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Research Article
Received 30 April 2013, Accepted 1 November 2013 Published online 3 December 2013 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/sim.6052
Model-based dose finding under model
uncertainty using general
parametric models
José Pinheiro,
a
Björn Bornkamp,
b
*
†
Ekkehard Glimm
b
and
Frank Bretz
b
The statistical methodology for the design and analysis of clinical Phase II dose-response studies, with related
software implementation, is well developed for the case of a normally distributed, homoscedastic response consid-
ered for a single timepoint in parallel group study designs. In practice, however, binary, count, or time-to-event
endpoints are encountered, typically measured repeatedly over time and sometimes in more complex settings
like crossover study designs. In this paper, we develop an overarching methodology to perform efficient multi-
ple comparisons and modeling for dose finding, under uncertainty about the dose-response shape, using general
parametric models. The framework described here is quite broad and can be utilized in situations involving for
example generalized nonlinear models, linear and nonlinear mixed effects models, Cox proportional hazards
models, with the main restriction being that a univariate dose-response relationship is modeled, that is, both
dose and response correspond to univariate measurements. In addition to the core framework, we also develop a
general purpose methodology to fit dose-response data in a computationally and statistically efficient way. Sev-
eral examples illustrate the breadth of applicability of the results. For the analyses, we developed the R add-on
package DoseFinding, which provides a convenient interface to the general approach adopted here. Copyright
© 2013 John Wiley & Sons, Ltd.
Keywords:
binary data; dose-response; count data; MCP-Mod; parametric; time-to-event data
1. Introduction
Finding the right dose is a critical step in pharmaceutical drug development. The problem of selecting
the right dose or dose range occurs at almost every stage throughout the process of developing a new
drug, such as in microarray studies [1], in vitro toxicological assays [2], animal carcinogenicity studies
[3], Phase I studies to estimate the maximum tolerated dose [4], Phase II studies covering dose ranging
and dose-exposure-response modeling [5], Phase III studies for confirmatory dose selection, and post-
marketing studies to further explore dose-response in specific subgroups defined by region, age, disease
severity, and other covariates [6–8]. In recent years, considerable effort has been spent on improving the
efficiency of dose finding throughout drug development [9–11]. Despite these efforts, however, improper
dose selection for confirmatory studies, due to lack of sufficient dose-response knowledge for both effi-
cacy and safety at the end of Phase II, remains a key driver of the ongoing pipeline problem experienced
by the pharmaceutical industry [12, 13].
Statistical analysis methods for late development dose finding studies can be roughly categorized into
modeling approaches to characterize the dose-response relationship [14, 15] and multiple test proce-
dures for dose-response signal detection [16] or confirmatory dose selection [17,18]. Hybrid approaches
combine aspects of multiple testing with modeling techniques to overcome the shortcomings of either
approach [19, 20]. More recently, considerable research has focused on extending these methods to
a
Janssen Research & Development, Raritan, NJ, U.S.A.
b
Novartis Pharma AG, Basel, Switzerland
*Correspondence to: Björn Bornkamp, Novartis Pharma AG, CH-4002 Basel, Switzerland.
†
E-mail: bjoern.bornkamp@novartis.com
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Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
J. PINHEIRO ET AL.
response-adaptive designs that offer efficient ways to learn about the dose-response through repeated
looks at the data during an ongoing study [21–24].
Most statistical methodology for dose-response analysis has been introduced in the context of a nor-
mally distributed, homoscedastic endpoint, with a parallel group design, in which each patient receives
only one treatment. In practice, however, one often faces more complex study designs (e.g., cross-over
designs) or situations where a heteroscedastic or non-normally distributed endpoint is measured (e.g.,
binary, count, and more rarely time-to-event data). One approach is to extend the existing methodo-
logy using generalized nonlinear models or generalized nonlinear mixed-effects models. However, these
extensions are typically specific to each new situation. In addition, general purpose software for these
types of models is not available, and a case-by-case implementation requires a major coding effort. In
this paper, we describe an overarching hybrid approach, combing multiple comparisons and modeling, to
the analysis of dose-response data for general parametric models and general study designs, that allows
for a straightforward computer implementation.
More specifically, we extend the Multiple Comparison Procedures and Modeling (MCP-Mod)
approach [20], which was originally introduced for normal, homoscedastic, independent data. This
approach provides the flexibility of modeling for dose-response and target dose estimation, while
accounting for model uncertainty through the use of multiple comparison and model selection/averaging
procedures. The approach can be described in two main steps (Figure 1). At the trial design stage, the
clinical team needs to decide on the core aspects of the trial design, as in any other trial. Specific for
MCP-Mod is that a candidate set of plausible dose-response models is pre-specified at this stage, based
on available pharmacological data/dose-response information from similar compounds and so on. This
gives rise to a set of optimal contrasts used to test for the presence of a dose-response signal consistent
with the corresponding candidate models. In case of large model uncertainty, this candidate set should
be chosen to cover a sufficiently diverse set of plausible dose-response shapes.
The trial analysis stage consists of two main steps: the MCP and the Mod steps. The MCP step focuses
on establishing evidence of a drug effect across the doses, that is, detecting a statistically significant dose-
response signal for the clinical endpoint and patient population investigated in the study. This step will
typically be performed using a multiple contrast test, adjusting for the fact that multiple candidate dose-
response models are being considered. If a statistically significant dose-response signal is established,
one proceeds with determining a reference set of significant dose-response models by discarding the non-
significant models from the initial candidate set. Out of this reference set, a best model (or a weighted
average of models, when using model averaging) is selected for dose estimation in the last stage of
the procedure. The selected dose-response model is then employed to estimate target doses using inverse
regression techniques and possibly incorporating information on clinically relevant effects. The precision
of the estimated doses can be assessed using, for example, bootstrap methods.
The original MCP-Mod method proposed by [20] was intended to be used with parallel group designs
with a normally distributed, homocedastic response. Although that covers a good range of dose finding
designs utilized in practice, the restrictions of the original method create serious practical limitations
Figure 1. Overview of MCP-Mod approach.
