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**Bayesian Adaptive Methods**

Carrie Deis Nadine Dewdney

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**Overview Phase I clinical trials Standard Designs Adaptive Designs**

Bayesian Approach Traditional vs. Bayesian Hybridization FDA Guidance Conclusion

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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)

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**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

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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

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**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

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**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

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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

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**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

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**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

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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

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**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

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Bayesian Approach

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**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

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**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

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**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

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**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

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**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

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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

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**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

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**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

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**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

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**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

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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

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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

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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:

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