Bayesian Statistics in Clinical TrialsCase Studies: AgendaBayesian Statistics in Clinical TrialsCase Studies: Agenda

Review of when Bayesian statistics are useful in the clinical trial setting

Five Examples (Case Studies):

– Two-arm Non-inferiority trial with a binary endpoint

– Continual Reassessment Method (CRM)

– Phase 2 adaptive dose-ranging trial

– Seamless Phase 1/2 trial (outline)

– Accrual and event endpoint monitoring in a Phase 3 trial

Review of Design Considerations for a Bayesian Trial

Review of Benefits of the Bayesian Approach in Clinical Trials (not exhaustive)Review of Benefits of the Bayesian Approach in Clinical Trials (not exhaustive)

Can incorporate information from previous studies through the use of prior distributions and historical controls

– Can effectively reduce the sample size required to achieve trial objectives

Can be useful for incorporating decision criteria into your adaptive clinical trial

– Response-adaptive randomization

– Stopping rules

Is particularly useful for trial monitoring purposes

– Use predictive probability to forecast future trial outcome(s)

Can make probability statements about trial results

– p-values are not probabilities of trial hypotheses

When are Bayesian Trials Most Often Used?When are Bayesian Trials Most Often Used?

When it is appropriate to use historical data/evidence in your trial

In medical device trials where often there exists prior evidence for a control arm

– FDA/CDRH has issued guidance documentation “for the use of Bayesian Statistics in Medical Device Clinical Trials:http://www.fda.gov/cdrh/osb/guidance/1601.html

When the sponsor organization wants to make internal decisions based on the results of a clinical trial

When a Bayesian design can provide improved efficiencies in the adaptive realm

Pedagogical Example 0Pedagogical Example 0

Suppose we conduct an experiment (clinical trial) by allocating n=50 subjects to a treatment (label it C).

– Assume a binary response, so number of responders, X, is a Binomial R.V.

– We observe X=23 (out of 50) responses

– We want to estimate p, the proportion of responders

Frequentist analysis:

– Point estimate:

– 95% Confidence Interval:

Pedagogical Example 0 (cont’d)Pedagogical Example 0 (cont’d)

Bayesian analysis:

– Suppose we have historical evidence on treatment C:previous study of size 20 with observed proportion 0.5

– Leverage the beta-binomial conjugacy and construct a Beta(10,10) prior on p.

– Posterior: Beta(10+X,10+n-X) = Beta(33,37)

– Posterior mean: 0.471

– 95% Credible Interval: (0.356, 0.588)

Notes:

– Posterior mean (0.471) is between data mean (0.46) and prior mean (0.5)

– Bayesian credible interval is narrower than frequentist CI

0.0 0.2 0.4 0.6 0.8 1.0

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Example 1Two-Arm Non-Inferiority Trial: Binary EndpointExample 1Two-Arm Non-Inferiority Trial: Binary Endpoint

Consider a two-arm trial comparing treatment C (Control, from previous example) to treatment E (Experimental)

– Binary endpoint

– Non-inferiority trial with Δ=0.1

Bayesian approach

– Beta(10,10) prior on treatment C (using historical evidence)

– Beta(1,1) prior on treatment E (uniform prior)

– Success criterion: =

Example 1No Adaptation: Operating CharacteristicsExample 1No Adaptation: Operating Characteristics

Fixed sample size: 50 per arm

Power calculated through simulation

More power with Bayesian approach due to inclusion of historical data through prior distribution

– Equivalently, fewer patients are required in current trial using Bayesian approach

With a “non-informative” prior for [Beta(1,1), or uniform(0,1)], achieve similar power levels using two approaches

Power

BayesianP(success)

Frequentist

0.7 0.5 95.4% 92.2%

0.65 0.5 87.5% 80.5%

0.6 0.5 71.3% 62.7%

0.5 0.5 27.5% 24.0%

0.4 0.5 4.3% 4.5%

Example 1Early Stopping for EfficacyExample 1Early Stopping for Efficacy

Enroll subjects in cohorts of size 20, maximum sample size 200

After each cohort, calculate the posterior probability of success,(Assume for simplicity that responses are available immediately)

If > 0.95, stop for efficacy

Average Sample Size

Beta(10,10) Beta(1,1)

0.7 0.5 38 46

0.65 0.5 51 59

0.6 0.5 73 83

0.5 0.5 188 194

0.4 0.5 ~200 ~200Include a futility rule!

