OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Uncertainty modeling in dose response using
Non-Parametric Bayes
Thomas A. Mazzuchi 1 ,
Lidia M. Burzala
2
October 22–23, 2007
Washington DC
1
2
George Washington University, Washington DC
Delft University of Technology, The Netherlands
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
PART I
Nonparametric Bayesian approach
PART II
Analysis of the data sets
CONCLUSIONS
REFERENCES
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
BAYES RULE
I
posterior ∝ prior × likelihood
I
Combination of two sources of information:
prior information + bioassay data
knowledge from experts,
previous experiment
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
LIKELIHOOD
I
si – number of events at xi , si ∼ Binomial(ni , pi )
where ni – number of subjects, pi – probability of response
I
The joint likelihood is a product:
L(s|p) =
M Y
ni
i=1
si
pisi (1 − pi )ni −si ,
where p = (p1 , . . . , pM ), M – number of dose levels
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
PRIOR
I
P = (P1 , . . . , PM ) has an ordered Dirichlet distribution with density
at p = (p1 , . . . , pM ):
M+1
P
αξi M+1
Γ
Y
(pi − pi−1 )αξi −1 ,
π(p1 , . . . , pM ) = M+1i=1
Q
Γ(αξi ) i=1
i=1
where
I
0 ≤ p1 ≤ · · · ≤ pM ≤ 1,
ξi = P0 (xi ) − P0 (xi−1 ), for i = 1, . . . , M + 1,
P0 (x0 ) ≡ 0, P0 (xM+1 ) ≡ 1
α – precision parameter and P0 – base distribution such that
E (P(xi )) = P0 (xi )
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
POSTERIOR
I
f (p1 , . . . , pM |s) = C
(M
Y
i=1
pisi (1
ni −si
− pi )
) M+1
Y
(pi − pi−1 )αξi −1
i=1
I
It becomes increasingly intractable as M increases, especially for
obtaining the marginals
I
Observe that:
I if α → ∞ then posterior approaches prior
I if for all i = 1, . . . , M + 1, ξi = P0 (xi ) − P0 (xi−1 ) = const
and α = ξ1i , then posterior approaches MLE
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
GIBBS SAMPLER
I
Sampling from the conditional distributions:
I
Zi |si , ni , p
and
Zi (1) = (Zi,1 , . . . , Zi,i ) ∼ Mult si , λ(1)
pi
λ(2)
Zi (2) = (Zi,i+1 , . . . , Zi,M+1 ) ∼ Mult ni − si , 1−p
,
i
where λ(1) = (p1 − p0 , . . . , pi − pi−1 ), λ(2) = (pi+1 − pi , . . . , 1 − pM )
I
p|s, Z
pi ∼ pi−1 + (pi+1 − pi−1 )Beta(δi , δi+1 ),
where δi = α(P0 (xi ) − P0 (xi−1 )) +
M
P
Zij
i=1
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
GIBBS SAMPLER PROCEDURE
I
(0)
(0)
Specify the initial values p (0) = p1 , . . . , pM
I
For each i = 1,. . . , M sample
from
(0)
(0)
Zi (1) ∼ Mult si , λ(1)
with λ(1) = (p1(0) , p2(0) − p1(0) , . . . , pi(0) − pi−1
),
(0)
pi
(0)
(0)
(0)
(0)
− pi , . . . , 1 − pM )
Zi (2) ∼ Mult ni − si , λ(2)(0) with λ(2) = (pi+1
1−pi
I
For each i = 1,. . . , M sample
from
(1)
(1)
(0)
(1)
pi ∼ pi−1 + pi+1 − pi−1 Beta(δi , δi+1 )
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
GIBBS SAMPLER PROCEDURE
(1)
(1)
I p (1) = p , . . . , p
1
M
I
For each i = 1,. . . , M sample
from
(0)
(0)
Zi (1) ∼ Mult si , λ(1)
with λ(1) = (p1(0) , p2(0) − p1(0) , . . . , pi(0) − pi−1
),
(0)
pi
(0)
(0)
(0)
(0)
− pi , . . . , 1 − pM )
Zi (2) ∼ Mult ni − si , λ(2)(0) with λ(2) = (pi+1
1−pi
I
For each i = 1,. . . , M sample
from
(1)
(1)
(0)
(1)
pi ∼ pi−1 + pi+1 − pi−1 Beta(δi , δi+1 )
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
GIBBS SAMPLER PROCEDURE
(1)
(1)
I p (1) = p , . . . , p
1
M
I
For each i = 1,. . . , M sample
from
(1)
(1)
Zi (1) ∼ Mult si , λ(1)
with λ(1) = (p1(1) , p2(1) − p1(1) , . . . , pi(1) − pi−1
),
(1)
pi
(1)
(1)
(1)
(1)
− pi , . . . , 1 − pM )
Zi (2) ∼ Mult ni − si , λ(2)(1) with λ(2) = (pi+1
1−pi
I
For each i = 1,. . . , M sample
from
(1)
(1)
(0)
(1)
pi ∼ pi−1 + pi+1 − pi−1 Beta(δi , δi+1 )
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Nonparametric Bayesian approach
Nonparametric Bayesian
GIBBS SAMPLER PROCEDURE
(1)
(1)
I p (1) = p , . . . , p
1
M
I
For each i = 1,. . . , M sample
from
(1)
(1)
Zi (1) ∼ Mult si , λ(1)
with λ(1) = (p1(1) , p2(1) − p1(1) , . . . , pi(1) − pi−1
),
(1)
pi
(1)
(1)
(1)
(1)
− pi , . . . , 1 − pM )
Zi (2) ∼ Mult ni − si , λ(2)(1) with λ(2) = (pi+1
1−pi
I
For each i = 1,. . . , M sample
from
(2)
(2)
(1)
(2)
pi ∼ pi−1 + pi+1 − pi−1 Beta(δi , δi+1 )
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
UNCERTAINTY MODELING IN DOSE RESPONSE
I
PURPOSE – analyze model’s suitability to recover the observational
probability
I
QUESTION – what is the uncertainty on the number of
experimental subjects responding in each experiment?
