Transcript Zhu,Wei

Controlled Optimal Designs for Dose Response Studies
---- and a Website for Optimal Designs
Xiangfeng Wu1, Wei Zhu1, Holger Dette2, Weng Kee Wong3
1Department
of Applied Mathematics and Statistics, State University of New York,
Stony Brook, NY 11794-3600, USA;
2Ruhr-Universität Bochum, Fakultät für Mathematik, 44780 Bochum, Germany;
3Department of Biostatistics, University of California, Los Angeles, CA, 90095-1772,
USA
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Introduction
We present the numerical algorithms for finding Bayesian multipleobjective optimal designs for dose-response studies subject to certain
constraints. Dose-response studies are routinely conducted in drug
design process. Due to safety, efficacy, and experimental design
considerations, practical constraints are often imposed on dose range,
dose levels, dose numbers, dose proportions and potential missing trials.
The resulting controlled optimal designs satisfying these constraints can
be readily adopted for optimal estimation of the parameters of interest
such as the median effective dose or the threshold dose. In addition, we
describe the methodologies and the implementation of a web based
interactive optimal design platform for the practitioners. We
demonstrate our results and methodology through the widely used logit
dose response model.
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Optimal Experimental Design
The fundamental idea of Optimal Experimental Design is to find the
best allocation rule such that the underlying model or other parameters
of interest can be estimated or predicted with the best precision for a
given sample size. The variances of parameter estimates and
predictions depend upon the experimental design and should be
made as small as possible. Unnecessarily large variances and
imprecise predictions resulting from a poorly designed experiment
could render an experiment inconclusive.
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Quantal Dose Response Model
The study of drug toxicity and efficacy is a crucial part of the drug development process.
Quantal dose-response experiment is routinely conducted to study the relationship
between the dose level of a drug and the probability of a response (1 if the subject is
responsive at the given dose and 0 otherwise). A commonly used model is the simple logit
model:
 ( x)
log
  (x )
1   ( x)
Here  ( x)  1/1  exp(  ( x   )) is the probability of a response at dosage x   , where  is
the dose interval. The parameter  is the slope in the logit scale and  is the dose level at
which the response probability is 0.5 and often referred to as the “median effective dose”,
and denoted by ED50 . More generally, we let ED100 denote the dose level x at which the
probability of observing a response is . This means that the percentile or ED100 is
equal to    /  , where   log  /1   .
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Optimal Design Criteria –(1)
Fisher’s Information Matrix. Let y be the indicator of a response at
dose x and let θ=(α,β). Since the probability of observing a response at
dose level x is π(y=1|x,θ)=π(x), it follows that the (i,j)th element of the
observed Fisher information matrix for design ξ is
 M ( , )ij
2
 
log   y | x,      dx 
i  j
where design  is a probability measure with weights
support point xi for i  1,..., m, wi  0, m wi  1
wi
on design
i 1
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Optimal Design Criteria – (2)
An optimal design is one that minimizes some properly chosen convex
function of the information matrixM ( ,.) In particular, we introduce
the following optimality criteria.
1. D-optimality. The D-optimal design would minimize the
generalized variance of the parameter estimates:
log M 1  ,  
2. C-optimality. Here our interest lies in estimating the parameters
c(θ) with minimum variance. Thus the c-optimal design would
minimize the function:
c T M 1  , c 
where c  is the gradient of c(θ).
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Bayesian Approach – (1)
An immediate issue relating to the construction of optimal designs for
the dose-response model is that the information matrix M or the design
criteria function depends on the unknown parameters of that model.
A common approach to the problem is to plug in the best guess of the
parameter values, and thus the name ‘locally optimal design’.
Alternatively, one can impose distribution assumptions on the unknown
parameters from previous studies or by elicitation, and use a Bayesian
approach.
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Bayesian Approach – (2)
The Bayesian Approach is optimizing the average of a function of the
information matrix over a prior distribution placed on the unknown
parameters. Specifically, the Bayesian D-optimal design minimizes:


