Adaptive group sequential designs for clinical trials: combining the advantages of adaptive and of classical group sequential approaches

A general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing. This allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs. The concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans.     

A Practical Simulation Method to Calculate Sample Size of Group Sequential Trials for Time-to-Event Data under Exponential and Weibull Distribution

Group sequential design has been widely applied in clinical trials in the past few decades. The sample size estimation is a vital concern of sponsors and investigators. Especially in the survival group sequential trials, it is a thorny question because of its ambiguous distributional form, censored data and different definition of information time. A practical and easy-to-use simulation-based method is proposed for multi-stage two-arm survival group sequential design in the article and its SAS program is available. Besides the exponential distribution, which is usually assumed for survival data, the Weibull distribution is considered here. The incorporation of the probability of discontinuation in the simulation leads to the more accurate estimate. The assessment indexes calculated in the simulation are helpful to the determination of number and timing of the interim analysis. The use of the method in the survival group sequential trials is illustrated and the effects of the varied shape parameter on the sample size under the Weibull distribution are explored by employing an example. According to the simulation results, a method to estimate the shape parameter of the Weibull distribution is proposed based on the median survival time of the test drug and the hazard ratio, which are prespecified by the investigators and other participants. 10+ simulations are recommended to achieve the robust estimate of the sample size. Furthermore, the method is still applicable in adaptive design if the strategy of sample size scheme determination is adopted when designing or the minor modifications on the program are made.     

Data adaptive interim modification of sample sizes for candidate-gene association studies

OBJECTIVES: The use of conventional Transmission/Disequilibrium tests in the analysis of candidate-gene association studies requires the precise and complete pre-specification of the total number of trios to be sampled to obtain sufficient power at a certain significance level (type I error risk). In most of these studies, very little information about the genetic effect size will be available beforehand and thus it will be difficult to calculate a reasonable sample size. One would therefore wish to reassess the sample size during the course of a study. METHOD: We propose an adaptive group sequential procedure which allows for both early stopping of the study with rejection of the null hypothesis (H0) and for recalculation of the sample size based on interim effect size estimates when H0 cannot be rejected. The applicability of the method which was developed by Müller and Sch?fer [Biometrics 2001;57:886-891] in a clinical context is demonstrated by a numerical example. Monte Carlo simulations are performed comparing the adaptive procedure with a fixed sample and a conventional group sequential design. RESULTS: The main advantage of the adaptive procedure is its flexibility to allow for design changes in order to achieve a stabilized power characteristic while controlling the overall type I error and using the information already collected. CONCLUSIONS: Given these advantages, the procedure is a promising alternative to traditional designs.     

One‐sided Simultaneous Confidence Intervals for Effective Dose Steps in Unbalanced Designs

An important issue in dose finding is whether a further dose increment leads to a relevant increase in efficacy. Clinical efficacy should not be considered by point zero null hypotheses. Instead, shifted hypotheses for the difference or the ratio can be used. Because the a priori definition of a relevance threshold is frequently difficult, confidence intervals should be used for a posteriori interpretation. Sample size estimation – a‐priori or by adaptive interim analysis‐ is inherent, because the effective dose steps are arbitrary in un‐designed studies. For simultaneous confidence intervals without order restriction the exact distributions under the null and the alternative hypothesis is proposed for the general unbalanced one‐way design.     

Improved Repeated Confidence Bounds in Trials with a Maximal Goal

Repeated confidence intervals can be computed at every interim analysis of a flexible group sequential design without the need to stop the trial with a pre‐planned stopping rule. Often, however, there is a maximal goal such that the trial is surely stopped if this goal is reached. This induces a maximal stopping rule, and repeated confidence intervals are strictly conservative, when adhering to it. A modification is proposed which uniformly improves the one sided repeated confidence interval in such a situation. It preserves the monitoring character, and leads to uniformly smaller intervals, when reaching the maximal goal at an interim analysis. The modification is worked out for two stage designs and is indicated for multi‐stage trials. The extent of the improvement is quantified for two simple scenarios.     

Flexible designs by adaptive plans of generalized Pocock- and O'Brien-Fleming-type and by self-designing clinical trials

Flexible designs are provided by adaptive planning of sample sizes as well as by introducing the weighted inverse normal combining method and the generalized inverse chi-square combining method in the context of conducting trials consecutively step by step. These general combining methods allow quite different weighting of sequential study parts, also in a completely adaptive way, based on full information from unblinded data in previously performed stages. So, in reviewing some basic developments of flexible designing, we consider a generalizing approach to group sequentially performed clinical trials of Pocock-type, of O'Brien-Fleming-type, and of Self-designing-type. A clinical trial may be originally planned either to show non-inferiority or superiority. The proposed flexible designs, however, allow in each interim analysis to change the planning from showing non-inferiority to showing superiority and vice versa. Several examples of clinical trials with normal and binary outcomes are worked out in detail. We demonstrate the practicable performance of the discussed approaches, confirmed in an extensive simulation study. Our flexible designing is a useful tool, provided that a priori information about parameters involved in the trial is not available or subject to uncertainty.     

