OPTIMAL EXPERIMENTAL DESIGNS FOR MULTILEVEL MODELS WITH COVARIATES

In this paper optimal experimental designs for multilevel models with covariates and two levels of nesting are considered. Multilevel models are used to describe the relationship between an outcome variable and a treatment condition and covariate. It is assumed that the outcome variable is measured on a continuous scale. As optimality criteria D-optimality, and L-optimality are chosen. It is shown that pre-stratification on the covariate leads to a more efficient design and that the person level is the optimal level of randomization. Furthermore, optimal sample sizes are given and it is shown that these do not depend on the optimality criterion when randomization is done at the group level.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

On linear regression designs which maximize information

Necessary and sufficient conditions are established when a continuous design contains maximal information for a prescribed s-dimensional parameter in a classical linear model. The development is based on a thorough study of a particular dual problem and its interplay with the optimal design problem, extending partial results and earlier approaches based on differential calculus, game theory, and other programming methods. The results apply in particular to a class of information functionals which covers c-, D-, A-, L-optimality, they include a complete account of the non-differentiable criterion of E-optimality, and provide a constructive treatment of those situations in which the information matrix is singular. Corollaries pertain to the case of s out of k parameters, simultaneous optimality with respect to several criteria, multiplicity of optimal designs, bounds on their weights, and optimality which is induced by admissibility.     

Relationships of optimality for individual factors of a design

The optimality of two-factor experimental designs is studied in the dual senses of estimating contrasts in the parameters for each of the factors. The outline of comparison employed allows one to judge the performance of different designs for estimating contrasts of one set of parameters directly with the performance of the complementary set without going through a common intermediary step of considering all the parameters. The results hold for a wide class of optimality criteria (not merely D-, A- and E-optimality), which must satisfy a functional equation obtained in connection with our method. Also we investigate the optimality of row–column designs which satisfy an ‘adjusted orthogonality’ condition. Our point of departure is the paper by Shah, Raghavarao and Khatri (1976) and that of Mitchell and John (1977).     

Adaptive grid semidefinite programming for finding optimal designs

We find optimal designs for linear models using a novel algorithm that iteratively combines a semidefinite programming (SDP) approach with adaptive grid techniques. The proposed algorithm is also adapted to find locally optimal designs for nonlinear models. The search space is first discretized, and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are points that maximize the dispersion function of the SDP-generated optimal design using nonlinear programming. The procedure is repeated until a user-specified stopping rule is reached. The proposed algorithm is broadly applicable, and we demonstrate its flexibility using (i) models with one or more variables and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the information matrix of the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally efficient because it is based on mathematical programming tools and so optimality is assured at each stage; it also exploits the convexity of the problems whenever possible. Using several linear and nonlinear models with one or more factors, we show the proposed algorithm can efficiently find optimal designs.     

The robustness of response surface designs with errors in factor levels

ABSRTACT

Since errors in factor levels affect the traditional statistical properties of response surface designs, an important question to consider is robustness of design to errors. However, when the actual design could be observed in the experimental settings, its optimality and prediction are of interest. Various numerical and graphical methods are useful tools for understanding the behavior of the designs. The D- and G-efficiencies and the fraction of design space plot are adapted to assess second-order response surface designs where the predictor variables are disturbed by a random error. Our study shows that the D-efficiencies of the competing designs are considerably low for big variance of the error, while the G-efficiencies are quite good. Fraction of design space plots display the distribution of the scaled prediction variance through the design space with and without errors in factor levels. The robustness of experimental designs against factor errors is explored through comparative study. The construction and use of the D- and G-efficiencies and the fraction of design space plots are demonstrated with several examples of different designs with errors.     

