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### Bayesian Methods for Benefit/Risk Assessment

### Future Work in BR Assessment

Ram C. Tiwari

Associate Director

Office of Biostatistics, CDER, FDA

Disclaimer

This presentation reflects the views of the author and should not be construed to represent FDA’s views or policies.

Benefit-risk Assessment

Outline

- Introduction
- 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

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

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

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-risk Assessment

GBR scores

Benefit-risk Assessment

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

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).

Benefit-risk Assessment

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

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

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

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

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 niremoved.

Benefit-risk Assessment

Back to our example: Hydromorphone

Benefit-risk Assessment

Illustration: Posterior Means and 95% Credible Intervals for BR_Linear Measure

without power prior

with power prior

Benefit-risk Assessment

Illustration: Posterior Means and 95% Credible Intervals for BR_Indicator Measure

without power prior

with power prior

Benefit-risk Assessment

Illustration: Results

a. The model without power prior b. The model with power prior

Benefit-risk Assessment

Illustration: Posterior Means and 95% Credible Intervals for Power Prior Parameter

Benefit-risk Assessment

Illustration: Model Fit

Benefit-risk Assessment

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

Simulation study: Scenarios

Scenario 1: Treatment benefit does not outweigh risk compared to control

Scenario 2: Treatment benefit outweighs risk compared to control

Benefit-risk Assessment

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

Benefit-risk Assessment

Simulation study: Scenario 2

Treatment benefit outweighs risk compared to control

a. The model without power prior b. The model with power prior

Benefit-risk Assessment

Simulation study: Results

Scenario 1: Treatment benefit does not outweigh risk compared to control

Scenario 2: Treatment benefit outweighs risk compared to control

Benefit-risk Assessment

Simulation study: Model Fit

Benefit-risk Assessment

Benefit-risk Assessment

Future work in BR assessment

- Frequentist approaches:
- Bootstrap approach
- General linear mixed model (GLMM) approach
- Other Bayesian approaches:
- Normal priors
- Dirichlet process

Benefit-risk Assessment

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.

Benefit-risk Assessment

Bootstrap Approach-Results

Benefit-risk Assessment

General linear mixed model (GLMM) approach

- Within each arm (T or C), the ithsubject 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 γkis the longitudinal effect at kth visit, with

and, .

Benefit-risk Assessment

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.

Benefit-risk Assessment

General linear mixed model approach-Results

Benefit-risk Assessment

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 thatα0i |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

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.

Benefit-risk Assessment

Selected References

- Gelber RD, Gelman RS, Goldhirsch A. A quality-of-life oriented endpoint for comparing treatments. Biometrics. 1989;45:781-795
- Payne JT, Loken MK. A survey of the benefits and risks in the practices of radiology. CRC Crit Rev ClinRadiolNucl Med. 1975; 6:425-475
- Holden WL, Juhaeri J, Dai W. “Benefit-Risk Analysis: A Proposal Using Quantitative Methods,” Pharmacoepidemiology and Drug Safety. 2003; 12, 611–616. 154
- 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; 10:1349-1359.
- Norton, JD. A Longitudinal Model and Graphic for Benefit-risk Analysis, with Case Study. Drug Information Journal. 2011; 45: 741-747.
- Ibrahim, JG, Chen, MH. Power Prior Distributions for Regression Models. Statistical Science. 2000; 15: 46-60.

Benefit-risk Assessment

Benefit-risk Assessment

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