Repeated confidence intervals in self-designing clinical trials and switching between noninferiority and superiority

In self-designing clinical trials, repeated confidence intervals are derived for the parameter of interest where the results of the independent study stages are combined using the generalized inverse chi-square-method. The confidence intervals can be calculated at each interim analysis and always hold the predefined overall nominal confidence level. Moreover, the confidence intervals calculated during the course of the trial are nested in the sense that a calculated interval is completely contained in all the previously calculated intervals. During the course of the self-designing trial the sample sizes as well as the number of study stages can be determined simultaneously in a completely adaptive way. The adaptive procedure allows an early stop for significance. The clinical trial may be originally designed either to show noninferiority or superiority. However, in each interim analysis, it is possible to change the planning from showing superiority to showing noninferiority or vice versa. Since the repeated confidence intervals are nested, there is no risk to loose the noninferiority once showed when, after an interim analysis, the trial is continued in an attempt to reach superiority. A simulation study investigates the behavior of the considered confidence intervals. The performance of the derived nested repeated confidence intervals is also demonstrated in examples showing both kinds of switching during an ongoing trial.     

Information fraction estimation based on the number of events within the standard treatment regimen

For a Phase III randomized trial that compares survival outcomes between an experimental treatment versus a standard therapy, interim monitoring analysis is used to potentially terminate the study early based on efficacy. To preserve the nominal Type I error rate, alpha spending methods and information fractions are used to compute appropriate rejection boundaries in studies with planned interim analyses. For a one-sided trial design applied to a scenario in which the experimental therapy is superior to the standard therapy, interim monitoring should provide the opportunity to stop the trial prior to full follow-up and conclude that the experimental therapy is superior. This paper proposes a method called total control only (TCO) for estimating the information fraction based on the number of events within the standard treatment regimen. Based on theoretical derivations and simulation studies, for a maximum duration superiority design, the TCO method is not influenced by departure from the designed hazard ratio, is sensitive to detecting treatment differences, and preserves the Type I error rate compared to information fraction estimation methods that are based on total observed events. The TCO method is simple to apply, provides unbiased estimates of the information fraction, and does not rely on statistical assumptions that are impossible to verify at the design stage. For these reasons, the TCO method is a good approach when designing a maximum duration superiority trial with planned interim monitoring analyses.     

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.     

Implementation of group sequential logrank tests in a maximum duration trial

To control the Type I error probability in a group sequential procedure using the logrank test, it is important to know the information times (fractions) at the times of interim analyses conducted for purposes of data monitoring. For the logrank test, the information time at an interim analysis is the fraction of the total number of events to be accrued in the entire trial. In a maximum information trial design, the trial is concluded when a prespecified total number of events has been accrued. For such a design, therefore, the information time at each interim analysis is known. However, many trials are designed to accrue data over a fixed duration of follow-up on a specified number of patients. This is termed a maximum duration trial design. Under such a design, the total number of events to be accrued is unknown at the time of an interim analysis. For a maximum duration trial design, therefore, these information times need to be estimated. A common practice is to assume that a fixed fraction of information will be accrued between any two consecutive interim analyses, and then employ a Pocock or O'Brien-Fleming boundary. In this article, we describe an estimate of the information time based on the fraction of total patient exposure, which tends to be slightly negatively biased (i.e., conservative) if survival is exponentially distributed. We then present a numerical exploration of the robustness of this estimate when nonexponential survival applies. We also show that the Lan-DeMets (1983, Biometrika 70, 659-663) procedure for constructing group sequential boundaries with the desired level of Type I error control can be computed using the estimated information fraction, even though it may be biased. Finally, we discuss the implications of employing a biased estimate of study information for a group sequential procedure.     

Sequential tests for noninferiority and superiority

The problem of simultaneous sequential tests for noninferiority and superiority of a treatment, as compared to an active control, is considered in terms of continuous hierarchical families of one-sided null hypotheses, in the framework of group sequential and adaptive two-stage designs. The crucial point is that the decision boundaries for the individual null hypotheses may vary over the parameter space. This allows one to construct designs where, e.g., a rigid stopping criterion is chosen, rejecting or accepting all individual null hypotheses simultaneously. Another possibility is to use monitoring type stopping boundaries, which leave some flexibility to the experimenter: he can decide, at the interim analysis, whether he is satisfied with the noninferiority margin achieved at this stage, or wants to go for more at the second stage. In the case where he proceeds to the second stage, he may perform midtrial design modifications (e.g., reassess the sample size). The proposed approach allows one to "spend," e.g., less of alpha for an early proof of noninferiority than for an early proof of superiority, and is illustrated by typical examples.     

A group sequential test for survival trials: an alternative to rank-based procedures

A method of interim monitoring is described for survival trials in which the proportional hazards assumption may not hold. This method extends the test statistics based on the cumulative weighted difference in the Kaplan-Meier estimates (Pepe and Fleming, 1989, Biometrics 45, 497-507) to the sequential setting. Therefore, it provides a useful alternative to the group sequential linear rank tests. With an appropriate weight function, the test statistic itself provides an estimator for the cumulative weighted difference in survival probabilities, which is an interpretable measure for the treatment difference, especially when the proportional hazards model fails. The method is illustrated based on the design of a real trial. The operating characteristics are studied through a small simulation.     

