Human and Ecological Risk Assessment: An International
Journal
ISSN: 1080-7039 (Print) 1549-7860 (Online) Journal homepage: http://www.tandfonline.com/loi/bher20
Robust benchmark dose estimation using an
infinite family of response functions
Ishapathik Das
To cite this article: Ishapathik Das (2018): Robust benchmark dose estimation using an infinite
family of response functions, Human and Ecological Risk Assessment: An International Journal,
DOI: 10.1080/10807039.2018.1438172
To link to this article: https://doi.org/10.1080/10807039.2018.1438172
Published online: 01 Mar 2018.
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HUMAN AND ECOLOGICAL RISK ASSESSMENT
https://doi.org/10.1080/10807039.2018.1438172
Robust benchmark dose estimation using an infinite family
of response functions
Ishapathik Das
Department of Mathematics, Indian Institute of Technology Tirupati, Andhra Pradesh, India; Department of
Statistical Science, Duke University, Durham, North Carolina, USA
ABSTRACT
ARTICLE HISTORY
Researchers usually estimate benchmark dose (BMD) for dichotomous
experimental data using a binomial model with a single response
function. Several forms of response function have been proposed to fit
dose–response models to estimate the BMD and the corresponding
benchmark dose lower bound (BMDL). However, if the assumed
response function is not correct, then the estimated BMD and BMDL
from the fitted model may not be accurate. To account for model
uncertainty, model averaging (MA) methods are proposed to estimate
BMD averaging over a model space containing a finite number of
standard models. Usual model averaging focuses on a pre-specified list
of parametric models leading to pitfalls when none of the models in
the list is the correct model. Here, an alternative which augments an
initial list of parametric models with an infinite number of additional
models having varying response functions has been proposed to
estimate BMD for dichotomous response data. In addition, different
methods for estimating BMDL based on the family of response
functions are derived. The proposed approach is compared with MA in
a simulation study and applied to a real dataset. Simulation studies are
also conducted to compare the four methods of estimating BMDL.
Received 19 December 2017
Revised manuscript
accepted 5 February 2018
KEYWORDS
benchmark dose; binomial
models; model
misspecification; family of
response functions; interval
estimation
Introduction
One of the main goals in quantitative risk assessment is to estimate the risk function
R(d), which is the probability of adverse events, such as death, birth defect, weight
loss, cancer, or mutation exhibited in a subject exposed at dose level d. Suppose n
number of subjects are exposed to a dose level d and y number of adverse events are
observed. Then, the response y is distributed according to a binomial distribution with
parameter [n, R(d)], where R(d) is the probability of adverse events at dose level d. It
is usually observed that R(d) increases with doses d, and hence a binomial model with
a non-decreasing response function may be better to fit such dichotomous data than
any mathematical model. After estimating the risk function R(d), the extra risk function RE(d), defined as RE ðd Þ D Rð1d¡Þ ¡p0p0 is computed, where p0 is the risk at minimum dose
level usually called as background risk. The benchmark dose (BMD) is defined by the dose
CONTACT Ishapathik Das
ishapathik@iittp.ac.in
Department of Mathematics, Indian Institute of Technology
Tirupati, Tirupati – Renigunta Road, Settipalli Post, Tirupati, Andhra Pradesh 517506, India.
© 2018 Taylor & Francis Group, LLC
2
I. DAS
level having the extra risk RE ðBMDÞ D BMR, where BMR is called the benchmark
response, usually pre-specified in the range 0:01BMR0:1 (US EPA 2012). The benchmark dose lower bound (BMDL) is also determined using the risk function R(d). The accuracy of the estimation of BMD and BMDL is dependent upon the estimation of the risk
function R(d).
The benchmark analysis by estimating BMDs and corresponding BMDLs have been
accepted as one of the important tools for quantitative risk assessment by a variety of
agencies such as the U.S. Environmental Protection Agency (US EPA), the European
Food Safety Authority (EFSA), the Organisation for Economic Cooperation and
Development (OECD). Several agencies including the US EPA, the EFSA, the OECD
provide guidance on BMDs (OECD 2008; US EPA 2012; EFSA 2017) and the
estimation of BMDs is dramatically increased over the past few decades for quantitative risk assessment. Methods of estimating BMD and BMDL are discussed by several
researchers such as Crump (1984); Bailer et al. (2005); Morales et al. (2006); Wheeler
and Bailer (2007, 2009); West et al. (2012) to name just a few. Crump (1984) introduced methods of estimating BMD and BMDL by proposing four models for discrete
responses and three models for continuous responses. There are eight models
(Wheeler and Bailer 2007, 2009; West et al. 2012) that have been identified as
standard models for estimating BMD and BMDL. One of the models from the set of
standard models may be chosen for fitting the data sets. However, the responses may
be generated from the model other than the chosen model. Researchers (Wheeler and
Bailer 2007, 2009; West et al. 2012) have shown that the estimation of BMD and
BMDL are significantly affected if the assumed model is incorrect. So, there is a
recent rise in developing methods of accounting for model uncertainty in BMD
estimation.
