This document summarizes a Bayesian adaptive dose selection procedure with overdispersed count endpoint. The procedure uses Bayesian model averaging to model dose-response relationships with a monotonicity constraint. It evaluates doses at an interim analysis based on exclusion and efficacy criteria calculated using predictive probabilities from importance sampling. Simulations examine the procedure's performance under different dose-response scenarios and initial allocations.

1. A Bayesian
Adaptive Dose
Selection
Procedure with A Bayesian Adaptive Dose Selection
Overdispersed
Count
Endpoint
Procedure with Overdispersed Count
Luca Pozzi Endpoint
Introduction
Bayesian
Model
Averaging
Luca Pozzi
Dose-Response
Framework
Application
University of California, Berkeley
Study Layout
Decision Rules
Computations
Simulations
Results
Conclusions
p.luc@stat.berkeley.edu
References
Thanks
August 1st, 2011 - Joint Statistical Meeting, Miami Beach, FL

2. Motivating Example: Problem Setting
A Bayesian
Adaptive Dose
Selection Objective: Lowest Effective Dose (LED), i.e. dose whose efficacy
Procedure with
Overdispersed is at least 50% better than Placebo (Dose 1) and
Count
Endpoint
at most 20% worse than the highest dose (Dose 5);
Luca Pozzi Design of the Study: Start with initial allocation 1:a:b:c:1 then at
Introduction
interim stop or select the most promising dose d for a
Bayesian
second phase with only Placebo, Dose d and Dose 5;
Model
Averaging Endpoint: Overdispersed count data Y modeled by the negative
Dose-Response
Framework
binomial distribution (gamma-poisson mixture):
Application
Study Layout Y |λ ∼ Pois(λ)
Decision Rules
Computations
λ|(α, β) ∼ Gamma(α, β)
Simulations
Results
Dose-Response Relationship: Sigmoidal relationship
EMAX d
Conclusions
e.g. EMAX -model: E (d ) = E0 1 − ED50 +d
.
References
Strong prior information available for Placebo (E0 ) and
Thanks
highest dose (EMAX );

3. Modeling Dose-Response Relationship
A Bayesian
Adaptive Dose
Selection
Procedure with 1st challenge: Modeling
Overdispersed
Count
Endpoint Too few doses to adopt Parametric Dose-Response model.
Luca Pozzi (Adaptive design will start with only one lower dose)
Introduction
Bayesian Strategy: Semiparametric Specification
Model
Averaging
Dose-Response
The mode of action of the drug and Ph.III outcomes suggest
Framework
Application
that a monotonicity constraint holds for the dose-response
Study Layout
relationship:
Decision Rules
Computations
Simulations
Results
Mm = µj ≡ E[Yij ] : E0 = µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 = EMAX
Conclusions
References
Thanks

4. Modeling Approach
A Bayesian
Adaptive Dose Bayesian Model Averaging: Ingredients
Selection
Procedure with
Overdispersed
Count 1 A set of mutually exclusive models M = {M1 , ..., MM }.
Endpoint
To each model corresponds a probability distribution
Luca Pozzi
f (y |θ(m) , Mm );
Introduction
2 One set of priors g (θ(m) |Mm ) on θ(m) for each Mm ;
Bayesian
Model
Averaging
3 A vector of prior model probabilities
Dose-Response π = (π1 , ..., πM ), πm = P{Mm }, (e.g. πm = 1
M
), ∀ m = 1, ..., M.
Framework
Application
Study Layout
Decision Rules We have then
Computations
Simulations
Results
M
P{success|y } = P{success|Mm , y }P{Mm |y }
Conclusions
References
m=1
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5. Bayesian model
A Bayesian
Adaptive Dose
Selection Gamma-Poisson mixture
Procedure with
Overdispersed
Count
Endpoint
Luca Pozzi
Yij |λij ∼ dpois(λij ) (i -th patient-j -th dose group);
λij |αj , β ∼ dgamma(αj , β)
Introduction
Bayesian
So Yij marginal distribution is a dnegbin(αj , β)
Model
Averaging
Dose-Response
Framework
Application
Priors
Study Layout
Decision Rules
Computations
Simulations
log(α1 ) ∼ N(µα , σ2 );
α
Results α|m ∼ fm λd (m)
N(0, σ2 )
Conclusions
log(β) ∼ β
References
Thanks

