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Bayesian Approach For Clinical Trials
Mark Chang, Ph.D.
Executive Director Biostatistics and Data managementAMAG Pharmaceuticals Inc.
MBC August 28, 2008, Boston, USA
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OutlinesBasics of Bayesian ApproachFrequentist Power versus Bayesian
PowerBayesianism for Different Phases of
TrialsBayesian Decision Approach – Classic
and AdaptiveBayesian Trial SimulationsSummary
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Frequentist & Bayesian ParadigmsMany 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.
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Frequentist Fixed versus Bayesian Distributional ParametersOur 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
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Weighting
average
Illustration of Bayesian ApproachPrior knowledge =>
Prior probability
Current data =>
Likelihood function
Probability of outcome => posterior probability
P(data)
H)P(H)|P(datadata)|P(H
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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)
Normal Posterior
(Bayesian)
(nµ0 + mxm) /(m+n)
(6.67)m+n
(240)
2/(m+n)
(0.017)
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The Key Components for A Bayesian ApproachParametric Statistical Model
Modeling the underline mechanicsPrior 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.
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Bayesian Approach Basics (1)
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Bayesian Approach Basics (2)
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Bayesian Approach Basics (3)
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Power with Uncertainty of Treatment Effect (Prior)Treatment difference is a fixed but unknown
valuePrior response rate = 10%, 20%, or 30% with 1/3
probability each.Power = 80% based on n = 784, average effect
size =20%, orPower = (0.29+0.80+0.99)/3 = 0.69?
Effect size 10% 20% 30%
Power 0.29 0.80 0.99
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Power, Power? Power!
Probability of showing p-value < alphaConditional 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.
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Some Common MisconceptsAlpha = 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 IntervalCoverage 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
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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 – IIIDetermine sample size based on utility
Phase IV Better and more informative trial design
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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
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Bayesian Decision Approach for Pivotal Trials
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Bayesian Decision Approach for Pivotal Trials (cont.)
Time and Financial Constraints: Nmax.
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Bayesian Adaptive DesignAdaptive versus staticConditional versus unconditionalDecision difference under repeated
experiments vs. one time event in lifeExpected 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)
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Basic Steps for A Bayesian Trial Design1. Identify trial objectives2. Select statistical model .3. Determine the priors for the model
parameters.4. Calculate likelihood function (joint
probability) based on simulated data.5. Calculate the posterior probability.6. Define utility function.7. Specify constraints8. Perform optimization to maximize the utility
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Bayecian Dose Response Trials Using ExpDesign Studio 5.0
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Bayecian Dose Response Trials Using ExpDesign Studio (Cont)
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Bayecian Dose Response Trials Using ExpDesign Studio (Cont)
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Advanced TechniquesHierarchical modelNon-conjugate distributions and MCMC
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SummaryDrug 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.
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ReferencesMark 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.