1

Modeling and Estimation of

Benchmark Dose (BMD)

for Binary Response Data

Wei Xiong

2

Outline

Benchmark dose (BMD) and datasets Statistical models

logistic probit multi–stage gamma multi–hit

Model fitting and analyses Conclusions

3

where,

ADI: acceptable daily

intake

SF: safety factor

NOAELADI

SF

In environment risk assessment,

NOAEL (no-observed-adverse-effect level)

is used to derive a safe dose,

4(Filipsson et al., 2003)Problem:

5

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

Benchmark dose (BMD)

6

Benchmark dose (BMD)

• Fit a model to all data

• Estimate the BMD

from a given BMR (10%)

• Derive “safe dose”

from BMD

(Filipsson et al., 2003)

Advantage: BMD uses all the

data information by fitting a model

7

Non–cancer data Ryan and Van (1981)

i di ri ni

1 24 0 30

2 27 0 30

3 30 4 30

4 34 11 30

5 37 10 30

6 40 16 30

7 45 26 30

8 50 26 30

30 mice in each dose group

drug: botulinum toxin in 10–15 gram

response: death (Y or N) within 24 hrs

8

Plot of Non-cancer Data

log(dose)

Pro

pn

of

Mic

e M

ort

alit

y

3.2 3.4 3.6 3.8

0.0

0.2

0.4

0.6

0.8

Plot of non-cancer Data

9

Cancer dataBryan and Shimkin (1943)

i d i r i n i

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)

10

Plot of cancer data

Plot of Cancer Data

log(dose)

Pro

pn

of

Mic

e B

ea

rin

g T

um

ors

2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

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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

12

Statistical models

( ) (1 ) ( ; , )P d F d

• Logistic • Probit• Multi–stage• Gamma multi–hit

Model form:

where,

1> >=0 is the background response as dose0

F is the cumulative dist’n function

13

Probit model

Assuming:

log(d) is approx. normally distributed

2log / 21( )

2

e d xP d e dx

14

Logistic model

( log )

1( )

1 e dP d

e

Assuming:

log(d) has a logistic distribution

15

1( ) (1 )[1 exp( )]

n jjj

P d d

Multi–stage model (Crump, 1981)

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

16

Gamma multi–hit model (Rai and Van, 1981)

1

0

1

0

( ) (1 )

d t

t

t e dtP d

t e 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

17

Model fitting

Models are fit by maximum likelihood method

Model fitting tested by Pearson’s 2 statistic

If p-value 10%, the model fits the data well and

the mle of BMD is obtained from the fitted model

22 2

1

( )~

(1 )

mii i

n pi

ii i

r n P

n P P

where, is estimated from the fitted model

iP

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BMDL by LRT

(Crump and Howe, 1985)

where,

and are model parameters

P is the log(BMD) at response = p

P

2P 1 2*0.05,1

2[ ( , ) ( , )]

= ( , ) ( , )D D

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The BMDL is the value P, which is lower than the

mle , so that, P

P

21 2*0.05,1

2[ ( , ) ( , )]

=

20

BMDL by Fieller’s Theorem (Morgan, 1992)

12

22

221 0.05 12

11 12 22 1122

( )( )1

2 ( )(1 )

P P

P P

Vc

c V

Z VV V V c V

Vc

Fieller’s Theom constructs CI for the ratio of R.V.

For logistic model,

the BMDL is derived as,

where,

211 12 22~ (0, 2 )N V V V

2 2(1 0.05) 22 /c Z V

21

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

22

BMDS logistic modeling for non–cancer data

(Pearson’s 2, p = 0.325 > 0.1)

0

0.2

0.4

0.6

0.8

1

25 30 35 40 45 50

Fra

cti

on

Aff

ec

ted

dose

Log-Logistic Model with 0.95 Confidence Level

05:48 07/02 2005

BMDL BMD

Log-Logistic

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0

0.2

0.4

0.6

0.8

1

10 20 30 40 50

Fra

cti

on

Aff

ec

ted

dose

Multistage Model with 0.95 Confidence Level

05:39 07/02 2005

BMDBMDL

Multistage

BMDS multi–stage modeling for non–cancer data

(Pearson’s 2, p = 0.0000)

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BMDS two–stage modeling for cancer data

(Pearson’s 2, p = 0.556)

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

Fra

cti

on

Aff

ec

ted

dose

Multistage Model with 0.95 Confidence Level

07:02 07/01 2005

BMDBMDL

Multistage

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BMDL=1.536 by LRT (Probit model for cancer data)

x0 values

Pro

file

De

via

nce

1.4 1.6 1.8 2.0

01

23

45

Lo

we

r C

L=

1.5

36

13

91

51

80

69

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Software Model

Logistic Probit

BMDS 30.042 30.039

S–plus 30.042 30.039

MLE of BMD (non–cancer data)

( p–value by Pearson’s 2 )

(0.325) (0.386)

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Summary of BMDL (non–cancer data)

Methods Software Model

Logistic Probit

LRTBMDS 28.143 28.296

S–plus 28.139 28.293

Fieller’s Theorem

S–plus 27.991 28.218

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Software Model

Logistic Probit Two–stage

Multi–hit

BMDS 7.168 7.203 4.867 6.334

S–plus 7.171 7.199

MLE of BMD (cancer data)

# p–value by Pearson’s 2

# 0.585 # 0.666 # 0.556 # 0.602

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Methods SoftwareModel

Logistic Probit Two–

stage

Multi–

hit

LRTBMDS 4.434 4.647 3.087 3.290

S–plus 4.434 4.647

Fieller’s

Theorem

S–plus 4.181 4.519

Summary of BMDL (cancer data)

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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 ?

31

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,

1 0.05/ 2

| | 1/(2 )

/

p P nZ

PQ n

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Fitted and re–parameterized model

log ( )1e

Pd

P

log ( )1ec

log ( ) ( )1e P

Pc d

P

Fitted logistic model

Re-parameterized logistic model

where,

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Abbott’s Formula

(1 )P c c BMR

where,

P – observed response

c – response at dose zero

BMR – benchmark response with

default value 10%