Bayesian Hierarchical Model Approach To Evaluate The Treatment Effect For The Target Region In Multi-Regional Clinical Trials

BackgroundMulti-regional clinical trial（MRCT）refers to the simultaneous clinical trials conducted in multiple countries or regions（hereinafter referred to as regions）under the same protocol.After the MRCT completed,the drug efficacy and safety of each region in MRCT can be evaluated simultaneously,so as to achieve the purpose of registration at the same time.However,the regional treatment effects in MRCT may be different for the differences of ethnic and other factors among the regions in MRCT,the biostatisticians have developed many approaches for the consistency assessment between the treatment effect for each region and the overall MRCT.The weighting parameter in the weighted Z-test approach can’t be explained,the discounting factor approach which improved from the weighted Z-test approach,overcomes the shortcoming of the original weighted Z-test approach mentioned above.However,the discounting factor approach is an approach of frequency school,the discounting factor D is a fixed value without considering the variation.In addition,how to specify the weighting parameter in each consistency assessment approach is a very critical issue and the weighting parameter is usually difficult to be specified correctly,too much or too little information of the non-target regions would be borrowed for the weighting parameter not be specified correctly which would lead to inflating type-Ⅰ error or resources wasting.There is still a lack of authoritative and recognized statistical approaches.ObjectiveThere are two objectives in this study,one is to consider the variation of D on the basis of the original discounting factor approach,and another one is to specify more correctly how much information of the non-target regions should be borrowed to evaluate the treatment effect for the target region.Learning from the idea of bridging study and adopting Bayesian hierarchical model,we proposed two approaches to evaluate the treatment effect for the target region based on the discounting factor and proportion prior,respectively,and explored the feasibility of the two approaches.Methods1.Firstly,a new definition of D was given based on the original definition of D:the proportion of the subjects from the non-target regions equivalent to the subjects from the target region.On the basis of the new definition,we took D as a random variable and put forward the specification strategy for D.The posterior distribution of the weighted Z test statistic was constructed after the posterior distribution of D was obtained.By Monte Carlo simulation,we studied the power and type-Ⅰ error changing with the posterior mean and variation of D,and the consistency assessment boundary C in the design of simultaneous global drug development program.The proposed approach was adjusted according to the results of the simulation study,we then studied the power and type-Ⅰerror changing with the posterior mean and variation of D,the sample size of the local clinical trial N_TL,the proportion of the sample size for the target region in MRCT f,adjusting coefficient p and the estimator of the proportion of the treatment effect for the target region and the non-target regions δ_f for the adjusted approach by Monte Carlo simulation.2.We constructed the proportion prior distribution of the treatment effect for the target region by weighting the information from the non-target regions with a proportion coefficient q,the posterior distribution of the treatment effect for the target region was then constructed combining the proportion prior distribution and the data obtained from the target region by formula derivation or Markov Chain Monte Carlo method.The following simulation studies were carried out:1)The posterior mean and standard deviation of the treatment effect for the target region（（?）₁,η）changing with the mean and standard deviation of q（γ,ξ）when q follows normal distribution;2)The probability density function curve of the prior and posterior distributions of the treatment effect for the target region with q following different distributions were drawn to show directly the influence of q on the evaluation of the treatment effect for the target region;3)Assuming q follows laplace distribution,normal distribution and uniform distribution with the same 95%confidence interval,respectively,we studied the posterior probability of a positive treatment effect for the target region P（δ₁>0）changing with the maximum separation distance κ and the function shape of q.Results1.In the unadjusted Bayesian approach to evaluate the treatment effect for the target region based on the discounting factor,the power and type-I error increased with the posterior mean of D increasing and decreased with C increasing,the power and type-I error were not affected by the variation of D.When the posterior mean of D in the unadjusted Bayesian approach with C=0.5 was equal to the value of D in the original discounting factor approach,the two approaches always had the same power and type-I error.2.In the adjusted Bayesian approach to evaluate the treatment effect for the target region based on the discounting factor,the power and type-Ⅰ error still increased with the posterior mean of D increasing,but decreased with the variation of D increasing,the adjusted approach achieved the expected effect,the adjustment degree decreased with N_TL,f,ρ,δ_f increasing.The power increased with N_TL increasing,the type-Ⅰ error decreased with N_TL increasing.The power increased with f increasing,when the posterior mean of D was small,the type-Ⅰ error decreased with f increasing,however,when the posterior mean of D was large,the type-Ⅰ error increased with f increasing.The power and type-Ⅰ error increased with ρ,δ_f increasing.3.When q followed normal distribution,（?）₁ decreased with γ decreasing,but ηdidn’t change with γ.（?）₁ and η changed towards the analysis results of the data obtained from the target region with ξ increasing,if ξ is large enough,（?）₁ and η were basically the same as the analysis results of the data obtained from the target region.Whenγ=1 and ξ approaches to 0,（?）₁ and η were basically the same as the analysis results of the data obtained from the target region and the non-target regions.4.The corresponding prior and posterior distributions of the treatment effect for the target region were different if the distribution of q was different.When q followed laplace distribution,normal distribution,uniform distribution with the same 95%confidence interval,respectively,P（δ₁>0）changed from the P（δ₁>0）that all information of the non-target regions was borrowed towards the P（δ₁>0）that none information of the non-target regions was borrowed,when q followed uniform distribution,it approached fastest and when q followed laplace distribution,it approached slowest.ConclusionThe two consistency assessment approaches proposed in this study both learn from the idea of bridging study and adopt Bayesian hierarchical model.We suggest that the C in the Bayesian approach based on the discounting factor be specified to be 0.5 to ensure that when none information of the non-target regions is borrowed,the analysis results is just equal to the analysis results of the data obtained from the target region.The adjusted Bayesian approach can better reflect the influence of the variation of the discounting factor on the power and type-Ⅰ error.The q in the Bayesian approach to evaluate the treatment effect for the target region based on proportion prior has a good practical significance and can better fit the actual situation by adjusting the distribution of q,then how much information of the non-target regions to borrow can be more correctly specified.