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A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
A Bayesian Adaptive Dose SelectionProcedure with Overdispersed Count
August 1st, 2011 - Joint Statistical Meeting, Miami Beach, FL
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Motivating Example: Problem Setting
Objective: Lowest Effective Dose (LED), i.e. dose whose efficacyis at least 50% better than Placebo (Dose 1) andat most 20% worse than the highest dose (Dose 5);
Design of the Study: Start with initial allocation 1:a:b:c:1 then atinterim stop or select the most promising dose d for asecond phase with only Placebo, Dose d and Dose 5;
Endpoint: Overdispersed count data Y modeled by the negativebinomial distribution (gamma-poisson mixture):{
Not feasible to use WinBUGS for Predictive calculations
Stategy: Importance Sampling
Sample from the posterior sample using weighted resampling:{(α, β)(1), ..., (α, β)(N)
}→ (α, β)∗
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Algorithm: Predictive Resample
1 Sample (α, β)(1), ..., (α, β)(k), ..., (α, β)(N);2 Select Dose d;3 for l = 1, ..., L draw (α, β)(l) from the posterior sample at
interim of size N;4 simulate one dataset Y∗(l)
d |(α, β)(l),Ad ;
SIR
5 compute p(Y (l)
d |(α, β)(k)), k = 1, ...,N;
6 compute wk =l(θk ;Y∗)∑j l(θj ;Y∗)
=p(Y∗(l)
d |(α,β)(k))∑j p(Y∗(l)
d |(α,β)(j));
7 compute by resampling PP(l)d [criterion] for each criteria;
In the end
PPd = mean
(1{ ⋂{criteria}
{PP(l)d [criterion] > c}
})L
l=1
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Posterior Probability of Success
Instead of the above predictive criterion we could require a dose tosatisfy an upper bound on the posterior power. By an argument ofconditional probability we can show it equivalent to a smoothedversion of the predictive criterion:
P{θ ∈ ΘE |Aj ,Y } = EY∗
[P
{θ ∈ ΘE |Aj ,Y ∗,Y
}|Aj ,Y
](2)
beingPPS = PY∗ {P{θ ∈ ΘE |Y ∗,Y ,Aj} ≥ c |Aj ,Y }
Markov inequality gives us the following:
Posterior Lower Bound
PPS ≤EY∗
[P
{θ ∈ ΘE |Aj ,Y∗,Y
}|Aj ,Y
]c
=P{θ ∈ ΘE |Aj ,Y }
c(3)
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Dose-Response Relationships
Optimistic Scenario
dose
α
0.00.2
0.40.6
0.81.0
d1 d2 d3 d4 d5
Flat a
dose
α
0.00.2
0.40.6
0.81.0
d1 d2 d3 d4 d5
Flat b
dose
α
0.00.2
0.40.6
0.81.0
d1 d2 d3 d4 d5
Pessimistic Scenario
dose
α
0.00.2
0.40.6
0.81.0
d1 d2 d3 d4 d5
Moderate Scenario
dose
α
0.00.2
0.40.6
0.81.0
d1 d2 d3 d4 d5
Borderline Moderate Scenario
dose
α
0.00.2
0.40.6
0.81.0
d1 d2 d3 d4 d5
red represents the real LED (“right” dose).
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Simulation Setup
Initial Allocation: assuming we start with 1 : a : b : c : 1:• a = 0, b = 1, c = 0, i.e. 1:0:1:0:1;• 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.
Predictive Probability Threshold: t =
0.40.50.6
Number of Patients: split the 250 patients between the firstand the second phase:• 1/3 at interim and 2/3 for the next phase;• half at interim and half for the next phase.
Size: 500 simulations with 500 simulated studies forprediction and N = 104 for the resampling.
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Operation Characteristics (Adaptive Design)
Moderate scenario
Let us consider P{ “right” dose} in the Moderate Scenario:
Operation Characteristics: Posterior vs. Predictive
2 3
Moderate 111−50 (t=0.4)
PPS LED
0.0
0.2
0.4
0.6
2 3
Moderate 111−50 (t=0.5)
PPS LED
0.0
0.2
0.4
0.6
0.8
2 3 4
Moderate 111−50 (t=0.6)
PPS LED
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 3 5
Moderate 111−50 (t=0.4)
Posterior LED
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 3 5
Moderate 111−50 (t=0.5)
Posterior LED
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
2 3 4 5
Moderate 111−50 (t=0.6)
Posterior LED
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Summarizing
1 The procedure succeeds in detecting the properties ofdifferent Scenarios.
2 The Adaptive Design, when using an appropriatethreshold, is more efficient than the non-adaptive one interms of number of patients and not inferior in terms ofsensitivity and specificity.
3 The BMA allows for correction of suboptimal interimdecisions about the allocation.
4 Increasing the threshold we require the dose to have ahigher margin of superiority (0.6 too strict).
5 The 1/3 - 2/3 proportion and the 1:0:1:0:1 allocation aredefinitely less efficient than the other configurations.
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Some References
1 D.Ohlssen, A.Racine, A Flexible Bayesian Approach forModeling Monotonic Dose-Response Relationships inClinical Trials with Applications in Drug Development,Computational Statistics and Data Analysis,(UnderRevision);
2 A.F.M Smith, A.E. Gelfand, Bayesian Statistics withoutTears, The American Statistician, (1992);
3 J.A.Hoeting, D.Madigan, A.E.Raftery , C.T.Volinsky,Bayesian Model Averaging: a Tutorial (with Discussion).Statistical Science, (1999);
4 A.Doucet, A.M.Johansen et al., A Tutorial on ParticleFiltering and Smoothing: Fifteen Years Later. Tech.Report U.B.C., (2008)
A BayesianAdaptive Dose
SelectionProcedure withOverdispersed
CountEndpoint
Luca Pozzi
Introduction
BayesianModelAveragingDose-Response
Framework
Application
Study LayoutDecision Rules
Computations
Simulations
ResultsConclusions
References
Thanks
Acknowledgements
Thank you for your attention!!!
Authors
Luca Pozzi, U.C. BerkeleyAmy Racine, NovartisHeinz Schmidli, NovartisMauro Gasparini, Politecnico di Torino
Special Thanks to
David Ohlssen, NovartisJouni Kerman, Novartis
Funding
American Statistical Association, San Francisco-Bay Area ChapterTravel Award.MIUR (Italian Ministry for University and Research), PRIN 2007prot. 2007AYHZWC ”Statistical methods for learning in clinicalresearch”.