To deal with model uncertainty in BMD and BMDL estimation, various model averaging
(MA) methods have been proposed by Kang et al. (2000); Bailer et al. (2005); Wheeler and
Bailer (2007); Shao and Small (2011); West et al. (2012); Piegorsch et al. (2013). The estimates of BMD and BMDL in model averaging methods are given by the weighted average of
the estimates of BMD and BMDL when individually modeled. Bayesian methods and Bayesian model averaging methods for estimating BMD and BMDL are also proposed by Morales
et al. (2006); Shao and Small (2012); Simmons et al. (2015). The model averaging approach
may solve the problems of model uncertainty when the true model generating responses can
be well approximated by some members of the model space containing the assumed models.
The model space used in MA is always finite. However, there is an infinite number of other
models which cannot be approximated by the members of the model space used in MA leading to the problem of model uncertainty. So, model averaging techniques only provide a partial solution to the problem of model uncertainty.
In this article, an infinite family of models containing some of the standard models
is used to find the risk function. The family of response models is parameterized by
two unknown parameters. There is an infinite number of response models which can
be represented by different values of parameters. Some standard response models correspond to some specific values of parameters. So, we may get better results for accounting model uncertainty in BMD and BMDL estimation using the proposed family of
models.
HUMAN AND ECOLOGICAL RISK ASSESSMENT
3
Methods
Binomial models
The binomial models are members of the generalized linear models (GLMs) described by
three components given below.
1. Distributional components: let y1, y2, …, yn be n random samples of adverse events at
dose levels d1, d2, …, dn, where for each i 2 {1, 2, …, n}, yi has binomial distribution
with parameter (ni, ri), ri 2 [0, 1], and yi D nyii has scaled binomial distribution belongs
to the exponential family having the form of probability mass function (pmf) given by
(Fahrmeir and Tutz 2001)
yi ui ¡ bðui Þ
wi C cðyi ; wi ; ’Þ ;
s yi jui ; wi ; ’ D exp
’
where ri D Rðdi Þ D E yi jdi , ui D log 1 ¡ri ri is the so-called natural parameters, b(ui) D
h
i
log [1 C exp (ui)], wi D ni, f D 1, and cðyi ; wi ; ’Þ D log yi !ðnni i¡! yi Þ! .
0
2. Linear predictor: hðdi Þ D f ðdi Þb, where f(di) is a vector function of di, and b is the
regression parameter vector.
3. Parametric response function: Rðdi Þ D h½a; hðdi Þ or g ½a; Rðdi Þ D hðdi Þ, where h is the
parametric response function and g (the inverse of h) is the parametric link function.
We usually assume that the inverse of h exists.
For dose–response studies, the linear predictor is usually assumed to be h(d) D b0 C b1d,
or h(d) D b0 C b1d C b2d2, and a single response function such as logistic, probit, log-log,
complementary log-log or some other response functions are used to fit the models. Here,
instead of a single response function, we use a family of response functions (parametric
response function) parameterized by a response parameter vector a to fit the models. So, we
denote the response function as hða; ¢Þ in place of h( ¢ ).
Several researchers (Stukel 1988; Czado 1997) proposed various families of response functions
to fit binomial models. One such family of response functions for binomial models is given by
Rðd Þ D Eðyjd Þ D h½a; hðd Þ D
exp½Gða; hÞ
;
1 C exp½Gða; hÞ
(1)
where h h(d), and Gða; ¢Þ is called a generating family. There are several Generating families
proposed in the literature (Stukel 1988; Czado 1989). Stukel (Stukel 1988) provides the following
generating family:
if h 0 (i.e., r 12),
8
expða1 hÞ ¡ 1
>
>
;
a1 > 0
>
>
<
a1
a1 D 0
Gða; hÞ D h;
>
>
> logð1 ¡ a1 hÞ
>
:¡
; a1 < 0;
a1
4
I. DAS
and for h < 0 (i.e., r < 12),
8
1 ¡ exp ð ¡ a2 hÞ
>
>
;
>
>
<
a2
Gða; hÞ D h;
>
>
log ð1 C a2 hÞ
>
>
:
;
a2
a2 > 0
a2 D 0
a2 < 0;
0
where h h(d), and r R(d). Note that, for a D ½0; 0 , we get the logistic response function.
So, the logistic response function is a member of this family. Also, several important
response functions such as the Probit, the complementary log–log response functions can be
approximated by the members of this family (Stukel 1988).
For estimating the risk function using the above models, we need to0 estimate the
0
0
for the comunknown parameters using an available data set. Let us denote d D b ; a
bined parameter vectors including the unknown regression parameter vector b and the
response parameter vector a. The unknown parameter vector d can be estimated using the
maximum likelihood estimation (MLE) methods given in Stukel (1988). Due to the estimation of the response parameters along with regression parameters, the variances of the estimated regression parameters are increased (Taylor 1988). The variance inflations of the
regression parameters are asymptotically zero if the response parameters are orthogonal to
the regression parameters (Cox and Reid 1987). Czado (Czado 1997) proposed some conditions on the family of response functions providing local orthogonality between response
and regression parameter vectors. A family of response functions L D fhða; ¢Þ : a 2 Vg
provides local orthogonality between response and regression parameter vectors around a
point h0 asymptotically, if the following conditions are satisfied:
1. there exists h0 and r0 such that
hða; h0 Þ D r0 ; 8 a 2 V;
(2)
@hða; hÞ
jðh D h0 Þ D s0 ; 8 a 2 V;
@h
(3)
and
2. there exists s0 such that
where V is denoted for the parameter space of a. Such a family L D fhða; ¢Þ : a 2 Vg satisfying conditions (2) and (3) is called (r0, s0) ¡ standardized at h0 (Czado 1997).