6. Monotonicity Constraints
A Bayesian
Adaptive Dose
We introduce the jump variables
Selection
Procedure with
Overdispersed
δk = log(αk ) − log(αk −1 ) ≥ 0
δk 0 iff αk > αk +1
Count
Endpoint
Luca Pozzi
and we put a truncated normal prior on
Introduction
4
Bayesian
Model δsum = δk = log(α1 ) − log(α5 ) ∼ T N(µsum , σ2 )
sum
Averaging
Dose-Response 1
Framework
Application
being T N a normal distribution folded around its mean:
Study Layout
Decision Rules
formally if Z ∼ N(0, 1) then X ∼ T N(ν, τ2 ) ⇐⇒ X = ν + τ|Z |
Computations
Simulations
Results
Conclusions
α1 ≥ α2 ≥ α3 ≥ α4 ≥ α5
References
δ1,m δ2,m δ3,m δ4,m
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7. Jump×Model Matrix
A Bayesian
Adaptive Dose
Selection
Procedure with
Overdispersed
 δ1,1 0 δ1,3 δ1,4 0 0 δ1,3 
Count  
Endpoint  0 0 0 
Luca Pozzi
 δ2,1 δ2,2 δ2,3 δ2,4 0 δ2,6 0 δ2,8 0

 0 


∆=

 

 δ3,1 δ3,2 δ3,3 0 δ3,5 δ3,6 0 0 δ3,9
 
0 
 

Introduction

 

δ4,1 δ4,2 0 0 δ4,5 0 δ4,7
 
Bayesian
0 0 0
Model
Averaging
Dose-Response
Framework
Application
Study Layout e.g. M5 : µ1 = µ2 = µ3 > µ4 > µ5
Decision Rules
Computations
Simulations
α1 = α2 = α3 > α4 > α5
Results δ3,5 δ4,5
Conclusions
References
Thanks

8. Criteria
A Bayesian
Adaptive Dose
Selection Futility-Success
Procedure with
Overdispersed
Count
Endpoint
Exclusion criterion P{µd /µ1 ≥ 0.7|data} ≥ 50%, i.e. Dose d is not
Luca Pozzi
superior to Placebo.
Introduction
Efficacy criterion is the intersection of the following events:
Bayesian (i) the dose is far enough from Dose 1
Model
Averaging
Dose-Response
Framework
P{µd /µ1 < 1|data} ≥ 95%
Application
Study Layout (ii) the dose is either at least 50% better
Decision Rules than Dose 1, or at most 20% worse
Computations
than Dose 5.
Simulations
Results P{µd /µ1 ≤ 0.5|data}
max ≥ 50%
P{µd /µ5 ≤ 1.2|data}
Conclusions
References
Thanks

9. Interim decision
A Bayesian
Adaptive Dose
Selection
Procedure with At Interim
Overdispersed
Count
Endpoint • if Dose 4 meets Exclusion criterion stop for futility: no
Luca Pozzi dose lower than Dose 5 is effective;
Introduction • if Dose 2 meets Efficacy criteria stop for success: Dose 2
Bayesian
Model
is the LED;
Averaging
Dose-Response
• otherwise, for each not futile Dose d calculate the
Framework
Application
Predictive Probability of Success (PPS) and allocate to
Study Layout the lowest dose for which
Decision Rules
P{{(i)|Ad , Y ∗ , Y } ∩ {(ii)|Ad , Y ∗ , Y } > 50%|Y } ≥ t
Computations
Simulations
(1)
Results
Conclusions with Ad = {allocate to Dose d }.
References
Thanks