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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J. PINHEIRO ET AL.
to its wider use in drug development. For example, binary, count, and time-to-event endpoints are not
covered by the original MCP-Mod formulation. Likewise, longitudinal patient data, like in crossover
studies, with its implicit within-patient correlation, cannot be properly analyzed with the original for-
mulation of the MCP-Mod methodology. In what follows, we extend the MCP-Mod methodology, to
perform dose-response modeling and testing in the context of general parametric models and for general
study designs, in a similar way as [25] extended certain simultaneous inference procedures. Note that,
even though we focus on extending the MCP-Mod approach, the results of this paper remain valid, in
particular, if only a multiple comparison or a modeling approach is to be applied.
We introduce the proposed extension in Section 2. In Section 3, we describe an efficient approach for
dose-response model fitting, which is evaluated in terms of asymptotic considerations and simulations.
In Section 4, we use case studies to illustrate the proposed methodology and its implementation with
the R add-on package DoseFinding [26]. In summary, the method is illustrated for binary, overdis-
persed count, time-to-event data (based on the Cox PH model), and longitudinal data with patient specific
random effects.
2. Generalized MCP-Mod
This section describes an extension of the original MCP-Mod approach that can be used whenever the
response variable can be described by a parametric model in which one of the parameters captures the
dose-response relationship. We show how the basic ideas and concepts of the original MCP-Mod can be
extended to this setting.
2.1. Basic concepts, notation, and assumptions
In the original description of MCP-Mod, the expected value of the response is utilized as the parame-
ter capturing the dose-response relationship. The key idea of the extended version of MCP-Mod is to
decouple the dose-response model from the expected response, focusing, instead, on a more general
characterization of dose-response via some appropriate parameter in the probability distribution of the
response. To be more concrete, assume that the probability model assumed for the patient responses y
includes parameters
x
1
;:::;
x
K
capturing the dose-response effect for the doses x
1
;:::;x
K
.Inaddi-
tion, this probability model might depend on nuisance parameters and covariates ´. The key features of
the extended version of MCP-Mod can then be formulated with respect to
x
, such as the following:
accounting for uncertainty in a dose-response model for
x
via a set of candidate dose-response
models,
testing of dose-response signal via contrasts based on dose-response shapes,
model selection via information criteria, or model averaging to combine different models, and
dose-response estimation and dose selection via modeling.
Note that this formulation covers situations where there are repeated measurements per patient (an
example will be presented in Section 4.1) or each patient receives different doses in a sequence. The
main restriction imposed by the proposed formulation is that a univariate dose-response relationship is
modeled using
x
, that is, dose and response need to correspond to univariate measurements.
Because all dose-response information (at dose x) is assumed to be captured by
x
, the interpretabil-
ity of this parameter is critical for communicating with clinical teams, choosing candidate dose-response
shapes and specifying clinically relevant effects for target dose estimation. To better illustrate this point,
consider a time-to-event endpoint that is assumed to follow a Weibull distribution. The Weibull distribu-
tion is typically parameterized by a scale parameter and shape parameter ˛, neither of which has an
easy clinical interpretation. For the purpose of MCP-Mod, modeling the model could be reparameter-
ized in terms of the median time to event D Œlog.2/
1=˛
= and ˛, and then one would use this as the
dose-response parameter.
When it comes to modeling the dose-response, one assumes that the dose-response parameters
x
at the different doses x are related through a dose-response model f.x;/,thatis,
x
D f.x;/.All
dose-response models of interest in this paper can be expressed as
f.x;/ D
0
C
1
f
0
x;
0
; (1)
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Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
J. PINHEIRO ET AL.
where f
0
denotes the so-called standardized model function and
0
its parameter vector. For example,
for f
0
.x;
0
/ D x, one obtains the linear model f.x;/ D
0
C
1
x and for f
0
.x;
0
/ D x=.x C
0
/
the Emax dose-response model, where
0
then denotes the ED
50
parameter; see [20,27] for more exam-
ples. Additive covariates (i.e., covariates on the intercept) may also be included in (1), at the model fitting
stage, but we leave them out for now, to keep the notation simple.
Assume that K distinct doses x
1
;:::;x
K
are utilized in a trial, with x
1
denoting placebo. Assume
further that M candidate models f
1
;:::;f
M
have been chosen to capture the model uncertainty
about
x
. We define the dose-response parameter vector associated to candidate model m as
m
D
m;x
1
;:::;
m;x
K
; where
m;x
i
D f
m
.x
i
; /; i D 1;:::;K;mD 1;:::;M.
For the purpose of obtaining estimates of the dose-response, we initially consider an analysis of vari-
ance (ANOVA) parametrization for the dose-response parameter
x
i
;iD 1;:::;K. That is, we allow
a separate parameter
x
i
to represent the dose-response at each dose level. Let
b
denote the vector of
estimated dose-response parameters under the ANOVA parametrization, obtained using the appropriate
estimation method for the general parametric model for example via maximum likelihood (ML), gen-
eralized estimating equations or partial likelihood. Note that these type of ANOVA estimates are easily
available from standard statistical software packages. The key assumption underpinning the generalized
version of MCP-Mod is that
b
has an approximate distribution N
.
; S
/
; where S denotes the variance-
covariance of
b
. The matrix S may or may not depend on x and on additional nuisance parameters that
need to be estimated to derive an estimate
b
S of S . The estimation of is performed in a separate second
stage based on
b
and an estimate
b
S of its covariance matrix S . Section 3 explains this in detail.
2.2. Implementation of the multiple comparison procedure step
The MCP step consists of testing a set of contrasts, representing the candidate models for the dose-
response relationship
x
. To that end, one needs to specify a set of candidate models shapes (e.g.,
linear, Emax, logistic, and quadratic) and derive optimal model contrasts c
op t
1
;:::;c
op t
M
representing
these models. These contrasts are optimal in the sense of optimizing the noncentrality parameter for the
underlying single contrast test provided a particular, pre-specified dose-response parameter vector
m
is
the true dose-response parameter.
