1. Bayesian Adaptive Methods
Carrie Deis
Nadine Dewdney

2. Overview Phase I clinical trials Standard Designs Adaptive Designs
Bayesian Approach
Traditional vs. Bayesian
Hybridization
FDA Guidance
Conclusion

3. Phase I
Conducted to determine toxicity for the dosing of the new intervention
First time the drug is tested in humans
Small number of patients, 20 to 50
Depending on the nature of the new drug, patients are usually healthy volunteers
A higher dose is assumed to be more effective
Goal is to find maximum tolerable dose (MTD)

4. Phase I Known prior to the start of the trial:
Starting dose
Toxicity profile and Dose-limiting toxicity
Target toxicity level
Dose Escalation Scheme
Starting dose commonly chosen as:
1/3 lowest toxic dose in dogs
1/10 of the LD10 in mice
Dose escalation is done incrementally
Increments are pre-determined
Modified Fibonacci sequence – increase rate diminishing as the dose gets higher

5. Standard Design
Patients are assigned to dose levels according to predefined rules
Allow for only escalation and de-escalation of dose
Doses selected such that, D1,…, DK would be close to MTD
MTD is determined statistically as the dose at which 1/3 of the subjects develop toxicity

6. Standard Design Subjects are randomized
The number of subjects, ri, developing toxicity would be observed
pi = ri/ni, is used to calculate the proportions exhibiting toxicity
Dose-response is modeled based on the probability of toxicity
The MTD would be fitted to this model

7. Standard Design Ethical concerns with the traditional approach
Patients might be treated excessively and unnecessarily at low doses
Too many patients may be treated at doses that are too high or too low
Highly likely most subjects are treated at low doses
Not clear that the estimated MTD is the correct dose

8. Adaptive Designs
Adjustments and modifications can be made after the trial has started
Does not affect the integrity of the trial
Goal is to improve upon the probability of success of the trial and correctly identify the clinical benefits of the intervention under investigation
Prospective adaptations include
Stopping a trial early for safety or lack of efficacy
Dropping the loser - Inferior treatments dropped
Sample size re-estimation

9. Adaptive Designs Modifications hypothesis might be necessary
Inclusion/exclusion criteria
Dose/regimen
Treatment duration
Endpoints
Several types of adaptive designs
Group sequential
Sample size adjustable design
Drop-the-losers design
Adaptive treatment allocation design
Bayesian adaptive methods

10. Bayesian Approach Based on Bayes Theorem:
Expresses how a subjective degree of belief should rationally change to account for evidence
Used as a statistical inferential tool in adaptive designs
Strength of the Bayesian approach
Decision on trial continuation is made as data accumulates
Sample size not determined in advance although a maximum size might be specified
Drawbacks
Analysis after each subject is treated

11. Bayesian Approach
Calculates the predictive probability that the patient will respond to treatment
Specifies a prior distribution then updates it as information becomes available
Uses the likelihood function and the prior distribution to obtain a posterior distribution
MTD is determined from the posterior distribution
Studies are based on costs and public health benefits

12. p(d) = exp (3 + ad)/[ 1 + exp(3 + ad)]
Bayesian Approach
Prior Distribution
Logistic model:
p(d) = exp (3 + ad)/[ 1 + exp(3 + ad)]
Power model:
p(d) = dexp(a)
p(d) is the probability of DLT
d is the dose
a is a model parameter

13. Bayesian Approach

14. Bayesian Approach Once the posterior distribution is calculated:
The MTD is revised based on the distribution of a
The mode of the posterior distribution is used to estimate the next dose
Each patient is treated at the dose which is closest to the MTD
Toxicity profile is updated after each patient is treated
The sequence is repeated until a precise estimate of parameter a is obtained or the sample size is exhausted

15. Traditional vs. Bayesian
Example: A dose-finding escalation design from an oncology trial
Traditional approach
The 3+3 traditional escalation rule (TER)
Bayesian approach
The continual reassessment method (CRM)
The objective is to determine the MTD for a new drug using the least amount of patients

16. Traditional vs. Bayesian
Results from animal studies:
The dose limiting toxicity rate was determined to be 1% for the starting dose of 25 mg/m2, 1/10 of the lethal dose
The MTD is estimated to be 150 mg/m2
The dose limiting toxicity rate is defined as 0.25
Selected Model: A logistic toxicity model
Dose sequence was chosen with interim factors = 2, 1.67, 1.33, 1.33, 1.33, 1.33, 1.33, 1.33, 1.33

