combining an Internal Pilot with an Interim Analysis for Single Degree of Freedom Tests

An internal pilot with interim analysis (IPIA) design combines interim power analysis (an internal pilot) with interim data analysis (two-stage group sequential). We provide IPIA methods for single df hypotheses within the Gaussian general linear model, including one and two group t tests. The design allows early stopping for efficacy and futility while also re-estimating sample size based on an interim variance estimate. Study planning in small samples requires the exact and computable forms reported here. The formulation gives fast and accurate calculations of power, Type I error rate, and expected sample size.     

A simulation-free group sequential design with max-combo tests in the presence of non-proportional hazards

Non-proportional hazards (NPH) have been observed in many immuno-oncology clinical trials. Weighted log-rank tests (WLRT) with suitable weights can be used to improve the power of detecting the difference between survival curves in the presence of NPH. However, it is not easy to choose a proper WLRT in practice. A versatile max-combo test was proposed to achieve the balance of robustness and efficiency, and has received increasing attention recently. Survival trials often warrant interim analyses due to their high cost and long durations. The integration and implementation of max-combo tests in interim analyses often require extensive simulation studies. In this report, we propose a simulation-free approach for group sequential designs with the max-combo test in survival trials. The simulation results support that the proposed method can successfully control the type I error rate and offer excellent accuracy and flexibility in estimating sample sizes, with light computation burden. Notably, our method displays strong robustness towards various model misspecifications and has been implemented in an R package.     

Bayesian randomized clinical trials: A decision‐theoretic sequential design

The authors propose a Bayesian decision‐theoretic framework justifying randomization in clinical trials. Noting that the decision maker is often unable or unwilling to specify a unique utility function, they develop a sequential myopic design that includes randomization justified by the consideration of a set of utility functions. Randomization is introduced over all nondominated treatments, allowing for interim removal of treatments and early stopping. The authors illustrate their approach in the context of a study to find the optimal dose of pegylated interferon for platinum resistant ovarian cancer. They also develop an algorithm to implement their methodology in a phase II clinical trial comparing several competing experimental treatments.     

A stochastically curtailed two-arm randomised phase II trial design for binary outcomes

Randomised controlled trials are considered the gold standard in trial design. However, phase II oncology trials with a binary outcome are often single-arm. Although a number of reasons exist for choosing a single-arm trial, the primary reason is that single-arm designs require fewer participants than their randomised equivalents. Therefore, the development of novel methodology that makes randomised designs more efficient is of value to the trials community. This article introduces a randomised two-arm binary outcome trial design that includes stochastic curtailment (SC), allowing for the possibility of stopping a trial before the final conclusions are known with certainty. In addition to SC, the proposed design involves the use of a randomised block design, which allows investigators to control the number of interim analyses. This approach is compared with existing designs that also use early stopping, through the use of a loss function comprised of a weighted sum of design characteristics. Comparisons are also made using an example from a real trial. The comparisons show that for many possible loss functions, the proposed design is superior to existing designs. Further, the proposed design may be more practical, by allowing a flexible number of interim analyses. One existing design produces superior design realisations when the anticipated response rate is low. However, when using this design, the probability of rejecting the null hypothesis is sensitive to misspecification of the null response rate. Therefore, when considering randomised designs in phase II, we recommend the proposed approach be preferred over other sequential designs.     

The application of bespoke spending functions in group‐sequential designs and the effect of delayed treatment switching in survival trials

For a group‐sequential trial with two pre‐planned analyses, stopping boundaries can be calculated using a simple SAS? programme on the basis of the asymptotic bivariate normality of the interim and final test statistics. Given the simplicity and transparency of this approach, it is appropriate for researchers to apply their own bespoke spending function as long as the rate of alpha spend is pre‐specified. One such application could be an oncology trial where progression free survival (PFS) is the primary endpoint and overall survival (OS) is also assessed, both at the same time as the analysis of PFS and also later following further patient follow‐up. In many circumstances it is likely, if PFS is significantly extended, that the protocol will be amended to allow patients in the control arm to start receiving the experimental regimen. Such an eventuality is likely to result in the diminution of any effect on OS. It is shown that spending a greater proportion of alpha at the first analysis of OS, using either Pocock or bespoke boundaries, will maintain and in some cases result in greater power given a fixed number of events. Copyright © 2009 John Wiley & Sons, Ltd.     

