Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.     

Penalized optimal designs for dose-finding

Optimal design under a cost constraint is considered, with a scalar coefficient setting the compromise between information and cost. It is shown that for suitable cost functions, by increasing the value of the coefficient one can force the support points of an optimal design measure to concentrate around points of minimum cost. An example of adaptive design in a dose-finding problem with a bivariate binary model is presented, showing the effectiveness of the approach.     

Finding maximum G-criterion values for central composite designs on the hypercube

For quadratic regression on the hypercube, G—efficiencies are often used in the selection process of an experimental design. To calculate a design's G—efficiency, it is necessary to maximize the prediction variance over the experimental design region. However, it is common to approximate a G—efficiency. This is achieved by calculating the prediction variances generated from a subset of points in the design space and taking the maximum to estimate the maximum prediction variance. This estimate is then applied to approximate the G—efficiency. In this paper, it will be shown that over the class of central composite designs (CCDs) on the hypercube. the prediction variance can be expressed in a closed-form. An exact value of the maximum prediction variance can then be determined by evaluating this closed-form expression over a finite subset of barycentric points. Tables of exact G—efficiencies will be presented. Design optimality criteria, quadratic regression on the hypercube, and the structures of the design matrix X, X'X, and (X'X)^?1 for any CCD will be discussed.     

Universally optimal designs under interference models with and without block effects

The concept of neighbor designs was introduced and defined by Rees (1967 Rees, D.H. (1967). Some designs of use in serology. Biometrics 23:779–791.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) along with giving some methods of their construction. Henceforth, many methods of construction of neighbor designs as well as of their generalizations are available in the literature. However, there are only few results on their optimality. Therefore, the purpose of this article is to give an overview of study on this problem. Recent results on optimality of specified neighbor balanced designs under various interference models with block effects are presented and then these results are compared with respective models where block effects are not significant.     

Marginally restricted sequential D‐optimal designs

In many experiments, not all explanatory variables can be controlled. When the units arise sequentially, different approaches may be used. The authors study a natural sequential procedure for “marginally restricted” D‐optimal designs. They assume that one set of explanatory variables (x₁) is observed sequentially, and that the experimenter responds by choosing an appropriate value of the explanatory variable x₂. In order to solve the sequential problem a priori, the authors consider the problem of constructing optimal designs with a prior marginal distribution for x₁. This eliminates the influence of units already observed on the next unit to be designed. They give explicit designs for various cases in which the mean response follows a linear regression model; they also consider a case study with a nonlinear logistic response. They find that the optimal strategy often consists of randomizing the assignment of the values of x₂.     

Constructing optimal designs for fitting pharmacokinetic models

We consider some computational issues that arise when searching for optimal designs for pharmacokinetic (PK) studies. Special factors that distinguish these are (i) repeated observations are taken from each subject and the observations are usually described by a nonlinear mixed model (NLMM), (ii) design criteria depend on the model fitting procedure, (iii) in addition to providing efficient parameter estimates, the design must also permit model checking, (iv) in practice there are several design constraints, (v) the design criteria are computationally expensive to evaluate and often numerical integration is needed and finally (vi) local optimisation procedures may fail to converge or get trapped at local optima.We review current optimal design algorithms and explore the possibility of using global optimisation procedures. We use these latter procedures to find some optimal designs.For multi-purpose designs we suggest two surrogate design criteria for model checking and illustrate their use.     

A procedure for optimal augmentation of multidimensional designs

The augmentation of an existing multidimensional design is discussed from the point of view of estimability of certain two-factor interactions which are nonestimable from the original design. A general procedure is proposed which achieves this with a minimal number of additional assemblies and which is optimal in a certain sense. The individual steps in this procedure are described in detail and illustrated by an example.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

D-optimal designs for beta regression models with single predictor

The problem of construction of D-optimal designs for beta regression models involving one predictor is considered for the mean-precision parameterization suggested by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions. J Appl Stat. 2004;31:799–815]. Here we use the logit link function for the mean sub-model. These designs are presented and discussed for unrestricted as well as restricted design regions by considering the precision parameter as (1) a known constant and (2) an unknown constant. Efficiency comparison of obtained designs with commonly used equi-weighted, equi-spaced designs is made to recommend designs for practical use. Real-life applications are given to show the usefulness of these designs.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

First-order optimal designs for non-linear models

This paper presents D-optimal experimental designs for a variety of non-linear models which depend on an arbitrary number of covariates but assume a positive prior mean and a Fisher information matrix satisfying particular properties. It is argued that these optimal designs can be regarded as a first-order approximation of the asymptotic increase of Shannon information. The efficiency of this approximation is compared in some examples, which show how the results can be further used to compute the Bayesian optimal design, when the approximate solution is not accurate enough.     

