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Bayesian Approach For Clinical Trials Mark Chang, Ph.D. Executive Director Biostatistics and Data management AMAG Pharmaceuticals Inc. [email protected] MBC August 28, 2008, Boston, USA Outlines Basics of Bayesian Approach Frequentist Power versus Bayesian Power Bayesianism for Different Phases of Trials Bayesian Decision Approach – Classic and Adaptive Bayesian Trial Simulations Summary Frequentist & Bayesian Paradigms Many believe that the probability concepts from Frequentist and Bayesian are different. However, from decision-making point of view, we do not differentiate them. Frequentist: type I and type II error for trial design and p-values, point estimate, and confidence intervals for analysis. Bayesianism: prior distribution about model parameter (e.g., population mean treatment effect), combined with evidence from a clinical trial (likelihood function) to form the posterior distribution - the updated knowledge about the parameter. Frequentist Fixed versus Bayesian Distributional Parameters Our action taken is not upon the truth because the truth is always a mystery. We make decision is upon what we know about the truth, or more precisely based on what we think the truth is. Semantic: Parameter => Fixed & Unknown Knowledge about => distribution Illustration of Bayesian Approach Prior knowledge => Prior probability Current data => Weighting average Probability of outcome => posterior probability Likelihood function P(data| H)P(H) P(H| data) P(data) Effects of Priors on Posterior – A Simple Example of Weighting Average Mean difference Sample size Standard variance Normal Prior µ0 (5) n (40) 2/n (0.1) Trial data (Frequentist) Xm (7) m (200) 2/m (0.02) (nµ0 + mxm) /(m+n) m+n (240) 2/(m+n) (0.017) Normal Posterior (Bayesian) (6.67) The Key Components for A Bayesian Approach Parametric Statistical Model Modeling the underline mechanics Prior distribution Probability distribution of model parameters using evidences before the experiment. Likelihood function Probability distribution of model parameters using evidences from the experiment. Posterior distribution Probability distribution of model parameters derived from the products of prior and likelihood function. Predictive probability Probability distribution of future patient’s outcomes based on posterior distribution. Utility function A single index measuring overall gains of the treatment, which could include efficacy, safety and etc. Bayesian Approach Basics (1) Bayesian Approach Basics (2) Bayesian Approach Basics (3) Power with Uncertainty of Treatment Effect (Prior) Treatment difference is a fixed but unknown value Prior response rate = 10%, 20%, or 30% with 1/3 probability each. Power = 80% based on n = 784, average effect size =20%, or Power = (0.29+0.80+0.99)/3 = 0.69? Effect size 10% 20% 30% Power 0.80 0.99 0.29 Power, Power? Power! Probability of showing p-value < alpha Conditional or unconditional probability? Only 5% Phase I trials are eventually get approved. About 40% Phase III trials get approved, but 80%95% power when the trials are designed. Some Common Misconcepts Alpha = 2.5% => control false positive drug in the market no more than 2.5%. If all test drugs in phase-III are effective, then type-I error rate = 0%. If all test drugs in phase-III are ineffective, then type-I error rate = 100% Confidence interval = Bayesian Credible Interval Coverage probability concerns a set of CIs with various lengths and locations. Maturity of data is a requirement of rejecting the null hypothesis of no treatment difference When Should Bayesian Approach be used Phase – I Safety response models with various doses or regiments Phase –II Efficacy and safety response models; Dose selection Phase – III Determine sample size based on utility Phase IV Better and more informative trial design Bayesian Approach for Multiple-Endpoint Problems All stepwise or sequential procedures in Frequentist use a sort of “composite endpoint”: Rejection Criterion for the k-th null hypothesis: pk< F(alpha, p1, p2,…,pk-1) Q(p1, p2,…,pk) < alpha Bayesian Decision Approach for Pivotal Trials Bayesian Decision Approach for Pivotal Trials (cont.) Time and Financial Constraints: Nmax. Bayesian Adaptive Design Adaptive versus static Conditional versus unconditional Decision difference under repeated experiments vs. one time event in life Expected utility of life insurance is negative, we buy it because we have one life and a death will great impact on family member. Flip a coin, if head, gain $1.5m; if tail, lose $1m. Do you play? (Think about playing one time versus many times) Basic Steps for A Bayesian Trial Design 1. 2. 3. 4. 5. 6. 7. 8. Identify trial objectives Select statistical model . Determine the priors for the model parameters. Calculate likelihood function (joint probability) based on simulated data. Calculate the posterior probability. Define utility function. Specify constraints Perform optimization to maximize the utility Bayecian Dose Response Trials Using ExpDesign Studio 5.0 Bayecian Dose Response Trials Using ExpDesign Studio (Cont) Bayecian Dose Response Trials Using ExpDesign Studio (Cont) Advanced Techniques Hierarchical model Non-conjugate distributions and MCMC Summary Drug development involves a sequence of decision process where Bayesian adaptive approach provides powerful solutions that traditional frequentist can not provide. Computer simulations for Bayesian adaptive design could provide predictions on trial outcomes under various scenarios and therefore allows us to select optimal design It is likely that a hybrid Frequenstist-Bayesian approach would be used before adoption of full Bayesian in larger scale for clinical trials. References Mark Chang, Classical and Adaptive Clinical Trial Designs Using ExpDesign Studio (Includes ExpDesign 5.0 software CD). John-Wiley, 2008. Mark Chang, Adaptive Design Theory and Implementations Using SAS and R, Chapman & Hall/CRC, 2007.