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Bayesian Methods for Benefit/Risk Assessment

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Bayesian Methods for Benefit/Risk Assessment

Ram C. Tiwari

Associate Director

Office of Biostatistics, CDER, FDA

Ram.Tiwari@fda.hhs.gov

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

Benefit-risk Assessment

Benefit-risk Assessment

- 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

- 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

- 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

- 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

Data was provided by Jonathan Norton.

Benefit-risk Assessment

Benefit-risk Assessment

- 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

- 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

- 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

- 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

- 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

- 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

- 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

Benefit-risk Assessment

without power prior

with power prior

Benefit-risk Assessment

without power prior

with power prior

Benefit-risk Assessment

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

Benefit-risk Assessment

- 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

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

Scenario 2: Treatment benefit outweighs risk compared to control

Benefit-risk Assessment

Treatment benefit does not outweigh risk compared to control

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

Benefit-risk Assessment

Treatment benefit outweighs risk compared to control

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

Benefit-risk Assessment

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

Scenario 2: Treatment benefit outweighs risk compared to control

Benefit-risk Assessment

Benefit-risk Assessment

Future Work in BR Assessment

Benefit-risk Assessment

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

Benefit-risk Assessment

- 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

Benefit-risk Assessment

- 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

- 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

Benefit-risk Assessment

- (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

- 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

Benefit-risk Assessment

- 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

Q & A

Benefit-risk Assessment