It can be shown that the optimal contrast for testing the hypothesis of a flat dose-response profile with
maximal power for a single candidate model shape
m
is given by
c
op t
/ S
1
0
m
00
m
S
1
1
1
0
S
1
1
; (2)
where
0
m
D
f
0
m
x
1
;
0
;:::;f
0
m
x
K
;
0
0
(see Appendix A for details). For convenience, we nor-
malize the contrast coefficients such that jjc
op t
jj D 1. As seen from Equation (2), the optimal model
contrasts only depend on the standardized dose-response model function (not the complete model func-
tion), and these are determined solely by the parameters
0
. Because of this, at the planning stage,
guesstimates for this parameter are needed to derive the optimal model contrasts. That means, for exam-
ple, for the Emax model, a guesstimate of the ED
50
parameter is needed. Further details on and strategies
for the specification of candidate models and corresponding guesstimates are given in [28].
Once data are available, these optimal contrasts (pre-specified at design stage) are then applied to
the previously described ANOVA estimates
b
.LetC
op t
D
c
op t
1
c
op t
M
represent the matrix con-
taining the optimal contrasts. The contrast estimates are then given by
C
op t
0
b
, being asymptotically
normally distributed with mean
C
op t
0
and estimated covariance matrix
C
op t
0
b
SC
op t
: It follows
that the asymptotic z-test statistic for the m
th
candidate model hypotheses H
0
W
c
op t
m
0
D 0 vs.
H
0
W
c
op t
m
0
>0is given by ´
m
D
c
op t
m
0
b
=
h
C
op t
0
b
SC
op t
i
1=2
m;m
; with ŒA
m;m
denoting the m
th
diagonal element of the matrix A. The test statistic used for establishing an overall dose-response signal
is the maximum ´
.M /
D max
m
´
m
of the individual model test statistics. Critical values for tests with
(asymptotically) exact level ˛ can be derived from the joint distribution of ´ D
.
´
1
;:::;´
M
/
, which can
be obtained from the joint distribution of the contrast estimates given previously and using
P
´
.M /
>q
D 1 P
´
.M /
6 q
D 1 P
.
´ 6 q1
/
: (3)
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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J. PINHEIRO ET AL.
The last term in Equation (3) is an integral over the multivariate normal distribution, which can be
evaluated using numerical integration. The mvtnorm package in R includes functions to numeri-
cally calculate quantiles and also probabilities of the underlying multivariate normal distribution ([29]).
Multiplicity adjusted p-values for the individual model contrast tests can be derived similarly.
If the optimal contrasts and the critical values also depend on S , one needs guesstimates for nuisance
parameters appearing in the covariance matrix at the planning stage, as well. This is a difference com-
pared with the normal homoscedastic setting without covariates, where S is proportional to a diagonal
matrix with the reciprocal of the group sample sizes on the diagonal. Once data become available, the
z-statistics for the model contrast tests are calculated and their maximum used for the dose-response test.
At this stage, one can obtain the estimated
b
S matrix from the observed data and use this to recalculate
optimal contrasts and the critical value for the test. Note that the guesstimates for the parameters
0
are
not recalculated on the basis of the observed data, as this would lead to a serious type I error rate infla-
tion. For the purpose of the multiple contrast test, the guesstimates pre-specified at the planning stage
for
0
should be used to derive the optimal contrasts.
2.3. Implementation of the Mod step
Once a dose-response signal is established, one proceeds to the Mod step, fitting the dose-response pro-
files and estimating target doses based on the models identified in the MCP step. There are many ways
to fit the dose-response models to the observed data, including approaches based on maximizing the
likelihood (ML) or the restricted likelihood. However, a direct ML approach requires the derivation of
the likelihood in every specific case, with a considerable amount of model-specific coding involved. We
therefore suggest an alternative two-stage approach to dose-response model fitting that utilizes genera-
lized least squares. This approach is described in more detail in Section 3. It relies on asymptotic results
but has the appeal of being of general purpose application, as it depends only on
b
and
b
S . In addition,
as shown later in the simulation study, its finite and large sample properties are similar to those of the
approaches based on the full likelihood.
Model selection can be based on the maximum z-statistics, or using information criteria, such as the
AIC or the BIC. Estimation of target doses is performed on the basis of the selected fitted model for the
dose-response parameter [27].
Alternatively, model averaging approaches can be used to avoid the need to select a single model. The
individual AIC and BIC values for the candidate models with significant contrast test statistics determine
the model averaging weights [30]. This applies both to dose-response and target dose estimation.
The generalization of MCP-Mod described in this section has focused on testing and estimation asso-
ciated with the full dose-response profile, that is, including the response at placebo and the entire dose
range utilized in the study. In practice, there are cases in which inference might focus on the placebo-
adjusted dose-response (or more generally a control-adjusted response), that is, the dose-response with
the placebo or control effect subtracted f
C
.x; / D f.x;/ f.0;/. This could become relevant, for
example, if covariates are added to the placebo response
0
in (1). In the context of time-to-event data,
the focus on placebo-adjusted dose-response will occur naturally when modeling the hazard ratio as a
function of dose. As shown in Supporting information S.1, all results presented in this section, including
the derivation of optimal contrasts, as well as the model fitting results described in the next section, apply
equally in the context of placebo-adjusted dose-response.
3. Nonlinear dose-response model fitting using a two-stage, generalized least
squares approach
In this section, we describe an efficient methodology for fitting nonlinear dose-response models that can
be used for the Mod step of the MCP-Mod methodology. The fitting is carried out in two stages: First, the
ANOVA estimates
b
and
b
S introduced in Section 2 are obtained. Second, the parameter is estimated
by fitting the dose-response model to the ANOVA estimates from the first step using a generalized least
squares (GLS) estimation criterion. In Section 3.1, we establish consistency and asymptotic normality
of this estimate. In Section 3.2, we assess the accuracy of the asymptotic results via a simulation study.