10. Decision Tree
A Bayesian
Adaptive Dose
LED = d4
Selection
Procedure with
Overdispersed
LED = d3
Count
A2
Endpoint
LED = d3
Luca Pozzi LED
LED = d2
Introduction
LED = d2
A3
Bayesian
Model
LED = d4
Averaging Begin Trial
LED = d2
Dose-Response
LED
Framework
Application
Study Layout
LED = d2
Decision Rules
LED
Computations
LED
Simulations
A4
Results
Conclusions
LED = d3
References
LED = d4
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11. Performing Predictive Probability Calculations
A Bayesian
Adaptive Dose
Selection
Procedure with
Overdispersed
Count
Endpoint 2nd challenge: Computational
Luca Pozzi
Not feasible to use WinBUGS for Predictive calculations
Introduction
Bayesian
Model
Stategy: Importance Sampling
Averaging
Dose-Response Sample from the posterior sample using weighted resampling:
Framework
Application
Study Layout
Decision Rules
(α, β)(1) , ..., (α, β)(N ) → (α, β)∗
Computations
Simulations
Results
Conclusions
References
Thanks

12. Algorithm: Predictive Resample
Sample (α, β)(1) , ..., (α, β)(k ) , ..., (α, β)(N ) ;
A Bayesian
Adaptive Dose 1
Selection
Procedure with 2 Select Dose d;
for l = 1, ..., L draw (α, β)(l ) from the posterior sample at
Overdispersed
Count 3
Endpoint
interim of size N;
∗(l )
simulate one dataset Yd |(α, β)(l ) , Ad ;
Luca Pozzi
4
Introduction
Bayesian SIR
Model
Averaging
(l )
Dose-Response 5 compute p Yd |(α, β)(k ) , k = 1, ..., N;
Framework
Application ∗(l )
l (θk ;Y ∗ ) p (Yd |(α,β)(k ) )
Study Layout 6 compute wk = ∗ = ∗(l ) ;
Decision Rules j l (θj ;Y ) j p (Yd |(α,β)(j ) )
Computations
(l )
Simulations 7 compute by resampling PPd [criterion] for each criteria;
Results
Conclusions
In the end
L
 
References
1 (l )
{PPd [criterion] c }

 

PPd = mean

 


Thanks  
{criteria} l =1

13. Posterior Probability of Success
A Bayesian
Adaptive Dose
Selection Instead of the above predictive criterion we could require a dose to
Procedure with
Overdispersed
satisfy an upper bound on the posterior power. By an argument of
Count conditional probability we can show it equivalent to a smoothed
Endpoint
version of the predictive criterion:
Luca Pozzi
Introduction P{θ ∈ ΘE |Aj , Y } = EY ∗ P θ ∈ ΘE |Aj , Y ∗ , Y |Aj , Y (2)
Bayesian
Model
Averaging being
Dose-Response
Framework
PPS = PY ∗ {P{θ ∈ ΘE |Y ∗ , Y , Aj } ≥ c |Aj , Y }
Application
Study Layout
Markov inequality gives us the following:
Decision Rules
Computations Posterior Lower Bound
Simulations
Results EY ∗ P θ ∈ ΘE |Aj , Y ∗ , Y |Aj , Y P{θ ∈ ΘE |Aj , Y }
Conclusions PPS ≤ = (3)
c c
References
Thanks

14. Dose-Response Relationships
A Bayesian
Adaptive Dose
Selection
Procedure with
Optimistic Scenario Flat a Flat b
Overdispersed
Count
1.0
1.0
1.0
Endpoint
0.8
0.8
0.8
0.6
0.6
0.6
Luca Pozzi
α
α
α
0.4
0.4
0.4
0.2
0.2
0.2
Introduction
0.0
0.0
0.0
d1 d2 d3 d4 d5 d1 d2 d3 d4 d5 d1 d2 d3 d4 d5
Bayesian dose dose dose
Model
Pessimistic Scenario Moderate Scenario Borderline Moderate Scenario
Averaging
Dose-Response
1.0
1.0
1.0
Framework
0.8
0.8
0.8
Application
0.6
0.6
0.6
α
α
α
Study Layout
0.4
0.4
0.4
Decision Rules
0.2
0.2
0.2
Computations
0.0
0.0
0.0
d1 d2 d3 d4 d5 d1 d2 d3 d4 d5 d1 d2 d3 d4 d5
Simulations dose dose dose
Results
Conclusions
References
red represents the real LED (“right” dose).
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15. Simulation Setup
A Bayesian
Adaptive Dose
Selection
Procedure with
Initial Allocation: assuming we start with 1 : a : b : c : 1:
Overdispersed
Count • a = 0, b = 1, c = 0, i.e. 1:0:1:0:1;
Endpoint
• a = 1, b = 1, c = 1, i.e. 1:1:1:1:1;
• a = 1, b = 2, c = 1, i.e. 1:1:2:1:1.
Luca Pozzi
Introduction
 0.4


