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 [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

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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


Bayesian methods for benefit risk assessment

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


Bayesian methods for benefit risk assessment

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


General linear mixed model approach results

General linear mixed model approach-Results

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


Bayesian methods for benefit risk assessment

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


Bayesian methods for benefit risk assessment

Q & A

Benefit-risk Assessment


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