3.1. Asymptotic results
Assume that we have dose-response estimates
b
D
b
x
1
;:::;b
x
K
0
obtained from an ANOVA-type
parameterization of , which allows for unrelated responses parameters at each of the K dose levels
1650
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
J. PINHEIRO ET AL.
(Section 2.1). These estimates are assumed to be asymptotically multivariate normal distributed with a
covariance matrix S consistently estimated by
b
S . The estimates
b
and
b
S are easily available from stan-
dard statistical packages (see Appendix B for examples using R). Next, we fit the nonlinear dose-response
model f.x;/ to the estimates
b
by minimizing the GLS criterion
b
‰./ D
.
b
f
.
x;
//
0
A
n
.
b
f
.
x;
//
(4)
with respect to to obtain the estimates
b
. In Equation (4), we have f .x; / D .f .x
1
; /;:::;
f.x
K
; //
0
and A
n
denotes a symmetric positive definite matrix. We assume that A
n
P
! A,where
P
! denotes convergence in probability. In practice, we will always use A
n
D
b
S
1
, but this would
unnecessarily restrict the discussion at this stage.
Let
0
denote the true value of the parameter . In the Supporting information S.2, we show that,
under mild regularity conditions,
b
is a consistent estimator of
0
,thatis,
b
P
!
0
. Furthermore, we
have the asymptotic multivariate normality
p
a
n
b
0
d
! N
0; B
.
0
/
0
M .
0
/B.
0
/
; (5)
where M ./ D F ./
0
A†AF ./ and B./ D .F ./
0
AF .//
1
, F ./ denotes the d k matrix
of partial derivatives
df . x
i
;/
d
h
(i D 1;:::;k, h D 1;:::;d), a
n
a nondecreasing sequence of values
increasing to infinity as the sample size n goes to infinity and a
n
S
P
! †.
Selecting A
n
D
1
a
n
S
1
would be a good choice, if S were known. In this case, previous formulas
simplify to
p
a
n
.
b
0
/
d
! N
0;
F .
0
/†
1
F .
0
/
0
1
, and the optimization criterion would be
b
‰./ D .
b
f .x; //
0
S
1
.
b
f .x; //. Because S is not known, we will typically use a consistent
estimate
b
S of it to calculate the estimate
b
by minimizing
b
‰./ D .
b
f .x; //
0
b
S
1
.
b
f .x; //; (6)
with respect to .
Note that this two-stage, GLS estimate is quite similar to the ML estimate: For normal homoscedas-
tic data, both approaches lead to exactly the same estimate, whereas, for example, in generalized linear
model settings, the two estimators have the same large-sample variance (see Chapter 4.3 in [33]).
The computational advantage of using this two-stage approach is that the target function in (6)
that is optimized numerically is low dimensional: The dimension is equal to the number of different
dose levels, and the target function can thus be evaluated quite efficiently, whereas the target function
in a full likelihood approach depends on the complete data set. This difference in speed becomes
relevant in clinical trial settings, as often extensive clinical trial simulations are used to evaluate
proposed study designs. Another advantage of the proposed method is its broad applicability to general
parametric models.
Model selection criteria are generally defined as 2 log.L/Cdim./,whereL denotes the likelihood
function evaluated at the ML estimate and a penalty for the number of parameters, which depends on
the model selection criterion. One approach to model selection is hence to use
b
‰
b
C dim./ to
compare different dose-response models fitted based on the same
b
and S . This criterion is motivated
by the fact that, for normally distributed homoscedastic data without covariates, these two approaches
are equivalent in terms of selecting the same model: The likelihood function can be split into the sum of
the deviation between the observed data and
b
and the deviation between
b
and f .x;
b
/. The deviation
of the individual data and
b
is identical across the different dose-response models, so that the criterion
only varies with the deviation between
b
and f .x;
b
/, which, in case of normal data, is equal to
b
‰.
b
/.
In situations beyond the normal case, both approaches might lead to slightly different results. However,
as outlined in the Supporting information S.3, the term (6) is roughly proportional to
2log-likelihood
of the ML estimate of , when discarding the contribution of the first stage ANOVA-type fit (which
is equal for all dose-response models considered). Hence, both approaches will lead to very similar
results also in non-normal situations. Subsequently, D 2 will be used, and we refer to this criterion
as gAIC.
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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J. PINHEIRO ET AL.
In the next subsection, we investigate the properties of the proposed asymptotic approximations via
simulations before illustrating it with applications in Section 4.
3.2. Simulations
In this section, we evaluate the asymptotic performance of the approximations provided in Section 3.1
for fitting a single nonlinear dose-response model. More specifically, we compare the proposed methods
with more traditional ML estimation by evaluating the dose-response estimation accuracy using simula-
tions. In addition, we assess the coverage probability of the resulting confidence intervals for the model
parameter .
3.2.1. Design of simulation study. Throughout the simulations, we assume five active dose levels 0.05,
0.2, 0.5, 0.8, 1 plus placebo. We investigate equal group sample sizes of 15, 30, 50, 100, 300, and 1000
patients for each dose. The lower range of the investigated sample sizes is realistic for typical Phase II
dose-response studies, whereas the sample sizes of 300 and 1000 are included to assess the asymptotic
behavior.
We investigate three types of data: binary data, overdispersed count data using a negative binomial
model, and time-to-event data using a Cox PH model for estimation. Regarding dose-response models,
we will utilize an Emax model f.x;/D
0
C
1
x=.
2
Cx/, a quadratic model f.x;/D
0
C
1
xC
2
x
2
and an exponential model f.x;/ D
0
C
1
.exp.x=
2
/ 1/. In the simulations, we set
2
D 0:05 for
the Emax model,
2
D 0:2 for the exponential model and
1
=
2
D5=8 for the quadratic model (see
Figure 2 for the underlying model shapes). The remaining parameters
0
and
1
are chosen such that
the power for testing the dose with the maximum treatment effect against placebo at the 5% one-sided
significance level is 80% for 30 patients per group. This ensures a realistic range of sample sizes (in
terms of the signal to noise ratio) is investigated in the simulations.
