Luca Pozzi JSM 2011

A BayesianAdaptive Dose

SelectionProcedure withOverdispersed

CountEndpoint

Luca Pozzi

Introduction

BayesianModelAveragingDose-Response

Framework

Application

Study LayoutDecision Rules

Computations

Simulations

ResultsConclusions

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A Bayesian Adaptive Dose SelectionProcedure with Overdispersed Count

Endpoint

Luca Pozzi

University of California, Berkeley

[email protected]

August 1st, 2011 - Joint Statistical Meeting, Miami Beach, FL



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Luca Pozzi

Introduction


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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):{

Y |λ ∼ Pois(λ)λ|(α, β) ∼ Gamma(α, β)

Dose-Response Relationship: Sigmoidal relationship

e.g. EMAX -model: E(d) = E0

(1 − EMAX d

ED50+d

).

Strong prior information available for Placebo (E0) andhighest dose (EMAX );



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Modeling Dose-Response Relationship

1st challenge: Modeling

Too few doses to adopt Parametric Dose-Response model.(Adaptive design will start with only one lower dose)

Strategy: Semiparametric Specification

The mode of action of the drug and Ph.III outcomes suggestthat a monotonicity constraint holds for the dose-responserelationship:

Mm ={µj ≡ E[Yij] : E0 = µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 = EMAX

}



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Modeling Approach

Bayesian Model Averaging: Ingredients

1 A set of mutually exclusive modelsM = {M1, ...,MM}.To each model corresponds a probability distributionf(y |θ(m),Mm);

2 One set of priors g(θ(m)|Mm) on θ(m) for eachMm;

3 A vector of prior model probabilitiesπ = (π1, ..., πM), πm = P{Mm}, (e.g. πm = 1

M ), ∀m = 1, ...,M.

We have then

P{success|y} =M∑

m=1

P{success|Mm, y}P{Mm |y}



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Bayesian model

Gamma-Poisson mixture

Yij |λij ∼ dpois(λij) (i-th patient-j-th dose group);λij |αj , β ∼ dgamma(αj , β)

So Yij marginal distribution is a dnegbin(αj , β)

Priors

log(α1) ∼ N(µα, σ2α);

α|m ∼ fm � λd(m)

log(β) ∼ N(0, σ2β)



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Luca Pozzi

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Monotonicity Constraints

We introduce the jump variables

δk = log(αk ) − log(αk−1) ≥ 0

δk , 0 iff αk > αk+1

and we put a truncated normal prior on

δsum =4∑1

δk = log(α1) − log(α5) ∼ TN(µsum, σ2sum)

being TN a normal distribution folded around its mean:formally if Z ∼ N(0, 1) then X ∼ TN(ν, τ2)⇐⇒ X = ν + τ|Z |

α1 ≥︸︷︷︸δ1,m

α2 ≥︸︷︷︸δ2,m

α3 ≥︸︷︷︸δ3,m

α4 ≥︸︷︷︸δ4,m

α5



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Jump×Model Matrix

∆ =

δ1,1 0 δ1,3 δ1,4 0 0 0 0 0 δ1,3

δ2,1 δ2,2 δ2,3 δ2,4 0 δ2,6 0 δ2,8 0 0δ3,1 δ3,2 δ3,3 0 δ3,5 δ3,6 0 0 δ3,9 0δ4,1 δ4,2 0 0 δ4,5 0 δ4,7 0 0 0

e.g. M5: µ1 = µ2 = µ3 > µ4 > µ5

α1 = α2 = α3 >︸︷︷︸δ3,5

α4 >︸︷︷︸δ4,5

α5



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Criteria

Futility-Success

Exclusion criterion P{µd/µ1 ≥ 0.7|data} ≥ 50%, i.e. Dose d is notsuperior to Placebo.