Designs in nonlinear regression by stochastic minimization of functionals of the mean square error matrix

We reconsider and extend the method of designing nonlinear experiments presented in Pázman and Pronzato (J. Statist. Plann. Inference 33 (1992) 385). The approach is based on the probability density of the LS estimators, and takes into account the boundary of the parameter space. The idea is to express the elements of the mean square error matrix (MSE) of the LS estimators as integrals of the density, express optimality criteria as functions of MSE, and minimize them by stochastic optimization. In the present paper we include prior knowledge about some parameters, we derive improved approximations of the density of estimators, and we consider not only linear optimality criteria like generalized A-optimality, but also D- and $L_s$-optimality criteria with an integer s. Of basic importance is the use of an accelerated method of stochastic optimization (MSO). This together with the important progress in computing allowed us to elaborate a realistic algorithm. Its performance is demonstrated by the search of a generalized D-optimum design in the 4-parameter growth curve model of microbiological experiments presented by Baranyi and Roberts (Internat. J. Food Microbiol. 23 (1994) 277; 26 (1995) 199).     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.     

Bayesian D-optimal Accelerated Life Test plans for series systems with competing exponential causes of failure

This paper provides methods of obtaining Bayesian D-optimal Accelerated Life Test (ALT) plans for series systems with independent exponential component lives under the Type-I censoring scheme. Two different Bayesian D-optimality design criteria are considered. For both the criteria, first optimal designs for a given number of experimental points are found by solving a finite-dimensional constrained optimization problem. Next, the global optimality of such an ALT plan is ensured by applying the General Equivalence Theorem. A detailed sensitivity analysis is also carried out to investigate the effect of different planning inputs on the resulting optimal ALT plans. Furthermore, these Bayesian optimal plans are also compared with the corresponding (frequentist) locally D-optimal ALT plans.     

Optimum Designs for Parameter Estimation in Linear Mixture Models with Synergistic Effects

In this article, we derive optimum designs for parameter estimation in a mixture experiment when the response function is linear in the mixing components with some synergistic effects. The D- and A-optimality criteria have been used for the purpose. The Equivalence Theorem has been used to check for the optimality of the proposed designs.     

Rotation designs for experiments in high-bias situations

Many experiments in the physical and engineering sciences study complex processes in which bias due to model inadequacy dominates random error. A noteworthy example of this situation is the use of computer experiments, in which scientists simulate the phenomenon being studied by a computer code. Computer experiments are deterministic: replicate observations from running the code with the same inputs will be identical. Such high-bias settings demand different techniques for design and prediction. This paper will focus on the experimental design problem introducing a new class of designs called rotation designs. Rotation designs are found by taking an orthogonal starting design D and rotating it to obtain a new design matrix D_R=DR, where R is any orthonormal matrix. The new design is still orthogonal for a first-order model. In this paper, we study some of the properties of rotation designs and we present a method to generate rotation designs that have some appealing symmetry properties.     

Optimal design in average for inference in generalized linear models

This paper considers the problem of optimal design for inference in Generalized Linear Models, when prior information about the parameters is available. The general theory of optimum design usually requires knowledge of the parameter values. These are usually unknown and optimal design can, therefore, not be used in practice. However, one way to circumvent this problem is through so-called “optimal design in average”, or shortly, “ave optimal”. The ave optimal design is chosen to minimize the expected value of some criterion function over a prior distribution. We focus our interest on the aveD _A-optimality, including aveD- and avec-optimality and show the appropriate equivalence theorems for these optimality criterions, which give necessary conditions for an optimal design. Ave optimal designs are of interest when e.g. a factorial experiment with a binary or a Poisson response in to be conducted. The results are applied to factorial experiments, including a control group experiment and a 2×2 experiment.     

Optimal designs for treatment comparisons represented by graphs

Consider an experiment for comparing a set of treatments: in each trial, one treatment is chosen and its effect determines the mean response of the trial. We examine the optimal approximate designs for the estimation of a system of treatment contrasts under this model. These designs can be used to provide optimal treatment proportions in more general models with nuisance effects. For any system of pairwise treatment comparisons, we propose to represent such a system by a graph. Then, we represent the designs by the inverses of the vertex weights in the corresponding graph and we show that the values of the eigenvalue-based optimality criteria can be expressed using the Laplacians of the vertex-weighted graphs. We provide a graph theoretic interpretation of D-, A- and E-optimality for estimating sets of pairwise comparisons. We apply the obtained graph representation to provide optimality results for these criteria as well as for ’symmetric’ systems of treatment contrasts.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

Goodness of exact experimental designs with respect to a certain family of criteria

In the paper we assume to be given an approximate optimum exact design with respect to one optimality criterion. We investigate the goodness of this design in the sense of a family of criteria that includes those of A-E-, and .D-optimality.     