Table I summarizes the three dose-response model specifications for each data type. For binary data,
Table I gives the the mean on the logit scale. For count data, the logarithm of the mean is as specified
in Table I, and the overdispersion parameter is 1. For time-to-event data, we use an exponential distri-
bution for data generation with the log-means specified in Table I and where observations larger than
10 are censored. The mean in the placebo group is 1, so that the log-mean is equal to the difference in
log-hazard rates. The Cox PH model is formulated relative to the control group, so that, in this case, the
Dose
Standardized Response
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
exponential
0.0 0.2 0.4 0.6 0.8 1.0
emax
0.0 0.2 0.4 0.6 0.8 1.0
quadratic
Figure 2. Dose-response models used for simulation.
Table I. Dose-response model parameters .
0
;
1
;
2
/ used for simulation.
Data type Quadratic Emax Exponential
Binary .1:734; 4:335; 2:7094/ .1:734; 1:8207; 0:05/ .1:734; 0:01176; 0:2/
Count .2; 2; 1:25/ .2; 0:84; 0:05/ .2; 0:005427; 0:2/
Time-to-event .0; 1:8876; 1:1797/ .0; 0:7928; 0:05/ .0; 0:005122; 0:2/
Normal .0; 2:61; 1:633/ .0; 1:097; 0:05/ .0; 0:007089; 0:2/
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placebo parameter is set to 0 when estimating the dose-response model. In addition, we include normally
distributed data with D 1 as a benchmark comparison, because in this case, the two-stage, GLS and
ML estimates coincide.
The two-stage approach from Section 2 is employed as follows: (i) an ANOVA-type model is fit to the
data by using either a generalized linear model (binary and count data) or a Cox PH model (time-to-event
data) with ‘dose’ treated as a factor and (ii) a dose-response model is fit to the resulting dose-response
estimates obtained via GLS (6). In the simulations, we compare this approach to nonlinear ML (binary
and count data) and maximum partial likelihood (time-to-event data) estimation using the same link
functions as earlier. For the model fitting step, we assume lower and upper bounds for the
2
parameter
of 0.001 and 5 for the Emax and 0.05 and 5 for the exponential dose-response model.
In addition, we assess the coverage probability for three different methods of constructing confi-
dence intervals for the dose-response model parameter . First, we use the GLS (6) together with the
asymptotic results from Section 3.1 (denoted in the succeeding text as GLS). Second, we use parametric
bootstrap confidence intervals by sampling from the multivariate normal distribution underlying the first
stage ANOVA-type estimates and then fitting the nonlinear model to each of these samples using the
GLS criterion from (6). The bootstrap confidence intervals are then calculated by taking the 5% and
95% quantiles of the observed sample. For each simulation, we used 500 bootstrap samples (denoted
in the succeeding text as GLS-B). Finally, we use the ML fits and calculate confidence intervals based
on the inverse of the Hessian matrix and the usual asymptotic normality assumptions (denoted in the
succeeding text as ML).
3.2.2. Results of simulation study. Simulations were run with 2000 replications, using the
DoseFinding package version 0.9–1. To illustrate the performance of the GLS and ML methods
with regard to dose-response estimation, we calculated the root mean squared estimation error averaged
over the available doses
r
1
6
P
x2D
.f .x;
b
/ f.x;//
2
,whereD Df0; 0:05; 0:2; 0:6; 0:8; 1g. Figures 4–6
in the Supporting information S.4 display the results. It is evident from these plots that both approaches
can hardly be distinguished in terms of the mean squared error, indicating that, in terms of dose-response
estimation, both methods perform almost identically, even for small sample sizes.
Next, we assess the coverage probability for the three different methods described at the end of
Section 3.2.1. Figure 3 displays the results only for the count data case, because the results for the
normal, binary and time-to-event data are nearly identical (see Figures 1–3 in the Supporting infor-
mation S.4). We observe that the asymptotic confidence intervals for the GLS and ML methods perform
very similarly for all three models under investigation (Emax, exponential, and quadratic). Both methods
achieve the nominal 90% coverage probability fairly well for the quadratic model, even for small sample
sizes. For the Emax model, the nominal coverage probability is achieved at roughly 50–100 patients per
group, whereas for the exponential model the coverage probability is achieved only at very large sample
sizes (the poor performance of standard asymptotic confidence intervals for nonlinear regression models
even for moderate sample sizes is well-known, see for example, [34]). The reason for the better perfor-
mance under the quadratic model is the fact that it is linear in the model parameters. One reason for why
the confidence intervals perform worse for the exponential model than for the Emax model might be that
the dose design used in the simulations allows an easier identification of the model parameters under the
Emax model, because there are more dose levels in the lower part of the dose range than the upper part.
A way to improve the asymptotic normal confidence intervals slightly for the nonlinear models is to use
transformations, for example, log transformations for positive parameters, this was not further pursued
in this simulation.
The parametric bootstrap approach GLS-B achieves the 90% nominal coverage probability fairly well
for all three dose-response models, even at sample sizes as small as 30 patients per group. The GLS-B
method thus performs always at least as well as the GLS and ML methods, and the general recommen-
dation is to use this in case of small sample sizes. The approach is computationally more expensive,
as it requires repeated fitting of the dose-response models. But the bootstrap based on the GLS two-
stage fitting is computationally much more efficient than a traditional bootstrap approach based on ML:
The GLS two-stage approach only depends on the ANOVA-type estimates and not the individual obser-
vations, which makes evaluation of
b
‰./ computationally much cheaper than evaluation of the full
likelihood function.
In summary, we conclude that both the GLS and ML methods perform similarly under the different
dose-response shapes, sample sizes, and data types investigated in the simulation study. This conclusion
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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J. PINHEIRO ET AL.
n
Coverage Probability
0.80
0.85
0.90
0.95
1.00
15 30 50 100 300 1000
emax
GLS
exponential
GLS
15 30 50 100 300 1000
quadratic
GLS
emax
ML
exponential
ML
0.80
0.85
0.90
0.95
1.00
quadratic
ML
0.80
0.85
0.90
0.95
1.00
emax
GLS−B
15 30 50 100 300 1000
exponential
GLS−B
quadratic
GLS−B
theta0 theta1 theta2
Figure 3. Empirical coverage probability (based on 2000 simulations) of 90% confidence intervals.
holds both for the coverage probabilities, as well as for the average estimation error. As mentioned pre-
viously, however, the GLS method is very general and computationally more efficient, which facilitates
the usage of computationally expensive techniques such as the bootstrap approach.