Example 1SummaryExample 1Summary

Inclusion of prior information can increase power, or equivalently, reduce the number of subjects required to achieve trial objective

Bayesian early-stopping rule for efficacy reduces expected sample size

To design the trial adequately, a more thorough simulation exercise would need to be undertaken

Additional Bayesian decisions to consider during interim analyses:

– Early stopping for futility

– Adaptive randomization: Modify the randomization probabilities for subsequent cohorts based on posterior (predictive) probability that one treatment is better than the other (Cook, 2006):

• If is the posterior probability that treatment i is the best, then allocate subjects to that treatment with probability:

• k indexes the treatments and λ is “tuning parameter”

Example 2Continual Reassessment Method (CRM)Example 2Continual Reassessment Method (CRM)

Origins in Phase 1 toxicity Oncology trials

Subjects are accrued in cohorts (often of size 3)

Adaptive design where estimate of MTD is “continually reassessed” by fitting a parametric model to the toxicity data

Objective: Find the dose that yields a desired response/toxicity level

Interim analysis: Fit a parametric model to the data and estimate probability (possibly Bayesian posterior) of response at each dose

Interim decision: Allocate subjects in next cohort to dose with (posterior) probability closest to the target level

Example: Sample size = 40; Cohort size = 4; Target toxicity = 25%

Assumed dose-toxicity model:

Average allocation across 100 simulationsBehavior (simulated) after 8th cohort

Highest allocation to dose 3, which has

true toxicity closest to target of 25%

Example 2Continual Reassessment Method (CRM)Example 2Continual Reassessment Method (CRM)

Example 3Phase 2 Adaptive Dose-Finding TrialExample 3Phase 2 Adaptive Dose-Finding Trial

Consider a Phase 2 dose-finding trial with:

– 4 active dose levels plus placebo

– Continuous endpoint (change from baseline to Week 2 measurement)

Primary objective

– The primary objective is to find the best (clinically effective) dose to carry forward into Phase 3

– Clinical effectiveness is defined by difference of 3 relative to placebo

– 400 subjects are required for this trial using standard design (90% power)

Adaptive Solution

– Based on data observed at various interim looks, modify the randomization probabilities to focus allocation on better performing doses using Bayesian decision criteria

Example 3Bayesian Adaptive DesignExample 3Bayesian Adaptive Design

Trial Start

Stage/Cohort 1

Placebo

Arm 3

Arm 1 • Analyze the data

• Make a decision

Interim Analysis 1

Stage/Cohort 2

• Analyze the data

• Make a decision

Interim Analysis 2

etc.Arm 2

Arm 4

Placebo

Arm 3

Arm 1

Arm 2

Arm 4

At each Interim Analysis:

– Analyze the data by fitting a Bayesian model

– Make a Decision:

• Stop the trial early for efficacy

• Stop the trial early for futility

• Otherwise, allocate more subjects to the clinically efficacious doses

Example 3Bayesian ModelExample 3Bayesian Model

Bayesian Model (Normal Dynamic Linear Model, NDLM):

– Normal and Inverse-Gamma priors

– Muller et al (2006)

NDLM is a “smoother” and will allow doses to “borrow strength”

Use Markov Chain Monte Carlo to derivejoint posterior distribution of parameters

Graphic shows true EDx (red) and NDLMfit (blue)

0 1 2 3 4 5 6 7

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Doses

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Example 3Bayesian Decision RulesExample 3Bayesian Decision Rules

Based on Bayesian model (NDLM) fit, let

– be the estimated effect of each dose (relative to placebo)

– be the smallest clinically relevant effect

Efficacy rule:

– Stop the trial early if for any j

Futility rule:

– Stop the trial early if for all j

Response-Adaptive randomization:

– Assign subjects to dose j in next cohort using randomization probabilities:

Maximum sample size = 500

Example 3Dose-Response ScenariosExample 3Dose-Response Scenarios

Scenario 1

Treatment Arm

Mean c

hange fro

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aselin

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Treatment ArmM

ean c

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Scenario 3

Treatment Arm

Mean c

hange fro

m b

aselin

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1 2 3 4 5

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Scenario 4

Treatment Arm

Mean c

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Null Scenario

Treatment Arm

Mean c

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Example 3Operating CharacteristicsExample 3Operating Characteristics