Exact confidence bounds following adaptive group sequential tests

Summary .  We provide a method for obtaining confidence intervals, point estimates, and p-values for the primary effect size parameter at the end of a two-arm group sequential clinical trial in which adaptive changes have been implemented along the way. The method is based on applying the adaptive hypothesis testing procedure of Müller and Schäfer (2001, Biometrics 57, 886–891) to a sequence of dual tests derived from the stage-wise adjusted confidence interval of Tsiatis, Rosner, and Mehta (1984, Biometrics 40, 797–803). In the nonadaptive setting this confidence interval is known to provide exact coverage. In the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage. In general, however, all that can be claimed theoretically is that the coverage is guaranteed to be conservative. Nevertheless, extensive simulation experiments, supported by an empirical characterization of the conditional error function, demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased. No procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting. The methodology is illustrated by an application to a clinical trial of deep brain stimulation for Parkinson's disease.     

Population-Enrichment Adaptive Design Strategy for an Event-Driven Vaccine Efficacy Trial

A population-enrichment adaptive design allows a prospective use for study population selection. It has the flexibility allowing pre-specified modifications to an ongoing trial to mitigate the potential risk associated with the assumptions made at design stage. In this way, the trial can potentially encompass a broader target patient population, and move forward only with the subpopulations that appear to be benefiting from the treatment. Our work is motivated by a Phase III event-driven vaccine efficacy trial. Two target patient subpopulations were enrolled with the assumption that vaccine efficacy can be demonstrated based on the combined population. It is recognized due to the nature of patients’ underlying conditions, one subpopulation might respond to the treatment better than the other. To maximize the probability of demonstrating vaccine efficacy in at least one patient population while taking advantage of combining two subpopulations in one single trial, an adaptive design strategy with potential population enrichment is developed. Specifically, if the observed vaccine efficacy at interim for one subpopulation is not promising to warrant carrying forward, the population may be enriched with the other subpopulation with better performance. Simulations were conducted to evaluate the operational characteristics from a selection of interim analysis plans. This population-enrichment design provides a more efficient way as compared to the conventional approaches when targeting multiple subpopulations. If executed and planned with caution, this strategy can provide a greater chance of success of the trial and help maintain scientific and regulatory rigors.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.     

Use of logrank tests and group sequential methods at fixed calendar times

This paper presents simulations to determine the operating characteristics of several group sequential boundaries when applied to the repeated analysis of survival data at equal intervals of calendar time. Because the group sequential boundaries of Pocock (1977, Biometrika 64, 191-199) and O'Brien and Fleming (1979, Biometrics 35, 549-556) were constructed on the assumption that each interim analysis provides an equal increment of statistical information, these boundaries are not theoretically appropriate for interim analysis at prescheduled calendar times. Nonetheless, our simulations show that these boundaries yield size and power near nominal levels for repeated logrank analyses at equal intervals of calendar time.     

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.     

Interim analysis incorporating short‐ and long‐term binary endpoints

Designs incorporating more than one endpoint have become popular in drug development. One of such designs allows for incorporation of short‐term information in an interim analysis if the long‐term primary endpoint has not been yet observed for some of the patients. At first we consider a two‐stage design with binary endpoints allowing for futility stopping only based on conditional power under both fixed and observed effects. Design characteristics of three estimators: using primary long‐term endpoint only, short‐term endpoint only, and combining data from both are compared. For each approach, equivalent cut‐off point values for fixed and observed effect conditional power calculations can be derived resulting in the same overall power. While in trials stopping for futility the type I error rate cannot get inflated (it usually decreases), there is loss of power. In this study, we consider different scenarios, including different thresholds for conditional power, different amount of information available at the interim, different correlations and probabilities of success. We further extend the methods to adaptive designs with unblinded sample size reassessments based on conditional power with inverse normal method as the combination function. Two different futility stopping rules are considered: one based on the conditional power, and one from P‐values based on Z‐statistics of the estimators. Average sample size, probability to stop for futility and overall power of the trial are compared and the influence of the choice of weights is investigated.     

Interim monitoring in sequential multiple assignment randomized trials

A sequential multiple assignment randomized trial (SMART) facilitates the comparison of multiple adaptive treatment strategies (ATSs) simultaneously. Previous studies have established a framework to test the homogeneity of multiple ATSs by a global Wald test through inverse probability weighting. SMARTs are generally lengthier than classical clinical trials due to the sequential nature of treatment randomization in multiple stages. Thus, it would be beneficial to add interim analyses allowing for an early stop if overwhelming efficacy is observed. We introduce group sequential methods to SMARTs to facilitate interim monitoring based on the multivariate chi-square distribution. Simulation studies demonstrate that the proposed interim monitoring in SMART (IM-SMART) maintains the desired type I error and power with reduced expected sample size compared to the classical SMART. Finally, we illustrate our method by reanalyzing a SMART assessing the effects of cognitive behavioral and physical therapies in patients with knee osteoarthritis and comorbid subsyndromal depressive symptoms.