4. Applications
Two examples on handling binary and placebo-adjusted data in the general framework presented in this
paper and using the DoseFinding R package are presented in Appendix B. In what follows, we will
concentrate on illustrating the general method for a longitudinal data analysis.
4.1. Longitudinal modeling of neurodegenerative disease
This example refers to a Phase II clinical study of a new drug for a neurodegenerative disease. The state
of the disease is measured through a functional scale, with smaller values corresponding to more severe
neurodeterioration. It is believed that the natural disease progression decreases this functional scale lin-
early (at least over the 1-year time-frame considered in this study). The goal of the drug is to modify
the disease, by reducing the rate of disease progression, which is measured by the linear slope of the
functional scale over time.
The trial design includes placebo and four doses: 1, 3, 10, and 30 mg, with balanced allocation of 50
patients per arm. Patients are followed up for 1 year, with measurements of the functional scale being
taken at baseline and every 3 months thereafter. The study goals are to (i) test the dose-response signal,
(ii) estimate the dose-response, and (iii) select a dose to be brought into the confirmatory stage of the
development program.
The functional scale response is assumed to be normally distributed, and on the basis of historical
data, it is believed that the longitudinal progression of the functional scale over the 1 year of follow up
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J. PINHEIRO ET AL.
Dose (in mg)
Change in slope of disease progression
0.0
0.5
1.0
1.5
2.0
01015202530
emax
01015202530
exponential
01015202530
linear
01015202530
quadratic
55
5
5
Figure 4. Candidate models for neurodegenerative disease example.
can be modeled by a simple linear trend. We use this example to illustrate the application of MCP-Mod
in the context of mixed-effects models.
We consider a mixed-effects model representation for the functional scale measurement y
ij
on patient
i at time t
ij
:
y
ij
D
.
ˇ
0
Cb
0i
/
C
x
i
Cb
1i
t
ij
C
ij
;
Œ
b
0i
;b
1i
0
N
.
0; ƒ
/
and
ij
N
0;
2
; all stoch. independent:
(7)
The dose-response parameter in this case is the linear slope of disease progression
x
.If
x
is repre-
sented by a linear function of dose x, the model in (7) is a linear mixed-effects (LME) model, or else
it becomes a nonlinear mixed-effects (NLME) model. In particular, under the ANOVA parametrization
discussed in Section 2.1, Equation (7) is an LME model with different slopes for each dose.
The research interest in this study focuses on the treatment effect on the linear progression slope. At
t D 1 year, this is numerically equal to the average change from baseline and thus easily interpretable.
At the planning stage of the trial, the following assumptions were agreed with the clinical team for the
purpose of design:
Natural disease progression slope D5 points per year.
Placebo effect D 0 (i.e., no change in natural progression).
Maximum improvement over placebo within dose range D2 points increase in slope over placebo.
Target (clinically meaningful) effect D 1:4 points increase in slope over placebo.
Guesstimates for the variance-covariance parameters were obtained from historical data: var
.
b
0i
/
D
100,var
.
b
1i
/
D 9 corr
.
b
0i
;b
1i
/
D0:5,andvar
ij
D 9. Under these assumptions, it
is easy to see that the covariance matrix of the ANOVA-type estimate
b
of the slopes D
0mg
;
1mg
;
3mg
;
10mg
;
30mg
0
is compound symmetric. With these concrete guesstimates, the
diagonal element is 0:1451 and the off-diagonal element 0:0092.
From discussions with the clinical team, the four candidate models displayed in Figure 4 were
identified:
Emax model with 90% of the maximum effect at 10 mg, corresponding to an ED
50
D 1:11
Quadratic model with maximum effect at 23 mg, corresponding to standardized model parameter
ı D0:022
Exponential model with 30% of the maximum effect occurring at 20 mg, corresponding to a
standardized model parameter ı D8:867
Linear model
For confidentiality reasons, the data from the actual trial cannot be used here, so we utilize a simulated
data set with characteristics similar to the situation with an Emax dose-response profile imposed on the
linear slopes
x
. Figure 5 shows the simulated data per dose, which is available in the DoseFinding
package, in the neurodeg data set.
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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Time (month)
Functional scale
50
100
150
200
024681012
0 mg 1 mg
0 2 4 6 8 10 12 0 2 4 6 8 10 12
3 mg
0 2 4 6 8 10 12 0 2 4 6 8 10 12
10 mg 30 mg
Figure 5. Simulated data for neurodegenerative disease example. Gray lines correspond to individual patient
profiles and black line to loess smoother.
In what follows, we illustrate the individual steps of MCP-Mod along with its implementation in
DoseFinding package (version 0.9–6).
The
b
vector is estimated via an LME fit of the data, which can be performed, for example, using the
lme function in the nlme R package, as illustrated in the succeeding text.
fm <- lme(resp ~ as.factor(dose):time, neurodeg, ~time|id)
muH <- fixef(fm)[-1] # extract estimates
covH <- vcov(fm)[-1,-1]
The estimated slopes are
b
D
.
5:099; 4:581; 3:220; 2:879; 3:520
/
0
with corresponding esti-
mated variance-covariance matrix with compound symmetry structure with diagonal elements 0:149 and
off-diagonal elements 0:0094.
The optimal contrasts corresponding to the candidate models are calculated using the formula in (2),
with
b
S given by the estimated variance-covariance matrix of
b
.TheDoseFinding package includes
the function optContr to calculate optimal contrasts based on (2) as follows.
doses <- c(0, 1, 3, 10, 30)
mod <- Mods(emax = 1.11, quadratic=-0.022, exponential = 8.867,
linear = NULL, doses = doses) # definition of
candidate shapes
contMat <- optContr(mod, S=covH) # calculate optimal contrasts
The MCTtest function in the DoseFinding package implements the optimal model contrast tests
for
b
based on the multiple comparison approach described in Section 2.2 and can be called as follows:
MCTtest(doses, muH, S=covH, type = "general", critV = T,
contMat=contMat)
The Emax, quadratic, and linear model contrasts are all significant with p-values < 0.025, but the
exponential model failed to reach significance. Therefore, the significance of a dose-response signal
is established, and we can move forward to estimating the dose-response profile and the target dose.
