Bayesian Approach For Clinical Trials

Mark Chang, Ph.D.

Executive Director Biostatistics and Data managementAMAG Pharmaceuticals Inc.

Mark.Chang@Statisticians.org

MBC August 28, 2008, Boston, USA

OutlinesBasics of Bayesian ApproachFrequentist Power versus Bayesian

PowerBayesianism for Different Phases of

TrialsBayesian Decision Approach – Classic

and AdaptiveBayesian Trial SimulationsSummary

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.

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

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

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)

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.

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

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

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.

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

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

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 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)

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

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 TechniquesHierarchical modelNon-conjugate distributions and MCMC

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.

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.