Application Study On Bayesian Interim Analysis In Group Sequential Clinical Trials

Research background and objectiveThe application of Bayesian statistics in the clinical trial has always been a magnet in the field of statistical researching. Compared with classical statistics, Bayesian statistics is featured by integration of prior information and sample likelihood function, a characteristics making it a more flexible statistical method offering enjoying marked statistical advantage. However, how to use the prior information for constructuring the Bayesian analysis model has become a major bottleneck of application in the clinical trial both home and abroad.From the perspective of clinical ethics and cost benefit, group sequential designing is a very important part in sequential designing. For the interim analysis of group sequential, the major classical statistical methods include Fixed Boundary Shape Methods and Error Spending Methods, with the former including Pocock method and O'Brien & Fleming method. Involving a shorter test cycle and fewer samples size, the advantages of the interim analysis in PhaseⅢclinical trials are being accepted and receiving emphasis in the academic world. In the meantime, it has been discovered that the analysis process for group sequential design shares the characteristics of Bayesian statistics; therefore, Bayesian interim analysis in group sequential designing has gradually become a hot topic of research. In the past two decades, Bayasian approaches have been extensively developed in monitoring clinical trials (PhaseⅠand PhaseⅡ), and the Bayesian interim analysis employed in group sequential design is gaining increasing attention. Bayesian interim analysis fall into two categories:hybrid classical-bayesian methods based on hypothesis test and fully bayesian analysis based on posterior probability. The former is predominantly featured by the application of prior distribution in the desiging phase and exclusion of prior information in final phase of the analysis, while in the latter, the Bayesian posterior probability is worked out on the basis of the prior distribution and likelihood function, and the statistical inference is conducted on the basis of the posterior probability. Most of the present group sequential Bayesian interim analyses are based on fully Bayesian method, but this method is criticized by some scholars, mainly for the great subjectivity inherent with prior distribution and thus unacceptable results. All the researches mentioned above provide the theoretical basis to some extent for the application of the Bayesian method in the interim analysis in the group sequential designing. However, the present Bayesian interim analyses are just based on one or a few examples, not only lacking the comprehensive and systematic quantitative comparison of the advantages and disadvantages of Bayesian methods and classical statistical metods, but also going without the quantitative evaluation of reliability of Bayesian interim analysis. This study, based on the group sequential designing and by means of MCMC and statistical simulation, attempts to compare the differences between the Bayesian interim anaylysis and the interim analysis of the classical statistics, reveal the advantages and disadvantages of the two methods and with the results gained with the classical interim analysis as the reference, provide some references for the application of Bayesian statistics in the interim analysis of the clinical trials by means of the quantitative assessment of the reliability of the Bayesian results. MethodsIn order to compare independent samples mean between treatment and control, superior hypothesis test was established. In line with the data requirements for group sequential designing,α,β,δ,σ, times of interim analysis, methods of calculating the boundary values and the stopping rules were determined in the designing phase, and the normal data conforming to the sample size of each phase were randomly generated according to methods of classical interim analysis. With the accumulation of the data, the P value for the classical method and the Bayesian posterior P value were worked out in each interim analysis, with the stopping rules for each group sequential test determined by the P value of the classical interim analysis.In the data accumulation, the simulations and calculations were conducted in two circumstances:the negative study, a circumstance in which the Bayesian interim analysis TypeⅠerrorsε, which are based on various prior distributions, and the Bayesian negative coincidence rate were worked out; the positive study, a circumstance in which the power of Bayesian interim analysis based on various prior distributions, the Bayesian positive coincidence rate, average sample size, average stage and stage distribution were calculatedIn the above simulating research, two frequently-used methods of classical interim analysis were selected:Pocock method and O'Brien & Fleming method. Among the prior distributions in the Bayesian statistics, those deemed more "objecdtive" were considered, including skeptical prior, enthusiastic prior, non-informative prior and handicap prior distributions. And for the critical factors needed in the classical group sequential designing with early stopping to reject, includingα,β,δ,σ, the number of interim analyses and the stopping rules, the TypeⅠerrorαwas set at 0.05; the TypeⅡerrorβ,0.1,0.15,0.2,0.3 and 0.4, respectively;δ,2 and 3, respectively, andσ,10 and 18. The number of stages was set at 5,3 and 2.ResultsComparison of TypeⅠerrors and the Bayesian negative coincidence rate:In Pocock and the O'Brien & Fleming design, the TypeⅠerrorsεin the Bayesian interim analysis based on the non-informative prior and enthusiastic prior distributions were all markedly higher