Two approaches can be used for model fitting in this example: the two-stage GLS nonlinear dose-
response fitting method described in Section 3 or mixed-effects modeling (linear and nonlinear) incor-
porating a parametric dose-response model for the progression slope
x
. We consider first the two-stage
GLS method, which is implemented in the fitMod function in DoseFinding, illustrated in the call
in the succeeding text for the Emax model. In the call, muH and covH are the estimates obtained from
the lme fit.
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J. PINHEIRO ET AL.
fitMod(doses, muH, S=covH, model="emax", type = "general",
bnds=c(0.1, 10))
One obtains estimates for 5:181, 2.180, and 1.187 for the parameters
0
;
1
;
2
of the Emax model. The
gAIC values (as discussed in Section 3.1) corresponding to the fits for the Emax, quadratic, and linear
models are, respectively, 10.66, 11.07, and 24.22, indicating the better adequacy of the Emax model.
Note that the DoseFinding package also includes an MCPMod function that performs MCTtest,
model selection, and model fitting in one step.
The mixed-effects model fit approach in this case is illustrated in the succeeding text for the Emax
model using the nlme function.
nlme(resp ~ b0 + (e0 + eM
*
dose/(ed50 + dose))
*
time, neurodeg,
fixed = b0 + e0 + eM + ed50 ~ 1, random = b0 + e0 ~ 1 | id,
start = c(200, -4.6, 1.6, 3.2))
For the parameters of the Emax model
0
;
1
;
2
, one obtains estimates for 5:179, 2.181, and 1.199.
The estimated fixed effects from the NLME model are very close to the estimates obtained via the GLS
two-stage method for this data set. The AIC values corresponding to the Emax, quadratic, and linear
models under the mixed-effects model fit are, respectively, 8352.60, 8353.10, and 8365.79 confirming
the Emax as the best fitting model. It is intriguing to see how similar the differences in AIC between the
different models are to the differences in gAIC values.
Estimates for the target dose, that is, the smallest dose producing an effect greater than or equal to the
target value of 1.4, can be obtained with either of the model fitting approaches. The resulting estimated
target doses are 2.13 under the two-stage GLS method and 2.15 under the NLME model. Alternatively,
model averaging methods could have been used to estimate the target dose and the dose-response profile.
5. Conclusions
The extended MCP-Mod methodology, together with its corresponding software implementation in
the DoseFinding package in R, greatly broaden the scope of application of the original MCP-Mod
approach. Most type of endpoints, and associated model-based analyses, utilized in dose finding studies
can be handled in the context of the extended approach. The major restriction of the proposed extension
is that a univariate dose-response relationship is modeled, and only additive covariates (i.e. covariates on
the intercept) can be included into the dose-response relationship.
In addition to the extended MCP-Mod approach, a two-stage estimation approach was introduced,
which allows to fit dose-response data in a computationally efficient way that can be used generically for
a wide range of dose-response modeling situations. From a finite sample and asymptotic viewpoint, it
was shown that this approach and an ML approach based on the full data lead to very similar results. For
clinical trial simulations at the design stage or bootstrap sampling, the two-stage approach has advan-
tages, as it is computationally more efficient. In some situations, however, one might nevertheless opt to
use the full data approach for the final analysis of the trial, for example, when the sample size per dose
gets very small, so that the multivariate normal approximations used in the two-stage approach will get
unreliable.
One interesting aspect of the extended MCP-Mod approach combined with the two-stage GLS esti-
mation is that it also allows to deal with missing data in principled way. Fully saturated analysis of
covariance type longitudinal models (for example, a multivariate normal model in case of continuous
data) are now commonly used as primary analyses in clinical trials to deal with missing values under a
missing at random assumption (see, e.g., [35, 36] and [37]). The MCP-Mod method fitted to estimates
b
and
b
S obtained from such a model (at a particular timepoint of interest) then inherits the missing data
assumptions of the underlying first stage model (see also [38] for a discussion of missing data in the
context of dose-finding trials).
Further extensions of MCP-Mod are possible and of interest in practice. An increasing number
of indications and drugs require treatment regimen selection (e.g., treatment frequency or sometimes
administration route), in addition to the more traditional dose selection. Different approaches can be
considered in the context of MCP-Mod, or its extension, discussed here. One could focus on estimating
an exposure-response relationship, for example, combing dose and regimen into one model covariate.
The much larger number of exposure values, compared with dose levels, could pose a problem for the
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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derivation of optimal model contrasts and for the MCP step, more generally. Dose-time response mod-
eling, where dose and time are the inputs of the nonlinear regression model provide another venue for
extending MCP-Mod. Further research is needed in those topics.
Model-based dose finding methods, such as the extended MCP-Mod, provide better understanding
of the dose-response relationship, generally translating into more accurate dose selection for confirma-
tory trials. Realizing the full potential that these methods have to offer, however, requires changes in
the way Phase II studies are traditionally designed. By and large, dose finding studies are planned as
mini Phase III trials, using hypothesis tests to select the dose, or doses, to bring forward to the confir-
matory stage. Relatively few doses (typically two active treatment arms and placebo) are used in such
Phase II studies, making it hard to entertain any type of modeling. The sample size derivation in this
type of studies is based on power calculations for detecting at least one dose significantly different from
placebo. The resulting number of subjects is typically inadequate for proper estimation of target doses
(and dose-response modeling). A discussion of a different balance in resource allocation between Phases
II and III, taking into account the overall probability of program success, is long overdue. Utilization of
larger number of doses (e.g., 4 or 5), coupled with larger sample sizes, would go along way in enabling
model-based methods to improve dose selection in Phase II and, as a result, the probability of success in
Phase III.