Predictive Probability Threshold: t =  0.5
Bayesian


Model 
Averaging  0.6

Dose-Response
Framework
Application Number of Patients: split the 250 patients between the first
Study Layout and the second phase:
Decision Rules
Computations • 1/3 at interim and 2/3 for the next phase;
Simulations • half at interim and half for the next phase.
Results
Conclusions Size: 500 simulations with 500 simulated studies for
References prediction and N = 104 for the resampling.
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16. Operation Characteristics (Adaptive Design)
A Bayesian
Adaptive Dose
Selection
Procedure with Moderate scenario
Overdispersed
Count
Endpoint
Let us consider P{ “right” dose} in the Moderate Scenario:
Luca Pozzi
1:0:1:0:1 1:1:1:1:1 1:1:2:1:1
0.4 0.5 0.6 0.4 0.5 0.6 0.4 0.5 0.6
Introduction 1/2 0.818 0.878 0.996 0.774 0.818 0.768 0.730 0.794 0.848
1/3 0.860 0.860 0.996 0.818 0.858 0.814 0.734 0.780 0.864
Bayesian
Model 1/2 0.670 0.762 0.894 0.658 0.732 0.722 0.670 0.724 0.730
Averaging 1/3 0.516 0.772 0.896 0.702 0.750 0.748 0.688 0.746 0.744
Dose-Response
Framework
Application
Study Layout
Decision Rules
Performances of not-adaptive design
Computations
Simulations Optimistic Moderate Flat(a) Flat(b) Pessimistic
P{right dose} 0.982 0.84 0.26675 0.3924688 0.94
Results
Conclusions
P{Success} 1.000 1.00 0.36775 0.6775810 0.06
References
Thanks

17. Operation Characteristics: Cheaper Solution
A Bayesian
Adaptive Dose
Selection
Procedure with
Overdispersed
Count
Endpoint
Luca Pozzi Expected number of patients (total sample size = 250)
Introduction
1:0:1:0:1 1:1:1:1:1 1:1:2:1:1
Bayesian 50%-50% 30%-70% 50%-50% 30%-70% 50%-50% 30%-70%
Model Optimistic 250 250 137.75 129.67 144.5 149.67
Averaging Moderate 250 250 232.25 236.67 237.75 238.67
Dose-Response
Flat (a) 191.88 178 180.5 151.17 168.12 151.5
Framework
Flat (b) 231.88 216.17 216.5 191.33 216.5 195.33
Application
Pessimistic 250 250 183 177.33 188.67 188.33
Study Layout
Decision Rules
Computations
Simulations
Results
Conclusions
References
Thanks

