Maximin Doptimal designs for binary longitudinal responses

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Maximin D-optimal designs for binary longitudinal
responses
Fetene B. Tekle, Frans E. S. Tan and Martijn P.
F. Berger Department of Methodology and
Statistics, Maastricht University, Maastricht,
NL
DEMA2008 11 August to 15 August 2008 Isaac
Newton Institute for Mathematical Sciences,
Cambridge, UK
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Outline

• Introduction
• Objectives
• Model and variance-covariances
• D-optimality
• Relative efficiency
• Maximin D-optimal designs
• Numerical study and results
• Concluding remarks

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• Introduction

Problem The optimality criteria for non linear
models depend on unknown parameter values
(Silvey, 1980 Atkinson and Donev, 1996),
designing experiments requires full knowledge of
the regression coefficients. Solutions locally
optimal (Fedorov and Müller, 1997)
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• Introduction (contd.)

• Solutions
• Sequential procedure (Wu, 1985 Sitter and
  Forbes, 1997 and Sitter and Wu, 1999),
• Bayesian procedure (Chaloner, 1989 Chaloner and
  Larntz, 1989 Atkinson et al., 1993 Chaloner and
  Verdinelli, 1995 Dette, 1996 and Han and
  Chaloner, 2004), and
• Maximin approach (Müller, 1995 Dette, 1997
  Müller and Pazman, 1998 Dette and Sahm, 1997,
  1998 King and Wong, 2000 Imhof, 2001 Ouwens et
  al., 2002, 2006 Braess and Dette, 2007).

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• Objectives

• Binary longitudinal responses (Mixed-effects
  logistic model)
• Investigate locally D-optimal designs
• Maximin approach with D-optimality
• Identify maximin D-optimal designs,
• Identify the optimum number of repeated
  measurements needed for a longitudinal study with
  binary responses.

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• Model and variance-covariances

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The model
(1)
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• Model and variance-covariances

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Estimation and variances
(2)
a) PQL method An approximate variance-covariance
matrix for the parameters estimator is (Breslow
and Clayton, 1993)
(3)
where V is the block-diagonal matrix with blocks
q x q matrices given by
(4)
where wi is a diagonal matrix of the conditional
variances of the responses given the random
effects, the q x r matrix z is made by having
as rows.
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• Model and variance-covariances

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b) Extended GEE method with serial correlations
(Zeger et al., 1998 Molenberghs and Verbeke,
2005)
(5)
where the working variance-covariance of the
responses vi is given by
(6)
The q x q matrix R(?) is the correlation matrix
of the responses over time (AR(1) structure is
assumed).
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• D-optimality

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A design is D-optimal if
(Atkinson and Donev, 1996)
where the design space of q x 1 time vectors be
denoted by
(7)
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• Relative Efficiency (RE)

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RE is used to compare two designs
(8)
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• Maximin D-optimal designs

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A design is maximin D-optimal for
, if it maximizes
the smallest relative efficiency over
The maximin D-optimal design maximizes the
minimum relative efficiency among all designs
with q time points and the corresponding maximin
efficiency is given by
(9)
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• Numerical study and Results

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• Locally D-optimal and maximin D-optimal designs
  within the time interval -1, 1.
• Random intercept, and random intercept and slope
  models with
• linear,
• quadratic and
• cubic terms of time.
• Two sets of intervals of parameters (0.0, 1.5)
  and (0.0, 3.0 for the maximin D-optimal designs.
• A MATLAB code with fmincon function

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• Optimal time points include the two end points of
  the study period (time interval) if a value of a
  regression parameter is small.
• As q increases, the optimal locations of the
  additional time points are close to the optimal
  allocations of a design with q p for the PQL
  method.
• When serial correlation is introduced, the
  optimal locations of the additional time points
  move further to the center and the design
  resembles an equi-distance design.

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Maximin designs that include the end points of
the time interval with regression parameter
values in (0.0, 1.5).
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• Concluding remarks

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• Optimal design points may not include the two end
  points of the study period (time interval),
• MMEs are large,
• A maximin D-optimal design with number of
  repeated measurements q is equal to the number of
  regression parameters p is the most efficient
  design compared to designs with more number of
  repeated measurements.
• Future work
• ? Control for additional experimental variable(s)
• ? Dropouts