D    E log M 1  ,    log M 1  ,  g   d

And the Bayesian c-optimal design minimizes:


c    E c T M 1  , c    c T M 1  , c g  d

Where g   denotes the prior probability density, can be
constructed .
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Multiple-Objective Optimal Design
Suppose there are m objectives in the study and each of them is represented by
a convex criterion i , i  1,..., m
Two equivalent approaches have been suggested for constructing a multipleobjective optimal design. One way is to find a compound optimal design and
the other is to find a constrained optimal design.
We consider compound optimal design. The compound optimal design is the
design which minimizes the convex combination
m

   
i 1 i i
where each i  0,1 is user-selected and

m
i 1
i  1
For each convex combination, the compound optimal design is found using the
same numerical algorithm for generating a single-objective optimal design.
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Design Efficiency
The design efficiency of an arbitrary design  relative to the
*
optimal design
is defined as
E ( )   ( * ) /  ( )
For the c-optimality criterion, this ratio compares the expected
variances for the estimated c( ) given by the two designs. Designs
with high efficiencies are desirable because they provide more
accurate estimates for a given sample size.
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Restricted Design Space
In the framework of does-response studies accomplished on human
subjects such as clinical trials prior to the launch of new drugs, we often
encounter the problem that the support points of an optimal design lie
outside a reasonable dosage range, i.e. dose levels are either below zero or
exceed safety levels such as the maximum tolerated dose of the drug.
Practitioners are therefore in need of efficient designs that take the
restricted design spaces into account in response to the drug toxicity
and/or efficacy considerations.
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More Design Constraints
 Constraints on the number of dose levels (e.g. because of the
usual modest sample size, one can not accommodate too many
dose levels).
 Potential missing trials (e.g. potential missing trials due to
toxicity/side effect or lack of efficacy).
 Constraints on dose levels (e.g. certain dose such as the placebo
must be included).
 Constraints on dose proportions (e.g. each dose level must have
at least 10% of the sample).
 We refer to optimal designs satisfying one ore more types of
these practical constrains as the controlled optimal designs.
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Numerical Approach
Using advanced math theory, one can sometimes, derive the optimal
designs analytically. With the development of advanced computing
software and hardware, however, we can always derive the optimal
designs numerically.
Computationally, the optimization of the design criterion function is a
nonlinear programming problem(NLP). In Practice, the controlled
optimal designs based on a fixed number of design points can be
found directly using an appropriate constrained nonlinear optimization
algorithm.
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Constrained Nonlinear Optimization
The Kuhn-Tucker (KT) equation. The KT equations are necessary conditions for
optimality for a constrained optimization problem. If the problem is a so-called
convex programming problem, that is, f (x) and Gi ( x), i  1,2,  m are convex
functions, then the KT equations are both necessary and sufficient for a global
solution point.
f ( x )   *i  Gi x *   0
m
*
i 1
*i  Gi x *   0
i  1,..., m
The solution of the KT equations forms the basis to many nonlinear
programming algorithms. These algorithms attempt to compute the Lagrange
multipliers directly.
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Sequential Quadratic Programming (SQP)-(1)
The principal idea is the formulation of a QP sub problem on a quadratic
approximation of the Lagrangian function.
m
L  x,    f  x     i  g i  x 
i 1
The Quadratic Programming (QP) Sub Problem
min imize 1 T
T


d
H
d


f
x
d
k
k
d n 2
g i xk  d  g i xk   0
T
g i xk  d  g i xk   0
T
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Sequential Quadratic Programming (SQP)-(2)
Updating the Hessian Matrix. At each major iteration a positive definite quasiNewton approximation of the Hessian of the Lagrangian function, H, is calculated
using the BFGS method.
H k 1
q k q kT
H kT H k
 Hk  T  T
qk sk sk H k sk
s k  x k 1  x k
n


q k  f ( x k 1 )   i  g i  x k 1    f  x k    i  g i  x k 
i 1
i 1


n
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Sequential Quadratic Programming (SQP)-(3)
At each major iteration of the SQP method, a QP problem of the following form is
solved, where Ai refers to the ith row of the m-by-n matrix A.
min imize
1 T


