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Luca Pozzi 5thBCC 2012
1. A Bayesian Adaptive Dose Selection Procedure with
Semi-Parametric Dose-Response Modeling
Luca Pozzi
University of California, Berkeley
p.luc@stat.berkeley.edu
January 24, 2012 - 5th Annual Bayesian Biostatistics Conference, Houston, TX
2. Predictive Probability in Clinical Trials
Berry et al. 2010, Chapter 4
“The Predictive Probability approach looks into the future based on
the current observed data to project whether a positive conclusion
at the end of the study is likely or not, and then makes a sensible
decision at the present time accordingly.”
3. Predictive Probability of Success (PPS)
For the ease of notation let us define:
Y = {Past Data} i.e. interim data;
Y ∗ = {Future Data} i.e. post interim data.
In our setting, as in Berry et al. (2010)
PPS = P{Success|Y } = P{Y ∗ ∈ YS |Y } = p (Y ∗ |Y )dY ∗ (1)
YS
being YS defined as
{Success} = YS = {Y ∗ : P{θ ∈ ΘE |Y ∗ , Y } > t }
for some efficacy domain ΘE and some threshold t, the predictive quantity (??)
then becomes
PPS = 1{P{θ ∈ ΘE |Y ∗ , Y } > t }p (Y ∗ |Y )dY ∗ (2)
where P{θ ∈ ΘE |Y ∗ , Y } = ΘE
p (θ|Y ∗ , Y )dθ.
4. Motivating Example: Problem Setting
Objective: Lowest Effective Dose (LED), i.e. dose whose efficacy is
at least 50% better than placebo (Dose 1) and
at most 20% worse than the highest dose (Dose 5);
Design of the Study: Start with initial allocation 1:a:b:c:1 then at interim
stop or select the most promising dose d for a second
phase with only placebo, Dose d and Dose 5;
Endpoint: Overdispersed count data Y modeled by the negative
binomial distribution (Gamma-Poisson mixture):
Y |λ ∼ Pois(λ)
λ|(α, β) ∼ Gamma(α, β)
Pharmacodynamic: Sigmoidal relationship
EMAX ·D
e.g. EMAX -model: E (D ) = E0 1 − D50% +D
.
Strong prior information available for placebo (E0 ) and
highest dose (EMAX );
5. 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 suggest that a
monotonicity constraint holds for the dose-response relationship:
Mm = µj ≡ E[Yij ] : E0 = µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 = EMAX
6. Modeling Approach
Bayesian Model Averaging: Ingredients
1 A set of mutually exclusive models M = {M1 , ..., MM }.
To each model corresponds a probability distribution
f (y |θ(m) , Mm );
2 One set of priors g (θ(m) |Mm ) on θ(m) for each Mm ;
3 A vector of prior model probabilities π = (π1 , ..., πM ), πm = P{Mm },
(e.g. πm = M ), ∀ m = 1, ..., M.
1
We have then:
M
P{success|y } = P{success|Mm , y }P{Mm |y }
m=1
7. Bayesian model
Gamma-Poisson:
for the i-th patient and the j-th dose group
Yij |λij ∼ dpois(λij )
λij |αj , β ∼ dgamma(αj , β)
So Yij marginal distribution is a dnegbin(αj , β)
Priors:
log(α1 ) ∼ N(µα , σ2 );
α
α|m ∼ fm λd (m)
log(β) ∼ N(0, σ2 )
β
8. 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
4
δsum = δk = log(α1 ) − log(α5 ) ∼ T N(µsum , σ2 )
sum
1
being T N a normal distribution folded around its mean: formally if
Z ∼ N(0, 1) then X ∼ T N(ν, τ2 ) ⇐⇒ X = ν + τ|Z |
α1 ≥ α2 ≥ α3 ≥ α4 ≥ α5
δ1,m δ2,m δ3,m δ4,m
10. Criteria
Futility-Success
Exclusion Criterion P{µd /µ1 ≥ 0.7|data} ≥ 50%, i.e. Dose d is not
superior 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% better than
Dose 1, or at most 20% worse than Dose 5.
P{µd /µ1 ≤ 0.5|data}
max ≥ 50%
P{µd /µ5 ≤ 1.2|data}
11. Interim Decision
At Interim
• if Dose 4 meets Exclusion Criterion stop for futility: no dose
lower than Dose 5 is effective;
• if Dose 2 meets Efficacy Criteria stop for success: Dose 2 is
the LED;
• otherwise, for each not futile Dose d calculate the Predictive
Probability of Success (PPS) and allocate to the lowest
dose for which
P (i) Ad , Y , Y ∗ ∩ (ii) Ad , Y , Y ∗ > 50% Y ≥ t (3)
with Ad = {allocate to Dose d }.
12. Decision Tree
LED = d3
LED = d4
A4
LED = d2
LED
LED = d4
LED = d5
LED
Start the Trial Decision at Interim A3
LED = d2
LED = d2
LED = d3
LED = d2
LED
A2
LED = d3
LED = d4
13. 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)∗
14. 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;
∗(l )
4 Simulate one dataset Yd |(α, β)(l ) , Ad ;
SIR
(l )
5 Compute p Yd |(α, β)(k ) , k = 1, ..., N;
∗(l )
l (θk ;Y ∗ ) p (Yd |(α,β)(k ) )
6 Compute wk = ∗ = ∗(l ) ;
j l (θj ;Y ) j p (Yd |(α,β)(j ) )
(l )
7 Compute by resampling PPd [criterion] for each criteria;
In the end:
L
1 (l )
{PPd [criterion] c }
PPd = mean
{criteria} l =1
16. 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.
0.4
Predictive Probability Threshold: t = 0.5
0.6
Number of Patients: split the 250 patients between the first and
the second phase:
• 30% at interim and 70% for the next phase;
• half at interim and half for the next phase.
Size: 500 simulations with 500 simulated studies for
prediction and N = 104 for the resampling.
20. Summary
1 The procedure succeeds in detecting the properties of
different Scenarios.
2 The Adaptive Design, when using an appropriate threshold, is
more efficient than the Non-Adaptive one in terms of number
of patients and not inferior in terms of sensitivity and
specificity.
3 The BMA allows for correction of suboptimal interim
decisions about the allocation.
4 Increasing the threshold we require the dose to have a
higher margin of superiority (0.6 too strict).
5 The 30%-70% proportion and the 1:0:1:0:1 allocation are
definitely less efficient than the other configurations.
21. Some References
1 ¨
S. Berry, B. Carlin, J. Lee, and P. Muller, Bayesian Adaptive
Methods for Clinical Trials, CRC Press, (2010);
2 D.Ohlssen, A.Racine, A Flexible Bayesian Approach for
Modeling Monotonic Dose-Response Relationships in Clinical
Trials with Applications in Drug Development, Computational
Statistics and Data Analysis,(Under Revision);
3 A.F.M Smith, A.E. Gelfand, Bayesian Statistics without Tears,
The American Statistician, (1992);
4 J.A.Hoeting, D.Madigan, A.E.Raftery , C.T.Volinsky, Bayesian
Model Averaging: a Tutorial (with Discussion). Statistical
Science, (1999);
22. Acknowledgements
Thank You for Your Attention!!!
Joint work with:
Heinz Schmidli, Novartis AG, Statistical Methodology
Mauro Gasparini, Politecnico di Torino, Department of Mathematics
Amy Racine, Novartis AG, Modeling Simulations
Special thanks to:
David Ohlssen, Novartis Pharma, Statistical Methodology
