1. Bayesian Statistics in Clinical Trials Case 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

2. 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

3. 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:
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

4. Pedagogical 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:

5. 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

6. Example 1 Two-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: =

7. Example 1 No 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
Bayesian P(success)
Frequentist
0.7
0.5
95.4%
92.2%
0.65
87.5%
80.5%
0.6
71.3%
62.7%
27.5%
24.0%
0.4
4.3%
4.5%

8. Example 1 Early 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 51 59
0.6 73 83
188
194
0.4
~200
Include a futility rule!

9. Example 1 Summary
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”

10. Example 2 Continual 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

11. Example 2 Continual Reassessment Method (CRM)
Example: Sample size = 40; Cohort size = 4; Target toxicity = 25%
Assumed dose-toxicity model:
Behavior (simulated) after 8th cohort
Average allocation across 100 simulations
Highest allocation to dose 3, which has true toxicity closest to target of 25%

12. Example 3 Phase 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

13. Example 3 Bayesian Adaptive Design
Trial Start
Stage/Cohort 1
Placebo
Arm 3
Arm 1
Analyze the data
Make a decision
Interim Analysis 1
Stage/Cohort 2
Interim Analysis 2
etc.
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

14. Example 3 Bayesian 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 derive joint posterior distribution of parameters
Graphic shows true EDx (red) and NDLM fit (blue)

15. Example 3 Bayesian 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

16. Example 3 Dose-Response Scenarios

17. Example 3 Operating Characteristics
Scenario 1:
Average sample size = 305
Pr(futility) = 0.5%
Pr(efficacy) = 73.9%
Pr(max) = 25.6%
Scenario 2:
Average sample size = 269
Pr(futility) = 0.1%
Pr(efficacy) = 79.5%
Pr(max) = 20.4%

18. Example 3 Operating 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%

19. Example 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

20. Example 5 Trial 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

21. Example 5 Accrual 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

22. Example 5 Event 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

23. Design 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)

24. References
Cook, J. (2006). Understanding the Exponential Tuning Parameter in Adaptively Randomized Trials. MDAnderson Technical Report 27. see also “AdaptiveRandomization” documentation on MDAnderson Cancer website:
Muller, P., Berry, D., Grieve, A., Krams, M. (2006). A Bayesian Decision-Theoretic Dose-Finding Trial. Decision Analysis. 3 (4)
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.

25. Summary 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.