Now, for estimating risk function R(d) using (r0, s0) ¡ standardized family at h0, we need
to estimate extra three parameters r0, s0, and h0. For avoiding estimating extra three parameters, Czado (1997) proposed to choose r0 D b0, s0 D 1, and h0 D b0. By choosing the values
such a way, the variance inflations of b are reduced
P as the values of h vary around the point
h0 D b0, when centered covariates (i.e., d D n1 niD 1 di D 0) are used (Czado 1997). For this,
if dose levels are not centered, we need to transfer the available dose levels as xi D di ¡ d, and
after estimating BMD/BMDL from the model, we make the inverse transformation to get
the estimates of BMD/BMDL in the true range of dose levels. For constructing (r0 D b0, s0 D
HUMAN AND ECOLOGICAL RISK ASSESSMENT
5
1) ¡ standardized family at h0 D b0, we adopt the methodologies given by Czado (1997).
Here, we use Stukel’s (Stukel 1988) generating family to construct the family of response
functions, and the (r0 D b0, s0 D 1) ¡ standardized at h0 D b0 generating family is given by
if hc 0 [logitðr Þb0 ],
8
expða1 hc Þ ¡ 1
>
>
;
a1 > 0
>
>
<
a1
a1 D 0
Gc ða; hÞ D b0 C hc ;
>
>
log
ð
1
¡
a
h
Þ
>
1
c
>¡
:
; a1 < 0;
a1
(4)
and for hc < 0 [logitðr Þ < b0 ],
8
1 ¡ expð ¡ a2 hc Þ
>
>
;
>
>
<
a2
Gc ða; hÞ D b0 C hc ;
>
>
logð1 C a2 hc Þ
>
>
:
;
a2
a2 > 0
a2 D 0
(5)
a2 < 0;
where h h(d), r R(d), hc D h ¡ b0, and logitðr Þ D log½r=ð1 ¡ r Þ. Hence, the risk function
R(d) using the binomial model with (r0 D b0, s0 D 1) ¡ standardized at h0 D b0 generating
family is given by
Rðd Þ D Eðyjd Þ D h½a; hðd Þ D
exp½Gc ða; hÞ
;
1 C exp½Gc ða; hÞ
(6)
where h h(d), and Gc ða; ¢Þ is given by Eqs. (4) and (5). In the next section, we provide an
expression for the benchmark dose (BMD) using the above model.
Benchmark dose estimation
The benchmark dose (BMD) is defined by the dose level having extra risk RE ðBMDÞ D BMR,
where BMR is the benchmark risk usually pre-specified as 0.01, 0.05, and 0.1. So, BMD is the
solution of the equation
RE ðBMDÞ
D
) RðBMDÞ
) BMD
D
D
RðBMDÞ ¡ p0
D BMR
1 ¡ p0
p0 C ð1 ¡ p0 ÞBMR D BMRE; say
R ¡ 1 ðBMREÞ;
(7)
where p0 is the background risk, i.e., the risk at the minimum dose level d1. We denote
0
0 0
d D ½b ; a for the joint parameter vector including the regression parameter vector b, and
the response parameter vector a. For a fixed value of BMR 2 ½0; 1, the BMD can be
expressed as a function of d, SðdÞ, say. From Eqs. (6) and (7), we get an expression for
6
I. DAS
BMD as
BMD D SðdÞ D S1 ðdÞIfLBMRb0 g C S2 ðdÞIfLBMR < b0 g ;
(8)
where LBMR D log 1 ¡BMRE
BMRE , and IfLBMRb0 g is the indicator function taking value 1 if
LBMRb0 , and 0 otherwise. The functions S1 ðdÞ and S2 ðdÞ are given by
8
log½a1 ðLBMR ¡ b0 Þ C 1
>
>
;
>
>
>
a1 b1
>
>
<
LBMR ¡ b0
;
S1 ðdÞ D
>
b1
>
>
>
>
1 ¡ exp½ ¡ a1 ðLBMR ¡ b0 Þ
>
>
;
:
a1 b1
a1 > 0
a1 D 0
a1 < 0;
and,
8
log½1 ¡ a2 ðLBMR ¡ b0 Þ
>
>
; a2 > 0
¡
>
>
>
a2 b1
>
>
<
LBMR ¡ b0
;
a2 D 0
S2 ðdÞ D
>
b1
>
>
>
>
exp½a2 ðLBMR ¡ b0 Þ ¡ 1
>
>
;
a2 < 0:
:
a2 b1
P
Note that we require centered dose levels (i.e., d D n1 niD 1 di D 0) for using the model
(6). If the dose levels are not centered, we make the transformation xi D di ¡ d to have the
centered dose levels. After estimating BMD from the model we make the inverse transformation to get the estimated value of BMD within the true range of dose levels.