I
NOTATION
I observational uncertainty – number of responses ∼ Bin(ni , si )
ni
I isotonic uncertainty – PAV algorithm; in case of violation of
monotonicity
I the Nonparameteric Bayesian CDF on the number of responses
generated from model
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
UNCERTAINTY MODELING IN DOSE RESPONSE
How to choose precision parameter and base distribution?
I
α=M +1
I
P0 (xi ) =
i
M+1
for each i = 1, . . . , M, then ξi = const =
1
M+1
Observe that for that parameter and base distribution posterior
approaches MLE.
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET I
(from BMD technical document)
POSTERIOR MEAN
I
posterior from model with α = 4
approaches MLE (green line)
I
posterior from model with α = 40
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET I
(from BMD technical document)
UNCERAINTY ANALYSIS
Model with α = 4
Model with α = 40
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – male
POSTERIOR MEAN
I
violation of monotonicity
I
posterior from model with α = 5
approaches MLE (green line)
I
posterior from model with α = 50
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – male
UNCERTAINTY ANALYSIS
Dose level 0
Dose level 15
Dose level 1.2
Dose level 82
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – female
POSTERIOR MEAN
I
violation of monotonicity
I
posterior from model with α = 5
approaches MLE (green line)
I
posterior from model with α = 50
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – female
UNCERTAINTY ANALYSIS
Dose level 0
Dose level 21
Dose level 1.8
Dose level 109
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – males and females
POSTERIOR MEAN
I
violation of monotonicity at almost
all dose levels
I
posterior from model with α = 8
approaches MLE (green line)
I
posterior from model with α = 80
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – males and females
UNCERTAINTY ANALYSIS
Dose level 0
Dose level 1.8
Dose level 1.2
Dose level 15
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET II
(Frambozadrine) – males and females
UNCERTAINTY ANALYSIS
Dose level 21
Dose level 82
Dose level 109
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET III
(Nectorine)
POSTERIOR MEAN
I
pooling data avoids violation of
monotonicity
I
posterior from model with α = 5
approaches MLE (green line)
I
posterior from model with α = 50
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET III
(Nectorine)
UNCERTAINTY ANALYSIS
Uncertainty distributions for dose levels: 10, 30 and 60 (starting from the left)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate) – B6C3F1 male mice
POSTERIOR MEAN
I
no violation of monotonicity
I
posterior from model with α = 4
approaches MLE (green line)
I
posterior from model with α = 40
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate)
UNCERTAINTY ANALYSIS
Uncertainty distributions for dose levels: 0, 27 and 41 (mg/kg per day)
(starting from the left)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate) – Crj:BDF1 male mice
POSTERIOR MEAN
I
violation of monotonicity
I
posterior from model with α = 5
approaches MLE (green line)
I
posterior from model with α = 50
approaches prior (black line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate) – Crj:BDF1 male mice
UNCERTAINTY ANALYSIS
Dose level 0
Dose level 14
Dose level 3.4
Dose level 36
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate) – combination of two types of mice
POSTERIOR MEAN
I
violation of monotonicity
I
posterior from model with α = 0.7
(green line) at dose 3.4, 14 and 36
approaches MLE better than
posterior from model with α = 7
(blue line)
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate) – combination of two types of mice
UNCERTAINTY ANALYSIS
Dose level 0
Dose level 14
Dose level 3.4
Dose level 27
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Analysis of the data sets
Analysis of the data
DATA SET IV
(Persimonate) – combination of two types of mice
UNCERTAINTY ANALYSIS
Dose level 36
Dose level 41
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
Conclusions
I
NPB model recovers the observational uncertainty very well in
case of precision parameter α = M + 1 (M - the number of
dose levels); exceptions are observed at zero dose level
I
Violation of monotonicity in the data produces unsatisfactory
fits
I
The PAV algorithm does not improve the model’s ability to
recover the uncertainty
I
Pooling data does not alter the uncertainty depending on the
data set
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OUTLINE
PART I
PART II
CONCLUSIONS
REFERENCES
References
1. Gelfand, A.E., Kuo, L. (1991). Nonparametric Bayesian
bioassay including ordered polytomous response. Biometrika
78, 3, 657-666.
2. Ramgopal, P., Laud, P.W., Smith, A.F. (1993).
Nonparametric Bayesian bioassay with prior constrains on the
shape of the potency curve. Biometrika 80, 3, 489-498.
3. Ramsey, F.L. (1972). A Bayesian approach to bio-assay.
Biometrics 28, 841-858.
4. Shaked, M., Singpurwalla, N.D. (1990). A Bayesian approach
for quantile and response probability estimation with
applications to reliability. Ann. Inst. Statist. Math. 42, 1-19.
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