Planning the duration of a survival group sequential trial with a fixed follow-up time for all subjects

To explore the operation characteristics of survival group sequential trials with a fixed follow-up period, the accrual time and total trial duration to ensure power and type I error rate requirements are explained and investigated for hazard ratios ranging from 1.3 to 3.0, with slow or high accrual rate, and in the presence or absence of censoring. Impacts of hazard rate, accrual rate, and competitive censoring on accrual time and subsequently on total trial duration are carefully illustrated. Real time for interim analyses, needed number of events, and recruited number of subjects at time of interim analyses are also tabulated.     

Group sequential monitoring based on the weighted log‐rank test statistic with the Fleming–Harrington class of weights in cancer vaccine studies

In recent years, immunological science has evolved, and cancer vaccines are now approved and available for treating existing cancers. Because cancer vaccines require time to elicit an immune response, a delayed treatment effect is expected and is actually observed in drug approval studies. Accordingly, we propose the evaluation of survival endpoints by weighted log‐rank tests with the Fleming–Harrington class of weights. We consider group sequential monitoring, which allows early efficacy stopping, and determine a semiparametric information fraction for the Fleming–Harrington family of weights, which is necessary for the error spending function. Moreover, we give a flexible survival model in cancer vaccine studies that considers not only the delayed treatment effect but also the long‐term survivors. In a Monte Carlo simulation study, we illustrate that when the primary analysis is a weighted log‐rank test emphasizing the late differences, the proposed information fraction can be a useful alternative to the surrogate information fraction, which is proportional to the number of events. Copyright © 2016 John Wiley & Sons, Ltd.     

Application of boundary calculation methodologies to group sequential testing in general parametric models

For clinical trials with interim analyses, there have been methodologies and software to calculate boundaries for comparing binomial, normal, and survival data from two treatment groups. Jermison & Turnbull (1991) extended Pocock (1977) and O' Brien- Fleming (1979) boundaries to t-tests, x²-tests and F-tests for comparing normal data from several treatment groups. This paper demonstrates that the above boundaries can be applied to a wide variety of test statistics based on general parametric settings. We show that asymptotically the x² boundaries as well as the corresponding nominal significance levels calculated by Jennison & Turnbull can be applied to interim analyses based on the score test, the Wald test, and the likelihood ratio test for general parametric models. Based on the results of this paper, currently available software in group sequential testing can be used to calculate. the nominal significance levels (or boundaries) for group sequential testing based on logistic regression, A NOVA, and other parametric methods.     

Impact of safety monitoring on error probabilities of binary efficacy outcome analyses in large phase III group sequential trials

In phase III clinical trials, some adverse events may not be rare or unexpected and can be considered as a primary measure for safety, particularly in trials of life-threatening conditions, such as stroke or traumatic brain injury. In some clinical areas, efficacy endpoints may be highly correlated with safety endpoints, yet the interim efficacy analyses under group sequential designs usually do not consider safety measures formally in the analyses. Furthermore, safety is often statistically monitored more frequently than efficacy measures. Because early termination of a trial in this situation can be triggered by either efficacy or safety, the impact of safety monitoring on the error probabilities of efficacy analyses may be nontrivial if the original design does not take the multiplicity effect into account. We estimate the actual error probabilities for a bivariate binary efficacy-safety response in large confirmatory group sequential trials. The estimated probabilities are verified by Monte Carlo simulation. Our findings suggest that type I error for efficacy analyses decreases as efficacy-safety correlation or between-group difference in the safety event rate increases. In addition, although power for efficacy is robust to misspecification of the efficacy-safety correlation, it decreases dramatically as between-group difference in the safety event rate increases.     

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.     

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.