Download Modeling and Estimation of Benchmark Dose (BMD)

Modeling and Estimation of
Benchmark Dose (BMD)
for Binary Response Data
 Wei Xiong
1
Outline
 Benchmark dose (BMD) and datasets
 Statistical models
 logistic

probit

multi–stage

gamma multi–hit
 Model fitting and analyses
 Conclusions
2
In environment risk assessment,
NOAEL (no-observed-adverse-effect level)
is used to derive a safe dose,
NOAEL
ADI 
SF
where,
ADI: acceptable daily intake
SF: safety factor
3
Problem:
(Filipsson et al., 2003)
4
Benchmark dose (BMD)
 BMD: Point estimate of the dose which induces a
given response (e.g. 10%) above unexposed
controls
 BMDL: 1–sided 95% confidence lower limit
for BMD
5
Benchmark dose (BMD)
• Fit a model to all data
• Estimate the BMD
from a given BMR (10%)
• Derive “safe dose”
from BMD
Advantage: BMD uses all the
data information by fitting a model
(Filipsson et al., 2003)
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Non–cancer data
Ryan and Van (1981)
i
1
2
3
di
24
27
30
ri
0
0
4
ni
30
30
30
4
5
6
34
37
40
11
10
16
30
30
30
7
8
45
50
26
26
30
30
30 mice in each
dose group
drug: botulinum toxin
in 10–15 gram
response:
death (Y or N)
within 24 hrs
7
Plot of non-cancer Data
0.6
0.4
0.2
0.0
Propn of Mice Mortality
0.8
Plot of Non-cancer Data
3.2
3.4
3.6
log(dose)
3.8
8
Cancer data
Bryan and Shimkin (1943)
i
di
ri
ni
1
3.9
0
19
2
7.8
3
17
3
15.6
6
18
4
31
13
20
5
62
17
21
6
125
21
21
17 to 21 mice in each
dose group
drug: carcinogenic
methylcholanthrene
in 10–6 gram
response:
tumor (Y or N)
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Plot of cancer data
0.6
0.4
0.2
0.0
Propn of Mice Bearing Tumors
0.8
1.0
Plot of Cancer Data
2
3
4
log(dose)
10
How to estimate BMD ?
 What models to be used
 ? Need to use different models for the cancer
and non-cancer data
 How to fit the model curve
11
Statistical models
•
•
•
•
Logistic
Probit
Multi–stage
Gamma multi–hit
Model form:
P(d )    (1   ) F (d ;  ,  )
where,
1> >=0 is the background response as dose0
F is the cumulative dist’n function
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Probit model
1 
P(d )   
2

   loge d

e
 x2 / 2
dx
Assuming:
log(d) is approx. normally distributed
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Logistic model
P(d )   
1 
1 e
 (   loge d )
Assuming:
log(d) has a logistic distribution
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Multi–stage model
(Crump, 1981)
P(d )    (1   )[1  exp( j 1  j d )]
n
j
Assuming:
1. Ordered stages of mutation, initiation or transformation
for a cell to become a tumor
2. Probability of tumor occurrence at jth stage is
proportional to dose by jd j
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Gamma multi–hit model
(Rai and Van, 1981)
d

P(d )    (1   )

0

0
t
 1  t
e dt
t  1et dt
Assuming: a tumor incidence is induced by at least   1 hits
of units of dose and  follows a Poisson distribution
The gamma model is derived from the Poisson dist’n of 
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Model fitting
 Models are fit by maximum likelihood method
 Model fitting tested by Pearson’s 2 statistic

m
2  
i 1
(ri  ni P i ) 2


ni P i (1  Pi )
~  n2 p

where,
Pi is estimated from the fitted model
 If p-value  10%, the model fits the data well and
the mle of BMD is obtained from the fitted model
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BMDL by LRT
(Crump and Howe, 1985)



2[ (  ,  )  ( P ,  )]



=D(P ,  )  D(  ,  )  
2
1 2*0.05,1
where,
 and  are model parameters
 P is the log(BMD) at response = p
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 The BMDL is the value P, which is lower than the

mle , so that,
P



2[ (  ,  )  ( P ,  )]
=
2
1 2*0.05,1
19
BMDL by Fieller’s Theorem
(Morgan, 1992)
Fieller’s Theom constructs CI for the ratio of R.V.
For logistic model,