q
d

d Hd  c T d
n
d 
2
Ai d  bi
Ai d  bi
i  1,..., me
i  me  1,..., m
The method used to solve the problem is an active set strategy (also known as a
projection method).
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Sequential Quadratic Programming (SQP)-(4)
Line Search and Merit Function. The solution to the QP sub problem produces a
vector d k , which is used to form a new iterate.
xk 1  xk  ad k
The step length parameter a k is determined in order to produce a sufficient decrease
in a merit function.
me
  x   f  x    ri  g i  x  
i 1
m
 r  max 0, g x 
i  me 1
i
i
1


ri  rk 1 i  max i , rk i  i 
i
2


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Web Based Design Software
Why Web Based? For one thing, we would like to enable the users to perform
statistical design and analysis from any location where an internet connection is
available, without any complicated software installation and setup procedures.
Client computers
Internet
Web Server
Design Software
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System Architecture
BINARY
STREAM
M-FILES
Optimal Design Engine
COMMON
GATEWAY
Matlab System
CLIENT SIDE
ACTIVE
SERVER
PAGE
Interface Web Application
HTML
IIS Web Server
Client web Browser (IE, etc.)
HTML
BINARY
STREAM
SEVER SIDE
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Implementation Technologies
Matlab Programming. MATLAB® is a high-performance language for technical
computing. It integrates computation, visualization, and programming in an easy-to-use
environment where problems and solutions are expressed in familiar mathematical
notation.
ASP. Net C Sharp Programming. ASP.NET is a technology for building powerful,
dynamic Web applications. ASP.NET makes building real world Web applications
dramatically easier.
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The Optimal Experimental Design Website
www.optimal-design.org is the interface to the interactive optimal experimental
design software. We also provide information of optimal design theories and useful
resources on this website.
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An Example of Controlled Optimal Design – (1)
Example: Consider the dose-response model on restricted dose space [0, 3.5] with
at least 10% of the subjects allocated to each dose level. Suppose that independent
uniform priors for alpha on the interval [1.5, 2.5] and beta on [0.9, 1.1] are selected.
Furthermore, the experimenter wish to find a 2-point D-optimal design.
D-optimal design for alpha ~ U[1.5, 2.5] beta ~U[0.9, 1.1]
CONSTRAINTS
D  D* 
 D*
Restricted dose
Range: [0, 3.5]
Dose level: ≥ 0.1
 0.784 3.217 


 1/ 2 1/ 2 
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An Example of Controlled Optimal Design – (2)
Inputting Design Parameters
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An Example of Controlled Optimal Design – (3)
The Output
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Future Work





Improving Optimization Algorithms
Incorporating Different types of Prior distributions
Optimal Designs for Other Dose Response Models
Incorporating Analysis of Experiments
Exploration of theoretical properties of controlled optimal designs such
as the derivation of the general equivalence theorem.
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Reference
A.C. Atkinson and A.N. Donev (1992). Optimum Experimental Designs.
Oxford Statistical Science.
V.V. Fedorov (1972). Theory of Optimal Experiments. Academic Press New
York and London.
W. Zhu and W.K. Wong. Bayesian optimal designs for estimating a set of
symmetrical quantiles. Statistics in Medicine. 2001;20:123-137
M.C. Biggs. "Constrained Minimization Using Recursive Quadratic
Programming," Towards Global Optimization (L.C.W. Dixon and G.P. Szergo,
eds.), North-Holland, pp 341-349, 1975.
M.J.D. Powell. "A Fast Algorithm for Nonlinearly Constrained Optimization
Calculations," Numerical Analysis, G.A.Watson ed., Lecture Notes in Mathematics,
Springer-Verlag, Vol. 630, 1978.
S. Biedermann, H. Dette and W. Zhu. Optimal Designs for Dose-Response
Models with Restricted Design Spaces. Journal of the American Statistical
Association. 101(474), 747 -759.
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Acknowledgement
This work is supported by the US National Institute of
Health grant no: 1R01 GM072876-01A1 “Cost Effective
Designs for Practitioners ”.
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