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

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 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 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) 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

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: =

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%

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!

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”

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

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%

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

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

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)

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

Example 3 Dose-Response Scenarios

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%

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%

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

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

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

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

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)

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: 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.

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.