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Bayesian Modeling Averaging
Approach to Model a Binary Outcome
for a Dose Ranging Trial
Bob Noble
GlaxoSmithKline
Director of Statistics and Programming
Statistics Leader of Virtual Proof of Concept Unit
Post-operative nausea and vomiting (PONV) often occurs following local,
regional, or general anesthesia and is the most frequently reported patient
complaint following anesthesia.
PONV is often of greater concern to patients than is the avoidance of postoperative pain .
In addition to anxiety and discomfort, PONV can lead to complications such as
fluid and electrolyte imbalances, surgical wound dehiscence, aspiration of
vomitus, and/or severe pulmonary morbidity that can lead to delayed
discharge from the recovery area or unscheduled hospital admission.
British Journal Of Anaesthesia (2015) 114 (3): 423-429
Kranke P, Thompson J, Dalby P, Novikova E, Johnson B, Russ S, Noble R, Brigandi R.
Comparison of vestipitant with ondansetron for the treatment of breakthrough
postoperative nausea and vomiting after failed prophylaxis with ondansetron.
The primary efficacy endpoint is number of subjects achieving
Complete Response after receiving study drug to treat
breakthrough PONV.
Complete Response is defined as no emesis and no further
rescue medication through 24 hours or discharge from the
hospital/clinic, whichever is sooner.
Doses of vestipitant (IP) 6mg, 12mg, 18mg, 24mg, 36mg.
Positive control 4mg ondansetron
A concern of the team in characterizing the dose response
over a wide range was that high doses of vestipitant may
cause nausea and vomiting. Solution: A non-monotonic dose
response model.
Information available for the efficacy of ondansetron lead the
team to a Beta(20,20) prior for the positive control arm. The
Bayes estimate using a Beta(20,20) prior will have smaller MSE
than the MLE for all n ≤ 20 on the interval 0.35 < p < 0.65.
Piecewise logistic regression model
1
pi 
1  exp   0  1dosei   2 (dosei  k ) I k  dosei

dose  k
pi 
1
1  exp   0  1dosei 
dose  k
pi 
1
1  exp   0   2 k   1   2 dosei ) 

Illustration
pˆ i 
1
1  exp  1.4  0.01dosei  0.07(dosei  20) I 20dosei

k=20

Model Version 1
Likelihood of data
ind
Yi ~ Bin (ni , pi ); i  0,1,2,3,4,5
Priors
p0 ~ Beta (20,20)
1
pi 
; i  1,...,5


1  exp   0  1dosei   2 (dosei  k ) I k  dose
 0 ~ N (0,100)
1 ~ N (0,100)
 2 ~ N (0,100)
k ~ U (6,36)
Model Version 2 (BMA framework)
M 1 : pi 
1
1  exp   0  1dosei   2 (dosei  7) I 7  dosei

M 2 : pi 
1
1  exp   0  1dosei   2 (dosei  8) I 8 dosei




1
1  exp   0  1dosei   2 (dosei  34) I 34 dosei

1
M 29 : pi 
1  exp   0  1dosei   2 (dosei  35) I 35 dosei

M 28 : pi 


1
M 30 : pi 
1  exp   0  1dosei 
M 31 : pi 
1
1  exp   0 
Posterior Probability of Model “i”
P( M i | data)  exp( 0.5 BIC i )
Posterior distribution of parameter p|dose
31
 ( p | dose)    ( p | dose, M i , data) P( M i | data)
i 1
Posterior Distribution of Logistic Regression Parameters
 (  i | M i , data) ~ MVN 3 ( ˆ i , ˆ i ); i  1,29
 (  30 | dose, M 30 , data) ~ MVN 2 ( ˆ 30 , ˆ 30 )
 (  31 | dose, M 31, data) ~ N ( ˆ31, ˆ 312 )
The posterior distribution of the response rate at a given dose for a given
model is just the sampled posterior values from the posterior of the
regression parameters plugged into the model.
The BMA estimate is the weighted (by the posterior probability of each
model) average of the individual model estimates.
Posterior Distributions by Dose
Study stopped early for futility (i.e. probability of success at
full enrollment was too low).
Actual n
Apparent n
Control
19
59
6mg
24
35.1
12mg
23
79.2
18mg
22
63.4
24mg
20
64.8
36mg
22
28.7
Total
130
330.2
Resulted in a project savings of more than $7MM and 9-12 months