Efficacy criterion is the intersection of the following events:

(i) the dose is far enough from Dose 1

P{µd/µ1 < 1|data} ≥ 95%

(ii) the dose is either at least 50% betterthan Dose 1, or at most 20% worsethan Dose 5.

max{P{µd/µ1 ≤ 0.5|data}P{µd/µ5 ≤ 1.2|data}

}≥ 50%



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Interim decision

At Interim

• if Dose 4 meets Exclusion criterion stop for futility: nodose lower than Dose 5 is effective;

• if Dose 2 meets Efficacy criteria stop for success: Dose 2is the LED;

• otherwise, for each not futile Dose d calculate thePredictive Probability of Success (PPS) and allocate tothe lowest dose for which

P{{(i)|Ad ,Y∗,Y } ∩ {(ii)|Ad ,Y∗,Y } > 50%|Y } ≥ t (1)

with Ad = {allocate to Dose d}.



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Introduction


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Decision Tree

Begin Trial

A3

A2

�LED = d2

A4

�LED = d4

�LED = d3

�LED = d3

�LED = d2

�LED = d2

�LED = d4

�LED = d4

�LED = d3

�LED�

�LED�

�LED�

�LED�

�LED = d2



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Luca Pozzi

Introduction


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Performing Predictive Probability Calculations

2nd challenge: Computational

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)∗



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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



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Luca Pozzi

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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)



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Luca Pozzi

Introduction


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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).



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Luca Pozzi

Introduction


Framework

Application


Computations

Simulations

ResultsConclusions

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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.



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Luca Pozzi

Introduction


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Operation Characteristics (Adaptive Design)

Moderate scenario

Let us consider P{ “right” dose} in the Moderate Scenario:

1:0:1:0:1 1:1:1:1:1 1:1:2:1:10.4 0.5 0.6 0.4 0.5 0.6 0.4 0.5 0.6

1/2 0.818 0.878 0.996 0.774 0.818 0.768 0.730 0.794 0.8481/3 0.860 0.860 0.996 0.818 0.858 0.814 0.734 0.780 0.864

1/2 0.670 0.762 0.894 0.658 0.732 0.722 0.670 0.724 0.7301/3 0.516 0.772 0.896 0.702 0.750 0.748 0.688 0.746 0.744

Performances of not-adaptive design

Optimistic Moderate Flat(a) Flat(b) PessimisticP{right dose} 0.982 0.84 0.26675 0.3924688 0.94P{Success} 1.000 1.00 0.36775 0.6775810 0.06



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Luca Pozzi

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Operation Characteristics: Cheaper Solution

Expected number of patients (total sample size = 250)

1:0:1:0:1 1:1:1:1:1 1:1:2:1:150%-50% 30%-70% 50%-50% 30%-70% 50%-50% 30%-70%

Optimistic 250 250 137.75 129.67 144.5 149.67Moderate 250 250 232.25 236.67 237.75 238.67

Flat (a) 191.88 178 180.5 151.17 168.12 151.5Flat (b) 231.88 216.17 216.5 191.33 216.5 195.33

Pessimistic 250 250 183 177.33 188.67 188.33



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Luca Pozzi

Introduction


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Operation Characteristics: Effect of Thresholds

2 3

Optimistic 010−30 (t=0.4)

PPS LED

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2 3


PPS LED

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2 3


PPS LED

0.0

0.2

0.4

0.6

0.8

2 3 5


Posterior LED

0.0

0.1

0.2

0.3

0.4

0.5

2 3 5


Posterior LED

0.0

0.1

0.2

0.3

0.4

0.5

2 3 5


Posterior LED

0.0

0.1

0.2

0.3

0.4

0.5



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Luca Pozzi

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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



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Luca Pozzi

Introduction


Framework

Application


Computations

Simulations

ResultsConclusions

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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.



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Luca Pozzi

Introduction


Framework

Application


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)



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Luca Pozzi

Introduction


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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”.

Luca Pozzi JSM 2011

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