Appendix A: Derivation of optimal contrasts
In this section the closed form solution for the optimal contrasts for the case of a general covariance
structure is derived. Optimality here refers to maximum power of the univariate contrast test, if a speci-
fied mean vector (with corresponding positive definite covariance matrix S ) is true, which means that
the non-centrality parameter
g.c; / D
c
0
p
c
0
Sc
;
needs to be maximized with respect to c, subject to c
0
1 D 0.
Writing C
0
D
1
K1
:
:
:I
K1
, the constrained maximization problem is equivalent to the uncon-
strained maximization of
.Qc
0
C
0
/
2
Qc
0
C
0
SC
0
0
Qc
with respect to
Q
c. This, however, is the solution to the generalized
eigenvalue problem
C
0
0
C
0
0
x DC
0
SC
0
0
x;
see e.g. [39], formula (2.66). As C
0
0
C
0
0
is of rank 1, it has only a single non-zero eigenvalue. Thus,
it is clear that
Q
c D const .C
0
SC
0
0
/
1
C
0
; with const ¤ 0 is the only solution to the generalized
eigenvalue problem. We further note that
Q
c D const .C
0
SC
0
0
/
1
C
0
implies that the optimal contrast
is given by c D C
0
0
Q
c D const S
1
.
0
S
1
1
1
0
S
1
1
1/, which can be verified, for example by using formulas
for the inverse of block partitioned matrices.
From the formula for the optimal contrast it is clear that the optimal solution is invariant with respect
to addition or multiplication of any scalar constant to the vector , which is why one can also use the
standardized mean vectors
0
m
instead of , which then gives the result in formula (2).
Appendix B. Binary and placebo-adjusted data
In this section, we will go through two concrete examples on how to use the DoseFinding R package
to apply MCP-Mod to binary data and placebo-adjusted normal data. Only the required R commands are
given here but not the output.
B.1. Binary endpoint
This example is based on trial NCT00712725 from clinicaltrials.gov. This was a randomized,
placebo-controlled dose-response trial for the treatment of acute migraine, with a total of seven active
doses ranging between 2.5 and 200 mg and placebo. The primary endpoint was ‘being pain free at 2 hours
postdose’, that is, a binary endpoint. The analysis presented here is a post hoc analysis.
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As a reasonable set of candidate models and contrasts, we select four different shapes of the Sigmoid
Emax model f.x;/ D E
0
C E
max
x
h
=
x
h
CED
h
50
, which cover a very wide range of mono-
tonic shapes and a quadratic model to safeguard against the possibility of a unimodal dose-response
relationship. The Mods function is used for that, and one can also plot the candidate shapes.
doses <- c(0,2.5,5,10,20,50,100,200)
models <- Mods(sigEmax = rbind(c(2.5, 1),c(10,1),c(50, 3),c(100,2)),
quadratic = -1/250, doses=doses)
plot(models)
The first stage of the two-step MCP-Mod approach consists of fitting a model with ANOVA-type
parametrization to the data to obtain estimates
b
and its asymptotic covariance matrix S . The logistic
regression model is used here, which means that the candidate models are formulated on the logit scale
(other scales could be used). The ANOVA logistic regression model can be fitted as follows.
## data from NCT00712725 study
dosesFact <- as.factor(doses) ## treat dose as categorical variable
N <- c(133, 32, 44, 63, 63, 65, 59, 58)
##
RespRate <- c(13,4,5,16,12,14,14,21)/N
## fit logistic regression (without intercept)
logfit <- glm(RespRate~dosesFact-1, family = binomial, weights = N)
muHat <- coef(logfit)
S <- vcov(logfit)
Now, all subsequent inference only depends on muHat and S obtained from the logistic regression.
The multiple contrast test from Section 2.2 using optimal trend contrasts can be produced as follows
MCTtest(doses, muHat, S=S, models = models, type = "general")
All contrasts are significant. The modeling step can now be performed using the fitMod function.
Here, for illustration, we fit the Sigmoid Emax model and the quadratic model.
modSE <- fitMod(doses, muHat, S=S, model = "emax", type="general")
modQuad <- fitMod(doses, muHat, S=S, model = "quadratic",
type = "general")
gAIC(modSE);gAIC(modQuad)
A comparison of the gAIC values reveals that the Sigmoid Emax model provides a better fit than the
quadratic model.
Earlier, we performed the different steps of the MCP-Mod procedure separately. One could alterna-
tively have used
MCPMod(doses, muHat, S=S, models=models, type = "general",
Delta = 0.2)
directly.
B.2. Fitting on placebo-adjusted scale
For this example, we use the IBScovars data set from the DoseFinding package, taken from [40].
The data are part of a dose ranging trial on a compound for the treatment of irritable bowel syndrome
with four active doses 1, 2, 3, 4 equally distributed in the dose range Œ0; 4 and placebo. The primary end-
point was a baseline adjusted abdominal pain score with larger values corresponding to a better treatment
effect. In total, 369 patients completed the study, with nearly balanced allocation across the doses.
The endpoint is assumed to be normally distributed, and it is of interest to adjust for the additive covari-
ate gender. While the DoseFinding package can deal with this situation exactly, here we illustrate
Copyright © 2013 John Wiley & Sons, Ltd. Statist. Med. 2014, 33 1646–1661
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J. PINHEIRO ET AL.
using the placebo-adjusted estimates. Note that, in the case of time-to-event data, one would proceed
similarly. Here, we only illustrate fitting an Emax model, using the MCTtest and MCPMod functions
is analogous to the calls in Section B.1, but using the additional argument placAdj = TRUE.We
plot the fitted model together with confidence intervals for the model fit and the ANOVA-type effect
estimates.
data(IBScovars)
anovaMod <- lm(resp~factor(dose)+gender, data=IBScovars)
drFit <- coef(anovaMod)[2:5] # placebo adjusted (=effect) estimates
at doses
S <- vcov(anovaMod)[2:5,2:5] # estimated covariance
dose <- sort(unique(IBScovars$dose))[-1] # vector of active doses
## now fit an emax model to these estimates
gfit <- fitMod(dose, drFit, S=S, model = "emax", placAdj = TRUE,
type = "general", bnds = c(0.01, 2))
plot(gfit, CI = TRUE, plotData = "meansCI")
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Supporting information
Additional supporting information may be found in the online version of this article at the publisher’s
web site.
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