A relaxation procedure for calculating (G–)minimax optimal designs

Summary: In nonlinear statistical models, standard optimality functions for experimental designs depend on the unknown parameters of the model. An appealing and robust concept for choosing a design is the minimax criterion. However, so far, minimax optimal designs have been calculated efficiently under various restrictive conditions only. We extend an iterative relaxation scheme originally proposed by Shimizu and Aiyoshi (1980) and prove its convergence under very general assumptions which cover a variety of situations considered in experimental design. Application to different specific design criteria is discussed and issues of practical implementation are addressed. First numerical results suggest that the method may be very efficient with respect to the number of iterations required.*Supported by a grant from the Deutsche Forschungsgemeinschaft. We are grateful to a referee for his constructive suggestions.     

The effect of small sample on optimal designs for logistic regression models

Asymptotic methods are commonly used in statistical inference for unknown parameters in binary data models. These methods are based on large sample theory, a condition which may be in conflict with small sample size and hence leads to poor results in the optimal designs theory. In this paper, we apply the second order expansions of the maximum likelihood estimator and derive a matrix formula for the mean square error (MSE) to obtain more precise optimal designs based on the MSE. Numerical results indicate the new optimal designs are more efficient than the optimal designs based on the information matrix.     

Robust sequential designs for approximately linear models

We consider the problem of the sequential choice of design points in an approximately linear model. It is assumed that the fitted linear model is only approximately correct, in that the true response function contains a nonrandom, unknown term orthogonal to the fitted response. We also assume that the parameters are estimated by M-estimation. The goal is to choose the next design point in such a way as to minimize the resulting integrated squared bias of the estimated response, to order n^-1. Explicit applications to analysis of variance and regression are given. In a simulation study the sequential designs compare favourably with some fixed-sample-size designs which are optimal for the true response to which the sequential designs must adapt.     

A note on D-optimal designs for models with and without an intercept

In this paper we give a sufficient condition under which theD-optimal design for a regression model without an intercept can be obtained from theD-optimal design for the corresponding model with an intercept by simply removing the origin from its support points. Examples are given to demonstrate the applications of the results.     

Robust sequential designs for nonlinear regression

The authors introduce the formal notion of an approximately specified nonlinear regression model and investigate sequential design methodologies when the fitted model is possibly of an incorrect parametric form. They present small‐sample simulation studies which indicate that their new designs can be very successful, relative to some common competitors, in reducing mean squared error due to model misspecifi‐cation and to heteroscedastic variation. Their simulations also suggest that standard normal‐theory inference procedures remain approximately valid under the sequential sampling schemes. The methods are illustrated both by simulation and in an example using data from an experiment described in the chemical engineering literature.     

Exactly optimal sampling designs for processes with a product covariance structure

The author considers the problem of finding exactly optimal sampling designs for estimating a second‐order, centered random process on the basis of finitely many observations. The value of the process at an unsampled point is estimated by the best linear unbiased estimator. A weighted integrated mean squared error or the maximum mean squared error is used to measure the performance of the estimator. The author presents a set of necessary and sufficient conditions for a design to be exactly optimal for processes with a product covariance structure. Expansions of these conditions lead to conditions for asymptotic optimality.     

Compound Regression and Constrained Regression: Nonparametric Regression Frameworks for EIV Models

Abstract

Errors-in-variable (EIV) regression is often used to gauge linear relationship between two variables both suffering from measurement and other errors, such as, the comparison of two measurement platforms (e.g., RNA sequencing vs. microarray). Scientists are often at a loss as to which EIV regression model to use for there are infinite many choices. We provide sound guidelines toward viable solutions to this dilemma by introducing two general nonparametric EIV regression frameworks: the compound regression and the constrained regression. It is shown that these approaches are equivalent to each other and, to the general parametric structural modeling approach. The advantages of these methods lie in their intuitive geometric representations, their distribution free nature, and their ability to offer candidate solutions with various optimal properties when the ratio of the error variances is unknown. Each includes the classic nonparametric regression methods of ordinary least squares, geometric mean regression (GMR), and orthogonal regression as special cases. Under these general frameworks, one can readily uncover some surprising optimal properties of the GMR, and truly comprehend the benefit of data normalization. Supplementary materials for this article are available online.