17. Traditional vs. Bayesian
Summary of simulation results for the designs
Method
Assumed True MTD
Mean Predicted MTD
Mean number of Patients
Mean number of DLTs
3+3 TER
100
86.7
14.9
2.8
CRM
99.2
13.4
150
125
19.4
2.9
141
15.5
2.5
200
169
22.4
186
16.8
2.2

18. Traditional vs. Bayesian
Both approaches underestimate the true MTD
However, the Bayesian approach was much closer to the true value for all dose levels
At all three dose levels the Bayesian approach required less patients
The mean number of DLTs for the Bayesian approach was either less than or equal to the traditional approach at all dose levels
The Bayesian CRM approach proved to be more favorable

19. Hybridization
The Bayesian approach can be used alone or as a hybrid with the classic approach
As a hybrid, the Bayesian approach is used to increase the probability of success
Example: Two-arm parallel design
Compares a test treatment and a control
Use data from 3 clinical trials with similar sample sizes
Prior probabilities for the effect size are 0.1, 0.25, and 0.4 with 1/3 probability for each trial

20. Hybridization The classic approach: The Bayesian approach:
Mean of the effect size, = 0.25, is used to calculate the sample size. For the design with β = 0.2:
The Bayesian approach:
The power of the effect size is
Φ is the c.d.f. of the standard normal distribution
Prior, π(ε), is the uncertainty of ε, the expected power

21. Hybridization Assuming, one-sided α = 0.025, Pexp =0.66+
Hybridization
Assuming, one-sided α = 0.025,
Pexp =0.66
With the hybrid approach the power is less than the 80% power stated in the frequentist approach, recall β = 0.2. In order to reach the expected power of 80%, the sample size needs to be increased
The Bayesian approach piece is used to increase the probability of success given that the final criterion is p ≤ α = 0.025

22. FDA Guidance – Medical Devices
Prior information and Assumptions
Criterion for success for safety and effectiveness
Justification for the proposed sample size
Prior probability of the study claim
This is the probability of the study claim before seeing any new data, and it should not be too high
Ensures the prior information does not overwhelm the current data, potentially creating a situation where unfavorable results from the proposed study get masked by a favorable prior distribution
Program Code

23. FDA Guidance – Medical Devices
Operating characteristics
Provide tables of the probability of satisfying the study claim, given “true” parameter values and sample sizes for the new trial
Provides an estimate of the probability of a type I error in the case where the true parameter values are consistent with the null hypothesis, or power in the case where the true parameter values are consistent with the alternative
Effective Sample Size
Quantifies the efficiency you are gaining from using the prior information and gauges if the prior is too informative

24. Conclusion
Bayesian full approach is more beneficial in Phase I studies
Inherent adaptive nature of the design
Conditions are more dynamic than other phases and the flexible nature of the Bayesian approach allows for unexpected changes
Produces a posterior probability which is useful in decision making and the transitioning from one phase to the next
Dose levels can be modified which could be beneficial for a phase I cancer study

25. Conclusion
Even without using a full Bayesian method, hybridization results in increased probability of success in trials
Maintaining the validity and integrity of the study and control of the type I error in applications of the method is important
Feasibility should be evaluated in order to prevent abuse of this method in applications such as endpoints or hypotheses changes
The FDA is cautious of the growing trend of Bayesian designs and continues to set guidelines for its use in Phase I trials

26. References
Chang, Mark (2008). Adaptive Design Theory and Implementation Using SAS and R. Boca Raton: Chapman & Hall/CRC
Berry, Scott M., Carlin, Bradley P., Lee, J.Jack, Muller, Peter (2011). Bayesian Adaptive Methods for Clinical Trials. Boca Raton: Chapman & Hall/CRC
Chow, Shein-Chung and Chang, Mark (2008). Adaptive Design Methods in Clinical Trials – A Review. Orphanet Journal of Rare Diseases, 3 11
Cook, Thomas D. and DeMets, David L. (2008). Introduction to Statistical Methods for Clinical Trials. Boca Raton: Chapman & Hall/CRC
The FDA Center for Drug Evaluation and Research, and Center for Biologics Evaluation and Research, Guidance for Industry: Adaptive Design Clinical Trials for Drugs and Biologics:

27. Questions