Bench test with Probabilistic
Inversion and Isotonic
Regression
Roger Cooke
Resources for the Future
&
Dept. Math Delft Univ of Technology
Oct. 22-23, 2007
Tentpole 1:
Observational Uncertainty
Give 49 rats dose 21[units] and observe 15 responses.
Sample independently 49 fresh rats from the same population, what is our
uncertainty on # responses?
Binomial
uncertainty
Bayes Uncertainty
With uniform prior
(normal
approximation)
1
Tentpole 2:
Uncertainty via parameter distribution
(doesn’t depend on dose)
Prob_Response(Dose) =
g + (1-g) × (1 – e-b1×dose)
Distribution over {g, b1}, the same for all doses
Tentpole 3:
Probability of Response does NOT
decrease as dose increases
Remove Non-monotonicity noise with
ISOTONIC REGRESSION: PAV (pair
adjacent violators)
#
R
e
s
p
o
n
s
e
Sample Independent binomials, if
increasing, then do nothing
if NOT increasing, then average
Build up Isotonic Uncertainty
Distribution
dose
2
Probabilistic Inversion
(Assume # subjects is same at each dose)
DR model:
ProbResp(dose) = f(d,α, β, γ,…)
#
R
e
s
p
o
n
s
e
Observational Uncertainty:
Isotonic
Observational Uncertainty:
binomial
dose
Probabilistic Inversion: an inverse DOES NOT
always exist, and NOT always unique
DR model:
ProbResp(dose) = f(d,α, β, γ,…)
#
R
e
s
p
o
n
s
e
dose
3
How do we do Probabilistic Inversion?
Iterative Proportional Fitting – variant
ProbResp(dose) = f(d,α, β, γ,…)
Start with wide distribution
#
R
e
s
p
o
n
s
e
Trim to Observ’l uncertainty
Re-weight to fit percentiles
dose
What is the difference with
Statistics as Usual?
4
Statistics as Usual
Recompute MLE estimates of
parameters, get (asymptotically) joint
normal parameter distribution
DR model:
ProbResp(dose) = f(d,α, β, γ,…)
Sample
Parameters,
Compute prob of
response for
each sample
Compute dist’n
over #
responses
#
R
e
s
p
o
n
s
e
ASSUME model is correct;
If we repeat experiments,
dose
DR model:
We get binomials
ProbResp(dose) = f(d,α, β, γ,…)
#
R
e
s
p
o
n
s
e
dose
5
Probabilistic Inversion: Equivalent Picture
DR model:
P
r
o
b
a
b
i
l
i
t
y
ProbResp(dose) = f(d,α, β, γ,…)
Dose 1
Dose 2
Dose 3
Dose 4
Nr Responses
BMD Tech. Guidance Doc
Perferred model, log logistic, β = 1:
6
BMD Tech. Guidance Doc
Probability
Dose 0
Dose 60
Dose 21
Number of responses
Probabilistic Inversion (fitting 5, 50, 95 %-tiles)
BMD preferred model is loglogistic, b=1
Fit not too good for
preferred model
7
Frambozadrine
Fambozadrine; males multistage: Nr*(gamma+(1-gamma)*(1-exp(-b1*dose)))
Dose zero
Dose 15
Dose 1.2
Dose 82
8
Fambozadrine; females multistage: Nr*(gamma+(1-gamma)*(1-exp(-b1*dose)))
Dose zero
Dose 1.8
Dose 109
Dose 21
Frambozadrine, males and females
Only found good fit with threshold model:
Nr*(g+(1-g)*i1{s,dose,∞}*(1-exp(-b1*max{t,dose}-b2*max2 {t,dose} -b3*max3{t,dose})))
Probability of Response, Threshold Model
1
probability
0.8
0.6
Prob_Resp
0.4
0.2
0
dose
0
20
40
60
80
100
120
Below threshold s, background rate g applies
Above threshold t a multistage model applies
Distributions for g, s, t, b1, b2, b3 found by PI
9
Frambozadrine, MF, doses 0, 1.2, 1.8, 15
Dose zero
Dose 1.2
Dose 15
Dose 1.8
Frambozadrine, MF, doses 21, 82, 109
Dose 21
Dose 82
Dose 109
10
Distribution of s, and t = 21 - s
Before PI
After PI
Distribution of g
Before PI
After PI
11
Distribution of B1
Before PI
After PI
Parameters’ Joint Distribution
12
Probability as function of dose
Nectorine
Prob(adenoma OR neuroblastom) =
Prob(adenoma) + Prob(neurblastoma) – Prob(adenoma) × Prob(neurblastoma)
13
BMD: logistic.
PI for multistage:NR*(g+(1-g)*(1-exp(-b1*dose)))
Dose 10
Dose 30
Dose 60
Persimonate
BARRIER MODEL:
NR*(g+(1-g)*(1-exp(-b1*i1{0,dose,t}-b2*i1{t,dose,s}-b3*i1{s,dose,∞ })))
b1 < b2 < b3
14
Barrier Model
NR*(g+(1-g)*(1-exp(-b1*i1{0,dose,t}-b2*i1{t,dose,s}-b3*i1{s,dose,∞ })))
b1 < b2 < b3
Probability of Response, Barrier Model
probability
1
0.8
0.6
Prob_Resp
0.4
0.2
0
0
20
40
60
80
100
120
dose
Distribution on intervals and coefficients from PI
Persimonate; Dose 0, 1.8, 9 ppm
Dose 0
Dose 1.8
Dose 9
15
Persimonate; Dose 18, 36, 45 ppm
Dose 18
Dose 36
Dose 45
Distributions for b1,b2, b3
Before PI
After PI
16
Probability as function of dose
CONCLUSIONS
PLUS
• Use Isotonic regression gets rid of ‘non-monotonic noise’
• PI CAN recover isotonic observational uncertainty
• Minus
• These methods are new and unfamiliar
• Many open questions
17