Asymptotic results
The asymptotic distributions of unknown parameters for q-dimensional multinomial
response models with a family of response functions are discussed in Das and Mukhopadhyay (2014). For q D 1, we get the binomial models using a family of response functions. So,
the similar results can be applicable for binomial models using a family of response functions. However, for making this article self-contained, we provide the required asymptotic
0
0
0 0
0
0 0
b ;a
b for the MLE of d D ½b ; a , lðdÞ for the logresults here. We denote b
d D ½b
@l
for the score function for the observed responses. Also, Jn is
likelihood function, and @d
denoted for the Fisher’s information matrix. The asymptotic results are given by the following Lemmas.
Lemma 1. The score function
mean 0 and variance Jn.
@l
@d
has an asymptotic multivariate normal distribution with
HUMAN AND ECOLOGICAL RISK ASSESSMENT
7
Proof. From the section “Methods”, the risk function is given by,
Rðd Þ D h½a; hðd Þ;
(9)
0
where hðd Þ D f ðd Þb, b is an unknown regression parameter vector and a is a vector of
unknown response parameters. Also,
0
hðd Þ D f ðd Þb D g ½a; Rðd Þ;
(10)
where g is the inverse of h.
&
Now, from the section “Methods”, the log-likelihood function for the sample y1, …, yn is
given by
lðdÞ
D
n
X
li ðdÞ
iD1
D
n X
iD1
(11)
yi ui ¡ bðui Þ ni C constant:
Thus, the score function is (Fahrmeir and Tutz 2001),
@lðdÞ
@d
n @X
y ui ¡ bðui Þ ni
@d i D 1 i
n
X
@ri ¡ 1 Var yi
yi ¡ ri ;
@d
iD1
D
D
(12)
and (Fahrmeir and Tutz 2001)
¡
@2 lðdÞ
0
@d@d
D
D
n
X
@ri iD1
@d
Var yi
Hn ; ðsayÞ:
¡1
n
@ri X
@2 ui ¡
yi ¡ ri ni
0
0
@d
i D 1 @d@d
(13)
From Eq. (13), we get the Fisher information matrix that is
X
n
@2 lðdÞ
@ri Var yi
D
Jn D ¡ E
0
@d
@d@d
iD1
¡1
@ri
0 :
@d
(14)
&
From Eq. (12), using the central limit theorem we have @l@dðdÞ has asymptotic normal distribu&
tion with mean 0 and variance Jn.
Lemma 2. The MLE of d, b
d has an asymptotic multivariate normal distribution with mean d
and variance Jn¡1.
8
I. DAS
Proof. By Taylor series expansion and approximating up to first-order term, we have
0D
@l bd
@d
2
@lðdÞ
@ lðdÞ b
C
D
d¡d ;
0
@d
@d@d
which gives (Fahrmeir and Tutz 2001, p. 439),
pffiffiffiffi
pffiffiffiffi
@lðdÞ pffiffiffiffi ¡ 1 @lðdÞ
D N Jn
C Op N ¡ 1=2 :
N bd ¡ d D N Hn¡ 1
@d
@d
Thus, the MLE of d, bd has an asymptotic normal distribution with mean d and variance
&
Jn¡1.
d DS b
Lemma 3. The estimate BMD
d is a consistent estimator for BMD.
Proof. The proof is trivial from the result that the MLE
of d, bd is a consistent estimator of d,
b
and SðdÞ is a continuous function of d. Hence, S d is a consistent estimator for SðdÞ, i.e.,
d is a consistent estimator for BMD.
BMD
&
In the next section, we provide confidence intervals for BMD to find BMDL from the proposed model.
Confidence intervals
Here, we provide four methods of constructing confidence intervals for BMD for a particular
value of BMR D BMR0 . The methods are discussed as follows:
1. Confidence interval using ML estimates: here, we use the asymptotic result of the distribution of bd for constructing the confidence interval for BMD. From Lemma 2, we
have bd has an asymptotic multivariate normal distribution with mean d and variance
0
d ¡ d has an asymptotic x2-distribution with p
S D Jn¡1. Hence, bd ¡ d S ¡ 1 b
degrees of freedom, where p is the order of the vector d. Hence, the 100(1 ¡ t)% confidence region for d is given by
0
C D fd 2 Rp : bd ¡ d S ¡ 1 bd ¡ d x2p;ð1 ¡ tÞ g;
(15)
where x2p, (1 ¡ t) is the (1 ¡ t)th quantile of the x2 distribution with p degrees of freedom. For BMR D BMR0 , we compute BMD D SðdÞ, for d 2 C using Eq. (8).Let us
denote
SL
SU
D
D
MinfSðdÞ : d 2 Cg; and
MaxfSðdÞ : d 2 Cg:
(16)
Now, from (16), we have d 2 C ) SðdÞ 2 ½SL ; SU , which implies PðSðdÞ 2 ½SL ; SU Þ Pðd 2 CÞ D 1 ¡ t. Hence, the 100(1 ¡ t)% conservative confidence interval for BMD
is given by [SL, SU].
