1. Bayesian Methods for Benefit/Risk Assessment
Ram C. Tiwari
Associate Director
Office of Biostatistics, CDER, FDA

2. Disclaimer
This presentation reflects the views of the author and should not be construed to represent FDA’s views or policies.
Benefit-risk Assessment

3. Benefit-risk Assessment

4. Outline Introduction Methodology Illustration and simulation study
Commonly-used Benefit-risk (BR) measures
Methodology
BR measures based on Global benefit-risk (GBR) scores and a new measure
Bayesian approaches
Power prior
Illustration and simulation study
Future work
Benefit-risk Assessment

5. Introduction
The benefit-risk assessment is the basis of regulatory decisions in the pre-market and post market review process.
The evaluation of benefit and risk faces several challenges.
Benefit-risk Assessment

6. Commonly used B-R measures
Various measures have been proposed to assess benefit and risk simultaneously:
Q-TWiST by Gelbert et al. (1989)
Ratio of benefit and risk by Payne (1975)
The Number Needed to Treat and the Number Needed to Harm by Holden et al. (2003)
Global Benefit Risk (GBR) scores by Chuang-Stein et al. (1991)
Benefit-risk Assessment

7. BR categories
A five-category multinomial random variable to capture the benefit and risk of a drug product on each individual simultaneously:
Table 1: Possible outcomes of a clinical trial with binary response data
Benefit
No benefit
No AE
Category 1
Category 3 AE
Category 2
Category 4
withdrawal
Category 5
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8. Example: Hydromorphone
Data was provided by Jonathan Norton.
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9. GBR scores
Benefit-risk Assessment

10. Methodology: BR measures
BR measures based on the global scores proposed by Chuang-Stein et al.
BR measures based on the global scores are for each arm (treatment and comparator) separately.
BR_Linear can take a continuous value on a scale of -4 to 4 (inclusive).
Benefit-risk Assessment

11. Methodology: New BR measure
A new indicator based measure is proposed:
BR_Indicator compares two arms simultaneously.
It takes a integer value between -6 to 6 (inclusive).
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12. Methodology: Dirichlet prior
Dirichlet distribution is used as the conjugate prior for multinomial distribution, and the posterior distribution of the five-category random variable is derived at each visit using sequentially updated posterior as a prior.
Benefit-risk Assessment

13. Methodology: Sequential Updating
Sequential updating of the posteriors are given by:
The posterior mean (i.e., Bayes estimate) and 95% credible interval for each of the four measures are obtained using a Markov chain Monte Carlo (MCMC) technique.
Benefit-risk Assessment

14. Methodology: Decision Rules
For a BR measure,
If the credible interval include the value zero, the benefit does not outweigh the risk;
If the lower bound of the credible interval is greater than zero, the benefit outweighs the risk;
If the upper bound of the credible interval is less than zero, the risk outweighs the benefit.
Benefit-risk Assessment

15. Methodology: Power Prior
Power prior (Ibrahim and Chen, 2000) is used through the likelihood function to discount the information from previous visits, and the posterior distribution of the five-category random variable is obtained using the Dirichlet prior for p and a Beta (1, 1) as a power prior for .
Benefit-risk Assessment

16. Methodology: Model Fit
The model fit of the two models (with and without power prior) is assessed through the conditional predictive ordinate (CPO) and the logarithm of the pseudo-marginal likelihood (LPML). The larger the value of LPML, the better fit the model is. Here, n(i) is the data with ni removed.
Benefit-risk Assessment

17. Back to our example: Hydromorphone
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18. Illustration: Posterior Means and 95% Credible Intervals for BR_Linear Measure
without power prior
with power prior
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19. Illustration: Posterior Means and 95% Credible Intervals for BR_Indicator Measure
without power prior
with power prior
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20. Illustration: Results
a. The model without power prior b. The model with power prior
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21. Illustration: Posterior Means and 95% Credible Intervals for Power Prior Parameter
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22. Illustration: Model Fit
LPML values
Treatment
Control
Model without power prior
Model with power prior
-6.432
-6.190
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23. Simulation study
Correlated longitudinal multinomial data are simulated using the R package SimCorMultRes.R, which uses an underlying regression model to draw correlated ordinal response.
Two scenarios are simulated:
The treatment arm is similar to the control arm in terms of benefit-risk;
The treatment arm is better than control arm in the sense that the benefit outweighs risk.
Benefit-risk Assessment

