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Bayesian Methods for Benefit/Risk Assessment. Ram C. Tiwari Associate Director Office of Biostatistics, CDER, FDA [email protected] Disclaimer. This presentation reflects the views of the author and should not be construed to represent FDA’s views or policies. Outline. Introduction

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bayesian methods for benefit risk assessment

Bayesian Methods for Benefit/Risk Assessment

Ram C. Tiwari

Associate Director

Office of Biostatistics, CDER, FDA

[email protected]

disclaimer
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
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
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
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
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

example hydromorphone
Example: Hydromorphone

Data was provided by Jonathan Norton.

Benefit-risk Assessment

gbr scores
GBR scores

Benefit-risk Assessment

methodology br measures
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
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
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
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
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
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
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
Back to our example: Hydromorphone

Benefit-risk Assessment

illustration posterior means and 95 credible intervals for br linear measure
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
Illustration: Posterior Means and 95% Credible Intervals for BR_Indicator Measure

without power prior

with power prior

Benefit-risk Assessment

illustration results
Illustration: Results

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

Benefit-risk Assessment

slide21

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

Benefit-risk Assessment

illustration model fit
Illustration: Model Fit

Benefit-risk Assessment

simulation study
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
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
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
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
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
Simulation study: Model Fit

Benefit-risk Assessment

future work in br assessment

Future Work in BR Assessment

Benefit-risk Assessment

future work in br assessment1
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
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
Bootstrap Approach-Results

Benefit-risk Assessment

general linear mixed model glmm approach
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
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

bayesian approaches with glmm
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
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
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

slide40

Q & A

Benefit-risk Assessment

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