The analytic construction of D-optimal designs for the two-variable binary logistic regression model without interaction

Candidate locally D-optimal designs for the binary two-variable logistic model with no interaction, which comprise 3 and 4 support points lying in the first quadrant of the two-dimensional Euclidean space, were introduced by Haines et al. (D-optimal designs for logistic regression in two variables. In: Lopez-Fidalgo J, Rodrigez-Diaz JM, Torsney B, editors. MODA8 – advances in model-oriented designs and analysis. Heidelberg: Physica-Verlag; 2007. p. 91–98). The authors proved algebraically the global D-optimality of the 3-point design for the special case in which the intercept parameter is equal to?1.5434. However for other selected values of the intercept parameter, the global D-optimality of the proposed 3- and 4-point designs was only demonstrated numerically. In this paper, we provide analytical proofs of the D-optimality of these 3- and 4-point designs for all negative and zero intercept parameters of the binary two-variable logistic model with no interaction. The results are extended to the construction of D-optimal designs on a rectangular design space and illustrated by means of two examples of which one is a real example taken from the literature.     

D-Optimal Two-Level Fractional Factorial Designs of Resolution III with Two Dispersion Factors

The problem of finding D-optimal designs, with two dispersion factors, for the estimation of all location main effects is investigated in the class of regular unreplicated two-level fractional factorial designs of resolution III. Designs having length three words involving both of the dispersion factors in the defining relation are shown to be inferior in terms of D-optimality. Tables of factors that are named as the two dispersion factors so that the resulting design is either D-optimal or has the largest determinant of the information matrix are provided. Rank-order of designs is studied when the number of length three words involving either one of the dispersion factors and the number of length four words involving both of the dispersion factors are fixed. Rank-order of designs when the numbers of aforementioned words are less than or equal to ten is given.     

D-optimal block designs with at most six varieties

For given positive integers v, b, and k (all of them ≥2) a block design is a k × b array of the variety labels 1,…,v with blocks as columns. For the usual one-way heterogeneity model in standard form the problem is studied of finding a D-optimal block design for estimating the variety contrasts, when no balanced block design (BBD) exists. The paper presents solutions to this problem for v≤6. The results on D-optimality are derived from a graph-theoretic context. Block designs can be considered as multigraphs, and a block design is D-optimal iff its multigraph has greatest complexity (=number of spanning trees).     

Convergence properties of sequential Bayesian D-optimal designs

We establish convergence properties of sequential Bayesian optimal designs. In particular, for sequential D-optimality under a general nonlinear location-scale model for binary experiments, we establish posterior consistency, consistency of the design measure, and the asymptotic normality of posterior following the design. We illustrate our results in the context of a particular application in the design of phase I clinical trials, namely a sequential design of Haines et al. [2003. Bayesian optimal designs for phase I clinical trials. Biometrics 59, 591–600] that incorporates an ethical constraint on overdosing.     

Optimal experimental design criterion for discriminating semiparametric models

This paper studies the optimal experimental design problem to discriminate two regression models. Recently, López-Fidalgo et al. [2007. An optimal experimental design criterion for discriminating between non-normal models. J. Roy. Statist. Soc. B 69, 231–242] extended the conventional T-optimality criterion by Atkinson and Fedorov [1975a. The designs of experiments for discriminating between two rival models. Biometrika 62, 57–70; 1975b. Optimal design: experiments for discriminating between several models. Biometrika 62, 289–303] to deal with non-normal parametric regression models, and proposed a new optimal experimental design criterion based on the Kullback–Leibler information divergence. In this paper, we extend their parametric optimality criterion to a semiparametric setup, where we only need to specify some moment conditions for the null or alternative regression model. Our criteria, called the semiparametric Kullback–Leibler optimality criteria, can be implemented by applying a convex duality result of partially finite convex programming. The proposed method is illustrated by a simple numerical example.