Equivalence theorem for Schur optimality of experimental designs

An experimental design is said to be Schur optimal, if it is optimal with respect to the class of all Schur isotonic criteria, which includes Kiefer's criteria of _Φp${\Phi }_p$-optimality, distance optimality criteria and many others. In the paper we formulate an easily verifiable necessary and sufficient condition for Schur optimality in the set of all approximate designs of a linear regression experiment with uncorrelated errors. We also show that several common models admit a Schur optimal design, for example the trigonometric model, the first-degree model on the Euclidean ball, and the Berman's model.     

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.     

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).     

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.     

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.     

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).     

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.     

Some sufficient conditions for the type 1 optimality of block designs

In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.     

On the optimality of chemical balance weighing designs

In this paper we consider the problem of optimally weighing n objects with N weighings on a chemical balance. Several previously known results are generalized. In particular, the designs shown by Ehlich (1964a) and Payne (1974) to be D-optimal in various classes of weighing designs where N≡2 (mod4) are shown to be optimal with respect to any optimality criterion of Type I as defined in Cheng (1980). Several results on the E-optimality of weighing designs are also given.     

Design of experiments in the presence of errors in factor levels

This paper is concerned with the statistical properties of experimental designs where the factor levels cannot be set precisely. When the errors in setting the factor levels cannot be measured, design robustness is explored. However, when the actual design could be measured at the end of the investigation, its optimality is of interest. D-optimality could be assessed in different ways. Several measures are compared. Evaluating them is difficult even in simple cases. Therefore, in general, simulations are used to obtain their values. It is shown that if D-optimality is measured by the expected value of the determinant of the information matrix of the experimental design, as has been suggested in the past, on average the designs appear to improve with the variance of the error in setting the factor levels. However, we argue that the criterion of D-optimality should be based on the inverse of the information matrix. In this case it is shown that the experiment could be better or worse than the planned one. It is also recognized that setting the factor levels with error could lead to an increased risk of losing observations, which on its own could reduce considerably the optimality of the experimental designs. Advice on choosing the design region in such a way that such a risk is controlled to an acceptable level is given.     

Addition of runs to a two-level supersaturated design

The purpose of this article is to introduce a new class of extended E(s²)-optimal two level supersaturated designs obtained by adding runs to an existing E(s²)-optimal two level supersaturated design. The extended design is a union of two optimal SSDs belonging to different classes. New lower bound to E(s²) has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(s²)-optimal SSDs are also included.     

A bidimensional class of optimality criteria involving φp and characteristic criteria

A new biparametric class of criteria for Optimal Experimental Design generalizing the families of φ_p and Characteristic Criteria is presented. Some properties of the Characteristic Criteria are provided: in particular, differentiability, monotonicity and convexity. A statistical interpretation is also offered. Optimal designs with respect to Characteristic Criteria are obtained for polynomial and compartmental models and an extension of the Michaelis-Menten model. A thorough discussion for the trigonometric model is given. The computed optimal designs are shown to be quite efficient for A-, D- and D_s -optimality. Thus, some of them appear as a natural compromise between different optimality criteria. A simulation in multiple linear regression confirms the quality of Characteristic designs for discovering significant differences between parameters.     

D–optimality of a class of saturated main–effect plans and allied results

This paper applies the theory of unimodular matrices to prove that all saturated main effect plans of an s₁ × s₂ factorial are equivalent from the point of view of D–optimality and are hence all D–optimal. The A– and E–optimal plans in this context have also been derived. An application in sequential experimentation has been considered     

Design selection criteria for discrimination/estimation for nested models and a binomial response

The aim of an experiment is often to enable discrimination between competing forms for a response model. We investigate the selection of a continuous design for a non-sequential strategy when there are two competing generalized linear models for a binomial response, with a common link function and the linear predictor of one model nested within that of the other. A new criterion, _TE$T_E$-optimality, is defined, based on the difference in the deviances from the two models, and comparisons are made with T$T$-, _Ds$D_s$- and D$D$-optimality. Issues are raised through the study of two examples in which designs are assessed using simulation studies of the power to reject the null hypothesis of the smaller model being correct, when the data are generated from the larger model. Parameter estimation for discrimination designs is also discussed and a simple method is investigated of combining designs to form a hybrid design in order to achieve both model discrimination and estimation. This method has a computational advantage over the use of a compound criterion and the similar performance of the designs obtained from the two approaches is illustrated in an example.     

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.     

Optimal designs for the logit and probit models for binary data

We consider optimal designs for a class of symmetric models for binary data which includes the common probit and logit models. We show that for a large group of optimality criteria which includes the main ones in the literature (e.g. A-, D-, E-, F- and G-optimality) the optimal design for our class of models is a two-point design with support points symmetrically placed about the ED50 but with possibly unequal weighting. We demonstrate how one can further reduce the problem to a one-variable optimization by characterizing various of the common criteria. We also use the results to demonstrate major qualitative differences between the F - and c-optimal designs, two design criteria which have similar motivation.     

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).     

A note on Läuter-type robust design for polynomial regression models

In this paper, we consider the problem of model robust design for simultaneous parameter estimation among a class of polynomial regression models with degree up to k. A generalized D-optimality criterion, the Ψ_α‐optimality criterion, first introduced by Läuter (1974) is considered for this problem. By applying the theory of canonical moments and the technique of maximin principle, we derive a model robust optimal design in the sense of having highest minimum Ψ_α‐efficiency. Numerical comparison indicates that the proposed design has remarkable performance for parameter estimation in all of the considered rival models.     

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.     

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.