18. Operation Characteristics: Effect of Thresholds
A Bayesian
Adaptive Dose
Optimistic 010−30 (t=0.4) Optimistic 010−30 (t=0.5) Optimistic 010−30 (t=0.6)
Selection
0.7
Procedure with
0.7
Overdispersed
0.6
0.8
0.6
Count
0.5
0.5
Endpoint
0.6
0.4
0.4
Luca Pozzi
0.3
0.4
0.3
0.2
0.2
0.2
Introduction
0.1
0.1
0.0
0.0
0.0
Bayesian
2 3 2 3 2 3
Model
Averaging PPS LED PPS LED PPS LED
Dose-Response
Optimistic 010−30 (t=0.4) Optimistic 010−30 (t=0.5) Optimistic 010−30 (t=0.6)
Framework
Application
0.5
0.5
0.5
Study Layout
0.4
0.4
0.4
Decision Rules
Computations
0.3
0.3
0.3
Simulations
0.2
0.2
0.2
Results
0.1
0.1
0.1
Conclusions
0.0
0.0
References 0.0
2 3 5 2 3 5 2 3 5
Posterior LED Posterior LED Posterior LED
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19. Operation Characteristics: Posterior vs. Predictive
A Bayesian
Adaptive Dose
Moderate 111−50 (t=0.4) Moderate 111−50 (t=0.5) Moderate 111−50 (t=0.6)
Selection
0.8
Procedure with
0.7
Overdispersed
0.6
0.6
Count
0.6
0.5
Endpoint
0.4
0.4
0.4
Luca Pozzi
0.3
0.2
0.2
0.2
Introduction
0.1
0.0
0.0
0.0
Bayesian
2 3 2 3 2 3 4
Model
Averaging PPS LED PPS LED PPS LED
Dose-Response
Moderate 111−50 (t=0.4) Moderate 111−50 (t=0.5) Moderate 111−50 (t=0.6)
Framework
0.7
Application
0.7
0.7
0.6
0.6
0.6
Study Layout
0.5
0.5
0.5
Decision Rules
0.4
Computations
0.4
0.4
0.3
0.3
0.3
Simulations
0.2
0.2
0.2
Results
0.1
0.1
0.1
Conclusions
0.0
0.0
References 0.0
2 3 5 2 3 5 2 3 4 5
Posterior LED Posterior LED Posterior LED
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20. Summarizing
A Bayesian
Adaptive Dose
Selection
Procedure with 1 The procedure succeeds in detecting the properties of
Overdispersed
Count different Scenarios.
Endpoint
Luca Pozzi 2 The Adaptive Design, when using an appropriate
threshold, is more efficient than the non-adaptive one in
Introduction
Bayesian
terms of number of patients and not inferior in terms of
Model
Averaging
sensitivity and specificity.
Dose-Response
Framework
3 The BMA allows for correction of suboptimal interim
Application
decisions about the allocation.
Study Layout
Decision Rules 4 Increasing the threshold we require the dose to have a
higher margin of superiority (0.6 too strict).
Computations
Simulations
Results 5 The 1/3 - 2/3 proportion and the 1:0:1:0:1 allocation are
Conclusions
definitely less efficient than the other configurations.
References
Thanks

21. Some References
A Bayesian
Adaptive Dose
Selection
Procedure with 1 D.Ohlssen, A.Racine, A Flexible Bayesian Approach for
Overdispersed
Count
Modeling Monotonic Dose-Response Relationships in
Endpoint
Clinical Trials with Applications in Drug Development,
Luca Pozzi
Computational Statistics and Data Analysis,(Under
Introduction Revision);
Bayesian
Model 2 A.F.M Smith, A.E. Gelfand, Bayesian Statistics without
Averaging
Dose-Response
Tears, The American Statistician, (1992);
Framework
Application 3 J.A.Hoeting, D.Madigan, A.E.Raftery , C.T.Volinsky,
Study Layout Bayesian Model Averaging: a Tutorial (with Discussion).
Decision Rules
Computations Statistical Science, (1999);
Simulations
4 A.Doucet, A.M.Johansen et al., A Tutorial on Particle
Results
Conclusions Filtering and Smoothing: Fifteen Years Later. Tech.
References Report U.B.C., (2008)
Thanks

22. Acknowledgements
A Bayesian
Adaptive Dose
Thank you for your attention!!!
Selection
Procedure with Authors
Overdispersed
Count Luca Pozzi, U.C. Berkeley
Endpoint
Amy Racine, Novartis
Luca Pozzi
Heinz Schmidli, Novartis
Introduction Mauro Gasparini, Politecnico di Torino
Bayesian
Model
Averaging
Special Thanks to
Dose-Response
Framework David Ohlssen, Novartis
Application
Jouni Kerman, Novartis
Study Layout
Decision Rules
Computations
Funding
Simulations American Statistical Association, San Francisco-Bay Area Chapter
Results Travel Award.
Conclusions
MIUR (Italian Ministry for University and Research), PRIN 2007
References
prot. 2007AYHZWC ”Statistical methods for learning in clinical
Thanks
research”.