24. Simulation study: Scenarios
Scenario 1: Treatment benefit does not outweigh risk compared to control Scenario 2: Treatment benefit outweighs risk compared to control
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25. Simulation study: Scenario 1
Treatment benefit does not outweigh risk compared to control
a. The model without power prior b. The model with power prior
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26. Simulation study: Scenario 2
Treatment benefit outweighs risk compared to control
a. The model without power prior b. The model with power prior
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27. Simulation study: Results
Scenario 1: Treatment benefit does not outweigh risk compared to control Scenario 2: Treatment benefit outweighs risk compared to control
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28. Simulation study: Model Fit
LPML values
Treatment
Control
Scenario 1:
Model without power prior
Model with power prior
-8.472
-7.667
Scenario 2:
-8.532
-8.393
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29. Future Work in BR Assessment
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30. Future work in BR assessment
Frequentist approaches:
Bootstrap approach
General linear mixed model (GLMM) approach
Other Bayesian approaches:
Normal priors
Dirichlet process
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31. Bootstrap Approach
Approximate underlying distribution using the empirical distribution of the observed data;
Resample from the original dataset;
Calculate the estimates and confidence intervals (CIs) of the BR measures based on the bootstrap samples;
Percentile bootstrap CIs;
Basic bootstrap CIs;
Studentized bootstrap CIs;
Bias-Corrected and Accelerated CIs.
Apply the decision rules.
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32. Bootstrap Approach-Results
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33. General linear mixed model (GLMM) approach
Within each arm (T or C), the ith subject falls in the jth category (vs. the first category) at kth visit can be modeled as,
where, α0 is the baseline effect assumed common across all categories, βj is the category effect, and γk is the longitudinal effect at kth visit, with
and,
Benefit-risk Assessment

34. GLMM approach
Note that different variance-covariance structures can be used for (γ1,γ2,…γ8), to model the longitudinal trend.
Compound-symmetry
Power covariance structure
Unstructured covariance structure
The estimates of the confidence intervals of the global measures can be derived from Monte Carlo samples, and the decision rules can be determined based on the confidence intervals.
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35. General linear mixed model approach-Results
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36. Bayesian approaches with GLMM
(α0 , βj ; j=1,…,5)~ independent Normal with means 0 and large variances;
Variance parameters~ IG
Dirichlet Process Approach: Let α0 to depend on subjects, that is, assume that α0i |G ~ iid G, with G~ DP(M, G0), M>0 concentration parameter and G0 a baseline distribution such as a normal with mean 0 and large variance. βj ; j=1,…,5 are independent normal with means 0, and large variances.
The posterior distributions of the probability and the global measures can be derived, and the decision rules can be determined based on the credible intervals.
Benefit-risk Assessment

37. Discussion
Quantitative measure of benefit and risk is an important aspect in the drug evaluation process.
The Bayesian method is a natural method for longitudinal data by sequentially updating the prior; Power prior can be used to discount information from previous visits.
Frequentist approaches such as bootstrapping method and general linear mixed model can be applied for benefit risk assessment.
Continuous research in longitudinal assessment of drug benefit-risk is warranted.
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38. Benefit-risk Assessment

39. Selected References
Gelber RD, Gelman RS, Goldhirsch A. A quality-of-life oriented endpoint for comparing treatments. Biometrics. 1989;45:
Payne JT, Loken MK. A survey of the benefits and risks in the practices of radiology. CRC Crit Rev Clin Radiol Nucl Med. 1975; 6:
Holden WL, Juhaeri J, Dai W. “Benefit-Risk Analysis: A Proposal Using Quantitative Methods,” Pharmacoepidemiology and Drug Safety. 2003; 12, 611–
Chuang-Stein C, Mohberg NR, Sinkula MS. Three measures for simultaneously evaluating benefits and risks using categorical data from clinical trials. Statistics in Medicine. 1991; 10:
Norton, JD. A Longitudinal Model and Graphic for Benefit-risk Analysis, with Case Study. Drug Information Journal. 2011; 45:
Ibrahim, JG, Chen, MH. Power Prior Distributions for Regression Models. Statistical Science. 2000; 15:
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40. Q & A
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