New classes of D-optimal edge designs

Edge designs are screening experimental designs that allow a model-independent estimate of the set of relevant variables, thus providing more robustness than traditional designs. In this paper we construct new classes of D-optimal edge designs. This construction uses weighing matrices of order n and weight k together with permutation matrices of order n to obtain D-optimal edge designs. One linear and one quadratic simulated screening scenarios are studied and compared using linear regression and edge designs analysis.     

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.     

A- and D-optimal designs for a log contrast model for experiments with mixtures

A- and D-optimal designs are investigated for a log contrast model suggested by Aitchison & Bacon-Shone for experiments with mixtures. It is proved that when the number of mixture components q is an even integer, A- and D-optimal designs are identical; and when q is an odd integer, A- and D-optimal designs are different, but they share some common support points and are very close to each other in efficiency. Optimal designs with a minimum number of support points are also constructed for 3, 4, 5 and 6 mixture components.     

Designs for weighted least squares regression, with estimated weights

We study designs, optimal up to and including terms that are O(n ^?1), for weighted least squares regression, when the weights are intended to be inversely proportional to the variances but are estimated with random error. We take a finite, but arbitrarily large, design space from which the support points are to be chosen, and obtain the optimal proportions of observations to be assigned to each point. Specific examples of D- and I-optimal design for polynomial responses are studied. In some cases the same designs that are optimal under homoscedasticity remain so for a range of variance functions; in others there tend to be more support points than are required in the homoscedastic case. We also exhibit minimax designs, that minimize the maximum, over finite classes of variance functions, value of the loss. These also tend to have more support points, often resulting from the breaking down of replicates into clusters.     

Bayesian D-optimal design for logistic regression model with exponential distribution for random intercept

ABSTRACT

Nowadays, generalized linear models have many applications. Some of these models which have more applications in the real world are the models with random effects; that is, some of the unknown parameters are considered random variables. In this article, this situation is considered in logistic regression models with a random intercept having exponential distribution. The aim is to obtain the Bayesian D-optimal design; thus, the method is to maximize the Bayesian D-optimal criterion. For the model was considered here, this criterion is a function of the quasi-information matrix that depends on the unknown parameters of the model. In the Bayesian D-optimal criterion, the expectation is acquired in respect of the prior distributions that are considered for the unknown parameters. Thus, it will only be a function of experimental settings (support points) and their weights. The prior distribution of the fixed parameters is considered uniform and normal. The Bayesian D-optimal design is finally calculated numerically by R3.1.1 software.     

Extensions of D-optimal minimal designs for symmetric mixture models

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide the desirable response characteristics in finished products. D-optimal minimal designs have been considered for a variety of mixture models, including Scheffé's linear, quadratic, and cubic models. Usually, these D-optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the lack-of-fit (LOF) tests. Also, the majority of the design points in D-optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. In this article, extensions of the D-optimal minimal designs are developed for a general mixture model to allow additional interior points in the design space to enable prediction of the entire response surface. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. We compare the proposed designs with Cornell (1986 Cornell, J.A. (1986). A comparison between two ten-point designs for studying three-component mixture systems. J. Qual. Technol. 18(1):1–15.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) two 10-point designs for the LOF test by simulations.     

On Exact K-optimal Designs Minimizing the Condition Number

A new design criterion based on the condition number of an information matrix is proposed to construct optimal designs for linear models, and the resulting designs are called K-optimal designs. The relationship between exact and asymptotic K-optimal designs is derived. Since it is usually hard to find exact optimal designs analytically, we apply a simulated annealing algorithm to compute K-optimal design points on continuous design spaces. Specific issues are addressed to make the algorithm effective. Through exact designs, we can examine some properties of the K-optimal designs such as symmetry and the number of support points. Examples and results are given for polynomial regression models and linear models for fractional factorial experiments. In addition, K-optimal designs are compared with A-optimal and D-optimal designs for polynomial regression models, showing that K-optimal designs are quite similar to A-optimal designs.     

Bayesian D-optimal designs for exponential regression models

We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

A new characterization of Elfving's method for high dimensional computation

We give a new characterization of Elfving's (1952) method for computing c-optimal designs in k dimensions which gives explicit formulae for the k unknown optimal weights and k unknown signs in Elfving's characterization. This eliminates the need to search over these parameters to compute c-optimal designs, and thus reduces the computational burden from solving a family of optimization problems to solving a single optimization problem for the optimal finite support set. We give two illustrative examples: a high dimensional polynomial regression model and a logistic regression model, the latter showing that the method can be used for locally optimal designs in nonlinear models as well.     