HUMAN AND ECOLOGICAL RISK ASSESSMENT
9
2. Confidence interval using LR test: here, we test the null hypothesis
H0 : RE ðd Þ D BMR0 vs: H1 : RE ðd Þ 6¼ BMR0 ;
(17)
where RE ðd Þ D Rð1d¡Þ ¡p0p0 , with p0 is the background risk.
Let D(d) be the deviance
(Fahrmeir and Tutz 2001) under nullhypothesis
and D b
d be the deviance of the fit
ted model. Then, Lðd Þ D Dðd Þ ¡ D b
d has an asymptotic x2-distribution with 1
degree of freedom. Let us denote
Lmin
D
Minfd 2 R : Lðd Þx2p;ð1 ¡ tÞ g; and
Lmax
D
Maxfd 2 R : Lðd Þx2p;ð1 ¡ tÞ g
(18)
Then, the 100(1 ¡ t)% confidence interval for BMD is given
h by
i ½Lmin ; Lmax .
3. Confidence interval using score test: let us denote u0 D @b@l b , where bd 0 is the MLE
0 d
b 20 be the estimated variance0 of u0 at d D bd 0 . Then,
of d under H0 given in Eq. (17). Let s
s 20 has an asymptotic x2 distribution with 1 degree of freedom (Fahrmeir
T ðd Þ D u20 =b
and Tutz 2001). Let us denote Tmin D minfd 2 R : T ðd Þx21;ð1 ¡ tÞ g, and
Tmax D maxfd 2 R : T ðd Þx21;ð1 ¡ tÞ g. Then, using score test, a 100(1 ¡ t)% confidence interval for BMD is ½Tmin ; Tmax .
Note that the above confidence intervals are by nature two-sided. To get one-sided
confidence interval (BMDL), some researchers (Nitcheva et al. 2005; Buckley et al.
2009) proposed an adjustment by doubling the significance level of the test and then
ignoring the upper limit. So, we construct 100(1 ¡ 2t)% two-sided confidence intervals
for BMD using the above methods and then the lower limits of that intervals are taken
as the one-sided 100(1 ¡ t)% lower confidence bound for BMD.
Since all of the above confidence intervals are constructed using asymptotic results,
here we provide a confidence interval using a bootstrap technique.
4. A bootstrap lower confidence bound: for constructing bootstrap lower confidence
bound, we generate l responses yk D [y1k, y2k, y3k, y4k]0 using the fitted model br i D h
0
b where yik has binomial distribution with parameter
b; b
½a
h ðdi Þ D f ðdi Þb,
h ðdi Þ with b
ðn;br i Þ for i D 1, …, 4 and k D 1, …, l. From the generated l data sets, we estimate
BMDs using the proposed method to have samples fBMD1 ; BMD2 ; …; BMDl g for
BMD. The 100(1 ¡ t)% bootstrap lower confidence limit for BMD is given by the tth
quantile of the sample fBMD1 ; BMD2 ; …; BMDl g.
Let us denote ML, LR, ST, and BT for the methods of estimating BMDL using ML estimates,
LR test, score test, and bootstrap technique, respectively. Example and simulation studies are provided to illustrate and test the performance of the proposed methods in next sections.
Example: Experiment on rats exposed to 1-bromopropane
For illustrating the proposed method, we present an example of estimating BMD using a real
data set on lung cancer incidence of rats exposed to 1-Bromopropane given in the NTP
Technical Report TR-569 (Program et al. 2011). In this study, four groups of rats with each
group contains 50 rats are exposed to four dose levels of 1-Bromopropane. After 2 years of
10
I. DAS
studies, the observed lung cancer incidence of rats at four dose levels 0 ppm, 62.5 ppm,
125 ppm, and 250 ppm is recorded as 1/50, 9/50, 8/50, and 14/50, respectively.
Wheeler and Bailer (Wheeler and Bailer 2012) also analyzed the same data set noting that “the data exhibit a linear or supra-linear response indicating that MA may not
be able to capture the true D-R relationship.” Here, we use the proposed method of
using a family of response functions (FR) to estimate BMD and corresponding BMDLs
using ML estimates (ML), likelihood ratio test (LR), score test (ST), and bootstrap technique (BT) as described in the section “Methods”. Wheeler and Bailer (2012) provide
the estimates of BMD and corresponding BMDLs using semi-parametric (diffuse),
semi-parametric (historical controls), model-averaging, and quantal-linear. In Table 1,
we report the estimated values of BMD using FR and estimated values of BMDLs using
four methods ML, LR, ST, and BT within bracket for BMR = 0.01, and 0.1. Also, we
report the estimated values of BMD and corresponding BMDLs in bracket using semiparametric (diffuse), semi-parametric (historical controls), model-averaging, and quantal-linear for BMR = 0.01, and 0.1 derived from Wheeler and Bailer (2012) in Table 1.