    ~ N (0,V11  2V12   2V22 )
the BMDL is derived as,


V12
c
P(
)( P 
)
1 c
V22


2
V

V11  2 P V12   P 2 V22  c(V11  12 )
V22
 (1  c)
Z10.05
where,

2
c  Z 2(10.05)V22 / 
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BMDL computation
 BMDS (benchmark dose software, US EPA)
provides the 4 models for BMDL using LRT
 S–Plus calculates BMDL using LRT and
Fieller’s Theorem
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BMDS logistic modeling for non–cancer data
(Pearson’s 2, p = 0.325 > 0.1)
Log-Logistic Model with 0.95 Confidence Level
1
Log-Logistic
0.8
0.6
0.4
0.2
0
BMDL
25
BMD
30
35
40
45
50
dose
05:48 07/02 2005
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BMDS multi–stage modeling for non–cancer data
(Pearson’s 2, p = 0.0000)
Multistage Model with 0.95 Confidence Level
1
Multistage
0.8
0.6
0.4
0.2
0
BMDL
BMD
10
20
30
40
50
dose
05:39 07/02 2005
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BMDS two–stage modeling for cancer data
(Pearson’s 2, p = 0.556)
Multistage Model with 0.95 Confidence Level
Multistage
1
0.8
0.6
0.4
0.2
0
BMDL BMD
0
20
40
60
80
100
120
dose
07:02 07/01 2005
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BMDL=1.536 by LRT
Lower CL= 1.5361391518069
3
2
1
0
Profile Deviance
4
5
(Probit model for cancer data)
1.4
1.6
x0 values
1.8
2.0
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MLE of BMD
(non–cancer data)
Software
BMDS
S–plus
Model
Logistic
Probit
30.042
30.039
(0.325)
(0.386)
30.042
30.039
( p–value by
Pearson’s 2 )
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Summary of BMDL
(non–cancer data)
Methods Software
BMDS
Model
Logistic Probit
28.143
28.296
S–plus
28.139
28.293
Fieller’s S–plus
Theorem
27.991
28.218
LRT
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MLE of BMD
(cancer data)
Software
Logistic
BMDS
7.168
# 0.585
S–plus
7.171
Model
Probit
Two–
stage
7.203
4.867
# 0.666
# 0.556
Multi–
hit
6.334
# 0.602
7.199
# p–value by Pearson’s 2
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Summary of BMDL
(cancer data)
Model
Methods Software
Logistic Probit
Two–
stage
Multi–
hit
BMDS
4.434
4.647
3.087
3.290
S–plus
4.434
4.647
Fieller’s S–plus
Theorem
4.181
4.519
LRT
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Conclusions
 Non–cancer data, BMD = 30.042 (logistic) and 30.039
(probit) in 10–15 gram; cancer data, BMD = 7.168
(logistic), 7.203 (probit), 4.867 (multi–stage) and 6.334
(multi–hit).
 Logistic and probit model fit both data sets well, multi–
stage and multi–hit fit only the cancer data well.
 BMDL obtained by Fieller’s Theorem seems to be smaller
than that by LRT, why ?
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Questions ?
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A note on qchisq( ) of 1–sided 95%
 > (qnorm(1 - 0.05))^2
 [1] 2.705543
 > qchisq(1 - 2 * 0.05, 1)
 [1] 2.705543
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95% CI for proportion in slides 21 & 22
 When n is large, nP  5 and n(1-P)  5, the sample
proportion p is used to infer underlying proportion P.
 p is approximately normal with mean P and
s.e.=sqrt(P(1-P)/n)
 Solving the following equation,
| p  P | 1/(2n)
 Z10.05/ 2
PQ / n
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Fitted and re–parameterized model
Fitted logistic model
P
log e (
)   d
1 P
Re-parameterized logistic model
P
log e (
)  c   (d   P )
1 P
where,

c  log e (
)
1 
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Abbott’s Formula
P  c  (1  c) BMR
where,
P – observed response
c – response at dose zero
BMR – benchmark response with
default value 10%
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