Bayesian optimal one point designs for one parameter nonlinear models

For nonlinear one parameter models and concave optimality criteria there always exists a locally optimal one point design. This can be proved by an application of Caratheodory's theorem (Läuter, Math. Operationsforsch. Statist. Ser. Statist. 5 (1974a) 625–636). If prior distributions with densities are used, this theorem gives no useful bound on the number of support points of a Bayesian optimal design. Chaloner (J. Statist. Plann. Inference, 37 (1993) 229–236) gave a sufficient condition on the support of the prior distribution for the existence of a Bayesian optimal one point design. In this article, a condition on the shape of the prior density is given, which is also sufficient for the existence of a Bayesian optimal one point design in nonlinear models with one parameter.     

Some d-optimal weighing designs for n≡ (mod 4)

A number of D-optimal weighing designs are constructed with the help of block matrices. The D-optimal designs (n,k,s)=(19,13,10), (19,14,7), (19,14,8), (19,15,7), (19,15,8), (19,17,6), (19,18,6), (23,16,8), (23,17,8), (23,18,8), (4n?1,2n+3,(3n+4)/2), (4n?1,2n+4,n+3), (4n?1,2n+4,n+2) where n≡0 mod 4 and a skew H_n exists, (31,24,8), (31,25,8) and many others are constructed. A computer routine leading to locally D-optimal designs is presented.     

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.     

Optimum designs for estimating EDp based on raw optical density data

For raw optical density (ROD) data, such as those generated in biological assays employing an ELISA plate reader, ED_p-optimal designs are identified for a family of homogeneous non-linear models with two parameters. In every case, the theoretical ED_p-optimal design is a design with one or two support points. These theoretical optimal designs might not be suitable for many practical applications. To overcome this shortcoming, we have specified ED_p-optimal designs within the class of k-point equally spaced and uniform designs. The efficiency robustness of these designs with respect to initial nominal values of the parameters have been investigated.     

On invariance and randomization in factorial designs with applications to D-optimal main effect designs of the symmetrical factorial

Under the setting of the columnwise orthogonal polynomial model in the context of the general factorial it is shown that (i) the determinant of the information matrix of a design relative to an admissible vector of effects is invariant under a permutation of levels; (ii) the unbiased estimation of a linear function of an admissible vector of effects can be obtained under equal probability randomization. These results extend the work on invariance and randomization carried out under the more restrictive assumption of the orthonormal polynomial model by Srivastava, Raktoe and Pesotan (1976) and Pesotan and Raktoe (1981). Moreover, the problem of the construction of D-optimal main effect designs in the symmetrical factorial is reduced to a study of a special class of (0,1)-matrices using the Helmat matrix model. Using this class of (0,1)-matrices and the determinant invariance result, some classes of D-optimal main effect designs of the s² and s³ factorial respectively are presented.     

Cyclic designs for a covariate model

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. We restrict our attention to classes of designs d for which the number of observations N to be taken is a multiple of V, i.e. N = V × R with R ≥2, and each treatment level is observed R times. Among these designs, called here equireplicated, there is a subclass characterized by the following: the allocation matrix of each treatment level (for short, allocation matrix) is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. We call these designs cyclic. Besides having easy representation, the most efficient cyclic designs are often D-optimal in the class of equireplicated designs. A known upper bound for the determinant of the information matrix M(d) of a design, in the class of equireplicated ones, depends on the congruences of N and V modulo 4. For some combinations of parameter moduli, we give here methods of constructing families of D-optimal cyclic designs. Moreover, for some sets of parameters (N, V,K = V), where the upper bound on ∣M(d)∣ (for that specific combination of moduli) is not attainable, it is also possible to construct highly D-efficient cyclic designs. Finally, for N≤24 and V≤6, computer search was used to determine the most efficient design in the class of cyclic ones. They are presented, together with their respective efficiency in the class of equireplicated designs.     

Folded over non-orthogonal designs

In this article, we use the notion of minimal dependent sets (MDS) to introduce MDS-resolution and MDS-aberration as criteria for comparing non-orthogonal foldover designs, and discuss the ideas and their usefulness. We also develop a fast isomorphism check that uses a cyclic matrix defined on the design before it is folded over. By doing so, the speed of the check for comparing two isomorphic designs is increased relative to merely applying an isomorphism check to the foldover design. This relative difference becomes greater as the design size increases. Finally, we use the isomorphism check to obtain a catalog of minimum MDS-aberration designs for some useful n$n$ and k$k$ and discuss an algorithm for obtaining “good” larger designs.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

Generalized neighbor designs with block size 3

A generalized neighbor design relaxes the equality condition on the number of times two treatments as neighbors in the design. In this article, we have considered the construction of some classes of generalized neighbor designs with block size k=3 by using the method of cyclic shifts. The distinguishing feature of this construction method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. A catalog of generalized neighbor designs with block size k=3 is compiled for v∈{5,6,…,18} treatments and for different replications. We provide the reader with a simpler method of construction, and in general the catalog that gives an open choice to the experimenter for selecting any class of neighbor 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.