From Table 1, we observe that the estimated values of BMD using FR are consistent with
the estimated values of BMD using semi-parametric (diffuse), semi-parametric (historical
controls), and quantal-linear for all values of BMR. As mentioned in Wheeler and Bailer
(2012), the estimated values of BMD using MA diverge and smaller than those using other
methods. We observe that the estimated values of BMDLs using FR are higher than those
reported by Wheeler and Bailer (2012). So, the proposed methods may provide better estimates of BMDL if the estimated confidence intervals have the expected coverage probabilities for small samples. So, we need to do simulation studies for verifying the coverage
probabilities of the proposed confidence intervals for small samples.
Simulation studies
In this section, we conduct simulation studies for testing the performance of the proposed
methods of estimating BMD and BMDL considering all types of possible cases of generating
datasets. Let us denote the proposed method of estimating BMD using the family of response
functions as FR. The model averaging method is usually denoted as MA. For testing the performance of FR compare to MA, we provide a simulation study by estimating BMD using
Table 1. Estimated values of BMD (ppm) and the corresponding BMDL (ppm) within bracket by four methods calculated using the family of response functions, and also the same derived from Wheeler and Bailer
(2012) by semi-parametric (diffuse), semi-parametric (historical controls), model-averaging, and quantallinear.
BMR
Method
Family of response functions
Semi-parametric
(Diffuse)
Semi-parametric
(Historical controls)
Model-averaging
Quantal-linear
0.01
0.1
8.6 (6.3, 6.2, 6.2, 7.4)
6.1 (2.1)
68.9 (61.8, 50.0, 49.8, 57.8)
56.6 (17.5)
6.6 (1.6)
97.1 (23.1)
1.1 (0.14)
7.8 (5.2)
51.1 (17.2)
81.5 (55.0)
HUMAN AND ECOLOGICAL RISK ASSESSMENT
11
Table 2. Six scenarios for the dose–response curves with true values for d, the probabilities of adverse
events at dose levels, and true BMDs corresponding to BMR = 0.01 and 0.1.
BMR
d D ½b0 ; b1 ; a1 ; a2 Scenario
1
2
3
4
5
6
’
[ ¡ 4.5031, 4.9075, 0.1170, 1.5162]0
[ ¡ 2.9252, 4.9961, 1.9078, ¡1.1403]0
[ ¡ 1.3677, 2.4678, 1.6912, ¡0.8872]0
[ ¡ 0.7784, 3.9106, 1.6554, ¡0.8438]0
[ ¡ 0.3852, 4.7828, 1.9908, ¡0.0870]0
[1.9190, 3.9682, 0.9064, 0.6930]0
R(d1)
R(d2)
R(d3)
R(d4)
0.01
0.1
0.0000
0.0176
0.1067
0.1374
0.0905
0.1909
0.0015
0.0276
0.1474
0.2060
0.2229
0.7202
0.0149
0.0760
0.2330
0.3829
0.5058
0.9000
0.0224
1.0000
0.9856
1.0000
1.0000
0.9999
0.4200
0.2475
0.0690
0.0430
0.0260
0.0039
0.8535
0.5418
0.4195
0.2914
0.1912
0.0365
FR and MA considering different simulation scenarios with varying sample sizes. Simulation
studies are also conducted for testing the performance of the proposed methods of estimating BMDL with respect to their coverage probabilities for small samples.
Comparison between FR and MA
We compare the proposed method FR with MA considering the following simulation set up
with experimental design consists of four dose levels as d1 D 0, d2 D 0.25, d3 D 0.5, and d4 D
1.00 mimicking the design considered by West et al. (2012). Six scenarios for dose–response
relationships have been considered to represent all types of possibilities of having probabilities of adverse events at dose levels varying from shallow to steep curves. The scenarios with
true parameter values and the probabilities of adverse events at dose levels are given in
Table 2.
For each scenario, responses (yi) are generated from binomial distribution with parameter
[n, R(di)], where n is the number of patients administered the dose level di, i 2 {1, …, 4}. For
testing the performance of the methods with varying sample sizes and BMR, we consider
d MA .
Table 3. Eight standard models used in MA for computing BMD
M
1
Name
Logistic
2
Probit
3
Quantal-linear
R(d)
1
1 C expð ¡ b0 ¡ b1 dÞ
1
b1
F(b0 C b1)d
1 ¡ exp ( ¡ b0 ¡ b1d)
4
Quantal-quadratic
g 0 C (1 ¡ g 0)(1 ¡ exp [b1d ])
5
Two-stage
1 ¡ exp ( ¡ b0 ¡ b1d ¡ b2d2)
2
BMD
¡b
0 BMR
log 1 C1 e¡ BMR
Notes
None
F ¡ 1 ½BMRð1 ¡ ’0 Þ C ’0 ¡ b0
b1
f0 D F(b0)
¡ logð1 ¡ BMRÞ
b1
b0 0, b1 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¡ logð1 ¡ BMRÞ
b1
¡ b1 C
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
2
b1 C 4b2 T
2b2
0 4 g 0 4 1, b1 0
bj 0, j D 0, 1, 2
T D ¡ logð1 ¡ BMRÞ
6
Log-logistic
7
Log-probit
8
Weibull
g0 C
1 ¡ g0
1 C expð ¡ b0 ¡ b1 log½dÞ
g 0 C (1 ¡ g 0)F[b0 C b1log (x)]
b0 b1
g 0 C ð1 ¡ g 0 Þ½1 ¡ expð ¡ e d Þ
exp
L ¡ b0
b1
h ¡1
i
Þ ¡ b0
exp F ðBMR
b1
h
i
exp logðTbÞ ¡ b0
1
0 4 g 0 4 1, b1 0
L D log 1 ¡BMR
BMR
0 4 g 0 4 1, b1 0
0 4 g 0 4 1, b1 0
T D ¡ logð1 ¡ BMRÞ
12
I. DAS
two different values of BMR D 0:01; and 0:1 and three different sample sizes n D 25, 50,
100 for each dose–response curve. This provides in total 6 curves £ 2 values of BMR £ 3
sample sizes = 36 different cases. For getting model averaging estimates of BMD, eight standard models (West et al. 2012) given in Table 3 are considered. The expression for model
d MA is given by
averaging estimates of BMD, denoted as BMD
d MA D
BMD
8
X
bk;
wk B
(19)
kD1
b k is the estimate of BMD using model k, and wk D P8exp ð ¡ 0:5Ak Þ
where B
kD1
exp ð ¡ 0:5Ak Þ
with Ak is the
L k C 2pk , where b
L k is the maxAkaike Information Criteria (Akaike 1973) given by Ak D ¡ 2b
imized log-likelihood value and pk is the number of parameters in model k.
