INTRODUCTION TO BAYESIAN SEQUENTIAL ANALYSIS

The increasing interest in Bayesian group sequential design is due to its potential to reinforce efficiency in clinical trials, shorten drug development time, and enhance the accuracy of statistical inference without compromising the integrity or validity of clinical trials. In a Bayesian trial, the prior information, and the trial results, as they emerge, are viewed as a continuous stream of information, in which inferences can be updated as new data become available. Bayesian sequential designs can be proposed for clinical trials with time-to-event outcomes and alpha spending functions are used to control the overall type I error rate. Alpha spending function distributes the type I error over the duration of a sequential test. Bayes factor are often adapted for decision-making at interim analyses and present Algorithms to form decision rules and to calculate power of the proposed tests. Also, a sensitivity analysis can be executed to evaluate the impact of different choices of prior parameters on choosing critical values. The power of tests, the expected event size of the proposed design, and therefore the quality of estimators can be studied through simulations and this can be compared with the frequentist group sequential design.

Sequential designs were first proposed by Armitage (1975) in which patients are recruited in pairs and the data analyzed as the results from each pair become available. Sequential designs were extended to group sequential designs, in which patients are enrolled in successive groups instead of pairs. These have been utilized for decades by the statistical and clinical trial communities. These designs allow for multiple interim analyses at the data as the clinical trial proceeds and giving the possibility of stopping the trial early due to efficacy, futility, or safety reasons. Early termination for an efficacy trial can occur when the superiority of the treatment under study is established, for futility when the establishment of a relevant treatment difference is not likely, or for safety concerns when unacceptable adverse events become evident. This design has the power to improve the efficiency of a clinical trial by reducing its duration without lowering any scientific and regulatory standards. These designs have been widely implemented in large, long-term trials, such as phase III trials of drug development for fatal diseases, particularly where the endpoints were progression-free survival time and/or overall survival time. The relatively long follow-up time and recruiting period for assessing the primary outcome allow for the implementation of multiple interim analyses and opportunities for increasing efficiency. For trials that need an accelerated approval, group sequential designs offer an advantage over conventional designs by reducing the time to market.

Group sequential designs can be implemented for both frequentist and Bayesian models. Bayesian inference gives a natural way to incorporate prior information into an ongoing trial. Bayesian designs have gained increasing popularity by the accumulation of historical data from previous studies and the advance of computational techniques. They have been used in early-phase trials to assess pharmacokinetics/pharmacodynamics and to find dose ranges, however, their use has been rapidly expanding to other applications. A major gain in efficiency using Bayesian sequential designs is for the studies involving rare diseases which has limited number of subjects and is surely an advantage over traditional designs. A major statistical challenge in group sequential design is the control of the study-wide overall type I error rate. The type I error is that the probability of rejecting the null hypothesis when it is true. When multiple testing is performed at each interim analysis of the study data, the overall type I error is inflated. Statistical adjustment is required to control the overall type I error rate. The methods to control the overall type I error include those developed by Pocock (1977) and O’Brien and Fleming (1979). Both require a fixed number of interim analyses at fixed times with a prescribed fraction of type I error spent at each analysis be specified in advance. Alpha spending methods, proposed by and extended by, offer a more flexible approach to control the overall type I error rate and have been widely used in frequentist group sequential designs. Alpha spending-function systematically allocates the type I error at various time points in the trial. Controlling the type I error rate in Bayesian sequential design is much complicated since the choices of prior distributions may also influence final decisions. Simulations which are usually needed to help decision making and to assess the operating characteristics of the study design under different scenarios proposed a Bayesian sequential design uses alpha spending function controlling the overall type I error. The basic idea is to regulate the overall type I error rate using alpha spending function.

The Bayesian designs will be used widely in the clinical trials, not only because of its ability to incorporate prior information as the trial progresses but also due to its ability to control the type I error while performing multiple interim analysis.

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