than 0.05, while the TypeⅠerrorsεbased on skeptical prior and handicap prior distributions were controlled around 0.05, with the Bayesian negative coincidence rate based on the enthusiastic prior distribution significantly lower than that in the other 3 distributions.Comparison in terms of power, Bayesian positive coincidence rate, average sample size and average numer of stages:In the O'Brien& Fleming design, when the power was 60% and 70%, the power of the skeptical prior distribution was lower than that of the handicap prior distribution in the design of five stages (For theδof 2, the power of skeptical piror were 34.00% and 49.33%, respectively, while the power of handicap piror 48.67% and 57.33%, respectively). When the power was above 80%, the power of the skeptical prior was higher than that of the handicap prior in the five-stage design, but slightly lower in the three-stage and two-stage design. And when the power was 80%, Bayesian power of the two kinds of priors mentioned above were markedly lower, relatively; when the power of O'Brien & Fleming increased from 85% to 90%, the Bayasian power of the two prior distributions neared the power of the O'Brien & Fleming method, with the Bayesian positive coincidence rate getting higher (coincidence rate of skeptical prior being around 98% and that of handicap prior, about 95%). The power of the non-informative prior distribution and that of the enthusiastic prior distribution were distinctly higher than that of the O'Brien & Fleming method. The alternative hypothesis being true, the average sample size and number of stages as required in the Bayesian methods with the various prior distributions were lower than those entailed by the O'Brien & Fleming method. And in case of larger average sample sizes and higher power, Bayesian interim analysis based on both Skeptical Prior and Handicap Prior had a greater possibility of early stopping for the test, compared with the O'Brien & Fleming method.In the five-stage Pocock design, the power of Skeptical Prior was slightly higher than that of Handicap Prior. When the power of the Pocock method was 80%, the Bayesian power of the two kinds of prior distributions mentioned above were maintained at low level, averaging around 65%, and when the Pocock's power increased from 85% to 90%, the difference in the power between the two kinds of prior distributions and the Pocock method was still large.In the three-stage Pocock design, the power of the Skeptical Prior was slightly lower than that of the Handicap Prior. When the Pocock's power increased from 85% to 90%, the difference in the power between the two kinds of distributions narrowed gradually, with the Bayesian positive coincidence rates of the two distributions being low, about 90%.In the two-stage Pocock design, the power of the Handicap Prior for various parameter combinations, was close to that of the Pocock method. And the Bayesian positive coincidence rate of the Handicap Prior approximated 100%, while the power of the Skeptical Prior was much lower than that of the Pocock method.In the Pocock designs, the average sample size and number of stages as required in the Bayesian interim analysis with the various prior distributions were smaller than those entailed by the Pocock method, but with the reduction of times of the interim analysis, the number of the saved samples in the Bayesian interim analysis based on the Skeptical Prior and Handicap Prior distributions could virtually be ignored, and these two types of analysis, compared with the Pocock method, could not increase the possibility of stopping the test ahead of time.ConclusionIn the condition of negative trial of O' Brien & Fleming and Pocock method, Bayesian interim analyses based on both the Non-informative Prior and Enthusiastic Prior are at the risk of increasing the TypeⅠerror, while in the Bayesian interim analysis based on the Skeptical Prior and Handicap Prior, the TypeⅠerrors can be well controlled. In negative trials, the results obtained through the Bayesian analysis employing the Enthusiastic Prior distribution are highly likely to be in disagreement with the results got through the analyses using the other three kinds of distributions.In the condition of positive trial of O'Brien & Fleming method, the coincidence of the conclusions obtained in Bayesian analyses using the Skeptical Prior and Handicap Prior distributions and the classical O'Brien & Fleming analyses increase with the enhancing of the power of O'Brien & Fleming method, which makes the conclusion more reliable. When the alternative hypothesis is true, the average sample size and number of stages as required in the Bayesian methods with the various prior distributions are smaller than those entailed by the O'Brien & Fleming method, and with a large-size sample and high power, the Bayesian interim analyses using the Skeptical Prior and Handicap Prior distributions, compared with the analyses adopting the O'Brien & Fleming method could markedly increase the possibility of ending the test ahead of time. After a positive study, major differences between the Non-informative and Skeptical Priors might suggest that confirmatory studies are warranted; in addition, they may explain the incomplete uptake of evidence-based therapies into clinical practice.In the condition of positive trial of Pocock method, with the reduction of the times of interim analysis, the power of the Handicap Prior distribution is approximate to the Pocock's power, but owing to the lack of advantages of the Bayesian interim analyses with the Skeptical Prior and Handicap Prior distributions in terms of the average sample sizes and average stages, the Bayesian interim analyses based on these two distributions do not have practical value for the group sequential design of Pocock method.