We generate l D 2000 data sets for each simulation set up. For some cases the simulated
responses produce virtually flat dose–response curve (Wheeler and Bailer 2009) which does not
give any finite estimate of BMD. So, we mimic the methodology given in Wheeler and Bailer
(2009) of screening the data sets using the Kendall correlation test (Kendall 1955). We regenerate
responses until the responses exhibit the Kendall p-value less than or equal to 0.15.
For each case described above, we estimate BMDs by FR and MA using 2000 simulated data
sets. The proposed method FR is compared with MA on the basis of observed absolute relative
d
B
MD ¡ BMD
(Wheeler and Bailer 2007).
median bias defined by the absolute value of median
BMD
The estimated values of BMD are used as a sample of size 2000 for computing absolute relative median bias by FR and MA. Smaller values of absolute relative median bias’ are desirable
for having better performance by a BMD estimation method. The absolute relative median
bias (ARMB) values by FR and MA for each simulation set-up are reported in Table 4.
Table 4. Comparison between FR and MA for accounting model uncertainty in BMD estimation. The
observed values of absolute relative median bias’ for FR and MA for different cases are reported in the columns FR and MA, respectively.
Sc
n
BMR
FR
MA
Sc
n
BMR
FR
MA
1
25
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.2043
0.0691
0.1083
0.0298
0.0433
0.0460
0.0600
0.0520
0.0997
0.0212
0.0336
0.0193
0.1059
0.0729
0.0773
0.0428
0.1487
0.0248
0.3149
0.1814
0.1384
0.1321
0.1392
0.1228
0.6192
0.0247
1.6802
0.4306
1.7606
0.4790
5.7751
0.4777
5.8792
0.5366
5.4006
0.4850
4
25
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.2480
0.1420
0.2925
0.1563
0.3092
0.1834
0.5322
0.3872
0.4079
0.2450
0.2973
0.2243
0.4644
0.3822
0.2919
0.2101
0.1474
0.1618
7.9541
0.8534
10.1225
1.2207
10.4497
1.2917
5.2402
0.7904
7.9412
1.3338
11.8273
2.4060
4.3147
2.1070
4.3990
2.1520
4.3084
2.5693
50
100
2
25
50
100
3
25
50
100
50
100
5
25
50
100
6
25
50
100
HUMAN AND ECOLOGICAL RISK ASSESSMENT
13
From Table 4, we see that the observed values of ARMB by FR are very close to those by
MA for all values of n and BMR in Scenario 1. So, FR and MA have comparable performance
with respect to their observed ARMB values for Scenario 1. Note that the chosen curve for
generating data sets in Scenario 1 has very slowly increasing probabilities of adverse events
at dose levels with R(d1) D 0, and R(d4) D 0.0224. So, it can be concluded that FR and MA
provide comparable performance for extremely shallow dose–response curves. If we move
toward less shallow dose–response curves (Scenarios 2–6), we see that the observed values of
ARMB by FR are smaller than those by MA. For example, in Scenario 4 with BMR D 0:01,
the values of ARMB are 0.5322, 0.4079, and 0.2973 by method FR, and 5.2402, 7.9412, and
11.8273 by method MA for sample sizes n D 25, 50, and 100, respectively. Also, for the same
Scenario with BMR = 0.1, the values of ARMB are 0.3872, 0.2450, and 0.2243 by method FR,
and 0.7904, 1.3338, and 2.4060 by method MA for sample sizes n D 25, 50, and 100, respectively. This shows that the values of ARMB by FR are smaller than those of ARMB by MA
for these cases. Hence, FR performs better than MA with respect to their observed ARMB
values for Scenarios 2–6. Also, it is noted that the values of ARMB by MA increase with sample sizes for some scenarios. This shows that the estimates by MA are asymptotically biased
when the true models are not included in the model space of MA to estimate BMD.