Scenario 1:– Average sample size = 305

– Pr(futility) = 0.5%

– Pr(efficacy) = 73.9%

– Pr(max) = 25.6%

Treatment Arm

Ave

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Placebo 1 2 3 4

Scenario 2:– Average sample size = 269

– Pr(futility) = 0.1%

– Pr(efficacy) = 79.5%

– Pr(max) = 20.4%

Treatment Arm

Ave

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Placebo 1 2 3 4

Example 3Operating CharacteristicsExample 3Operating Characteristics

Scenario 3:– Average sample size = 250

– Pr(futility) = 0%

– Pr(efficacy) = 83.4%

– Pr(max) = 16.6%

Scenario 4:– Average sample size = 244

– Pr(futility) = 0%

– Pr(efficacy) = 83.9%

– Pr(max) = 16.1%

Treatment Arm

Ave

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Placebo 1 2 3 4

Treatment Arm

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Example 4 (outline)Seamless Phase 1/2 TrialExample 4 (outline)Seamless Phase 1/2 Trial

Response-adaptive randomization in “Phase 1” based on laboratory parameters and biomarker(s)

– Adaptations are based on a Bayesian model

1 or 2 doses selected at end of Phase 1 that are biocomparable to active control

“Phase 2” stage uses efficacy endpoint for Proof-of-Concept (PoC)

– Efficacy measurements from Phase 1 are included in PoC assessment using Bayesian analyses

Example 5Trial Monitoring using Predictive ProbabilitiesExample 5Trial Monitoring using Predictive Probabilities

Two-arm Phase 3 trial with time-to-event endpoint

– Multi-national

– >100 sites

Monitoring Objective: Forecast Accrual and Events at various looks(Trial has been powered based on number of events, not number of subjects)

– Model accruals and events using parametric distributions

– Gather accrual and event information at an interim look

– Use Bayesian posterior predictive probabilities for forecasting

– Process does not affect trial design, only used for internal decision making purposes

Example 5Accrual MonitoringExample 5Accrual Monitoring

Assume accrual occurs according to a Poisson process

– Different rate for each site

– Different Gamma prior distribution for each site

Posterior predictive distribution at each site is Negative Binomial

Distribution of total number of future accruals is convolution of above distributions

Graphic shows past accrual with predicted means and 95% credible envelope

Time

Example 5Event MonitoringExample 5Event Monitoring

Assume events occur according to an exponential distribution

– Different parameter for two arms

– Different prior Gamma distribution for two arms

Compute posterior predictive distribution of number of events using numerical techniques

– Includes future accrual forecasts

Graphic shows past events with predicted means and 95% credible envelope Time

Design Considerations for a Bayesian TrialDesign Considerations for a Bayesian Trial

Bayesian clinical trials are inherently more complex to design than their classical counterparts

Homogeneity of responses/subjects between historical and current data

Simulation should be used to determine trial Operating Characteristics– Sensitivity to choice of prior– Power, type-I error– Expected sample size– Probabilities of trial outcomes

Computational issues (when posterior is not known completely)– Bayesian Central Limit Theorem– Laplace approximation– Importance sampling, rejection sampling– Markov Chain Monte Carlo (MCMC)

ReferencesReferences

Cook, J. (2006). Understanding the Exponential Tuning Parameter in Adaptively Randomized Trials. MDAnderson Technical Report 27.

see also “AdaptiveRandomization” documentation on MDAnderson Cancer website: http://biostatistics.mdanderson.org/SoftwareDownload

Muller, P., Berry, D., Grieve, A., Krams, M. (2006). A Bayesian Decision-Theoretic Dose-Finding Trial. Decision Analysis. 3 (4). 197-207.

Berry, D., et al (2001). Adaptive Bayesian Designs for Dose-Ranging Drug Trials. In Case Studies in Bayesian Statistics, Vol. V. Springer-Verlag, New York.

Spiegelhalter, D., Abrams, K., Myles, J. (2004). Bayesian Approaches to Clinical Trials and Health Care Evaluation. John Wiley and Sons, Chichester, UK.

Gilks, W., Richardson, S., Spiegelhalter, D. (1996). Markov Chain Monte Carlo in Practice. Chapman and Hall / CRC.

Gelman, A., Carlin, J., Stern, H., Rubin, D. (2004). Bayesian Data Analysis. Chapman and Hall / CRC.

SummarySummary

The Bayesian approach allows one to

– Formally synthesize and incorporate prior evidence into a design

– Incorporate flexible decision criteria into an adaptive design

– Take multiple “looks” at the data without incurring a penalty

– Monitor and forecast trial outcomes with predictive probabilities

Designing a Bayesian trial requires more up front planning to handle prior sensitivity issues, modeling assumptions, etc.