Comparison among four BMDL estimation methods
Here, we conduct simulation studies to compare four methods of estimating BMDL using
ML estimates (ML), likelihood ratio test (LR), score test (ST), and bootstrap technique (BT)
with respect to their observed coverage probabilities for small samples. We choose similar
simulation set-up considered in the section “Comparison between FR and MA” with the
experimental design d D [0.0, 0.25, 0.5, 1.0]0 and scenarios given in Table 2 to generate data
sets. We also consider two values of BMR D 0:01 & 0.1, and three sample sizes n D 25, 50,
and 100 for each scenario. For each simulation set up, we generate l D 1000 data sets which
are also screened by the Kendall correlation test (Kendall 1955) as discussed in the section
“Comparison between FR and MA”.
The simulated data sets are used to estimate 95% BMDL using the four methods
ML, LR, ST, and BT. After estimating BMDL using a method, we find an approximate
value of coverage probability given by Nl l , where Nl is the number of times the estimated
values of BMDL are less than or equal to BMD out of l datasets generated. The coverage
probabilities by four methods ML, LR, ST, and BT for each simulation set-up are reported
in Table 5.
From Table 5, we see that the observed coverage probabilities by LR and ST are greater
than 0.95 for all scenarios and BMR values with all sample sizes. The method BT fails to provide the expected coverage probabilities for Scenario 2 with n D 25, 50 and Scenarios 5 and 6
for all sample sizes when BMR = 0.01. The observed coverage probabilities by ML also
exceed the expected probability 0.95 for all the cases except for Scenario 5 with n D 50, 100,
when BMR = 0.1. We also studied the observed average lengths of the one-sided confidence
d
intervals (average of [BMD-BMDL])
by all four methods. it was observed that BT provides
d
the smallest values and ML and ST provide largest values for the average of (BMD-BMDL)
for all the cases considered. Hence, we conclude that LR is best among all the methods of
estimating BMDL with respect to coverage probabilities and lengths of the confidence
intervals.
14
I. DAS
Table 5. Comparison of four methods of estimating BMDL with respect to their coverage probabilities for
different simulation set-up. The observed coverage probabilities of ML, LR, ST, and BT are given against
the column ML, LR, ST, and BT respectively.
Methods
Methods
Sc
n
BMR
ML
LR
ST
BT
Sc
n
BMR
ML
LR
ST
BT
1
25
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.95
0.68
1.00
0.91
1.00
0.98
1.00
1.00
1.00
1.00
1.00
1.00
1.00
4
25
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
0.01
0.1
1.00
1.00
1.00
1.00
1.00
0.99
1.00
0.97
1.00
0.94
1.00
0.91
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
0.99
0.99
0.99
0.98
0.97
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.98
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.88
1.00
0.93
1.00
0.95
1.00
0.76
1.00
0.77
1.00
0.83
1.00
50
100
2
25
50
100
3
25
50
100
50
100
5
25
50
100
6
25
50
100
Computer programs used for computations
A language and environment for statistical computing called R (R Core Team 2017) are used
for all computations. The MLEs are computed by minimizing the negative of the log-likelihood using an R function “optim.” The Fisher information matrix is found by computing
gradient of a function using an R-package “numDeriv” (Gilbert and Varadhan 2016). An Rpackage “Kendall” (McLeod 2011) is used for computing Kendall rank correlation of a simulated data. An R-function “optimize” is also used to compute a minimum of a function
within a restricted domain.
Conclusions
For accounting model uncertainty in BMD estimation, a family of response functions for
binary response models is used to develop a method for estimating BMD. The family of
response functions provides local orthogonality between response and regression parameters to reduce the variance inflations of the estimated regression parameters. An infinite
number of response functions including some standard response functions are the members
of the family. For accounting model uncertainty in BMD estimation, the family of response
functions provides a better approach than model averaging method as the model space considered in MA to get model-averaged estimate usually contains only a finite number of models. Methods of estimating BMDL are also provided using the family of response functions.
The proposed method is illustrated by an example with a real data set observing that FR is
consistent with the existing results in the literature. By comparing FR with MA using simulation studies considering different simulation scenarios, we see that FR outperforms MA for
most of the scenarios. Simulation studies are also conducted to compare the four methods of
HUMAN AND ECOLOGICAL RISK ASSESSMENT
15
estimating BMDL and we see that the LR is best among the four methods of estimating
BMDL considering both the coverage probability as well as the length of the confidence
intervals.
If the true model generating a data cannot be approximated by any member of the proposed infinite family of models, then the results using the proposed method may not be
good. For this, we need to diagnose the model fitting with respect to some measures such as
deviance. In such cases, a non-parametric approach may be a better alternative. There are
other methods exist in the literature using Bayesian and non-parametric approach for
accounting model uncertainty in BMD estimation. However, the frequentist methods are
usually easy to implement and require less time for computations than the non-frequentist
approaches. We have compared FR with MA as both the methods are based on frequentist
approach to deal with the model uncertainty problems. In future, the proposed method may
be compared with other approaches such as non-parametric, Bayesian in estimating BMD to
test the performance of FR.
Funding
This work was supported by the University Grants Commission (F.NO.5-181/2016(IC)).
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