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.     

Optimal and efficient designs for 2-parameter nonlinear models

By Carathéodory's theorem, for a k-parameter nonlinear model, the minimum number of support points for any D-optimal design is between k and k(k+1)/2. Characterizing classes of models for which a D-optimal design sits on exactly k support points is of great theoretical interest. By utilizing the equivalence theorem, we identify classes of 2-parameter nonlinear models for which a D-optimal design is precisely supported on 2 points. We also introduce the theory of maximum principle from differential equations into the design area and obtain some results on characterizing the minimally supported nonlinear designs. Examples are given to demonstrate our results. Designs with minimum number of support points may not always be suitable in practice. To alleviate this problem, we utilize some geometric and analytical methods to obtain some efficient designs which provide more opportunity for the model checking and prevent biases due to mis-specified initial parameters.     

Minimum breakdown designs in blocks of size two

In scientific investigations, there are many situations where each two experimental units have to be grouped into a block of size two. For planning such experiments, the variance-based optimality criteria like A-, D- and E-criterion are typically employed to choose efficient designs, if the estimation efficiency of treatment contrasts is primarily concerned. Alternatively, if there are observations which tend to become lost during the experimental period, the robustness criteria against the unavailability of data should be strongly recommended for selecting the planning scheme. In this study, a new criterion, called minimum breakdown criterion, is proposed to quantify the robustness of designs in blocks of size two. Based on the proposed criterion, a new class of robust designs, called minimum breakdown designs, is defined. When various numbers of blocks are missing, the minimum breakdown designs provide the highest probabilities that all the treatment contrasts are estimable. An exhaustive search procedure is proposed to generate such designs. In addition, two classes of uniformly minimum breakdown designs are theoretically verified.     

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.     

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.     

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.     

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.     

Optimal designs for generalized linear models

Summary This paper solves some D-optimal design problems for certain Generalized Linear Models where the mean depends on two parameters and two explanatory variables. In all of the cases considered the support point of the optimal designs are found to be independent of the unknown parameters. While in some cases the optimal design measures are given by two points with equal weights, in others the support is given by three point with weights depending on the unknown parameters, hence the designs are locally optimal in general. Empirical results on the efficiency of the locally optimal designs are also given. Some of the designs found can also be used for planning D-optimal experiments for the normal linear model, where the mean must be positive. This research was carried out in part at University College, London as an M.Sc. project. Thanks are due to Prof. I. Ford (University of Glasgow) and Prof. A. Giovagnoli (University of Perugia) for their valuable suggestions and critical observations.     

D-optimal weighing designs and optimal chemical balance weighing designs for estimating the total weight

This paper provides D-optimal spring balance designs for estimating individual weights when the number of objects to be weighed in each weighing, B, is fixed. D-optimal chemical balance designs for estimating total weight under both homogeneous and nonhomogeneous error variances are found when the number of objects weighed in each weighing is ≥ B, a fixed number.

We indicate the restriction used in Chacko & Dey(1978) and Kageyama(1988), i.e. that chemical designs X be restricted to designs in which exactly “a” objects are replaced on the left pan and exactly “b” on the right pan in each of the weighings for a, b > 0, is unnecessary.     

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.     

Optimal designs for covariates’ models

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. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Robustness of subset response surface designs to missing observations

Experiments designed to investigate the effect of several factors on a process have wide application in modern industrial and scientific research. Response surface designs allow the researcher to model the effects of the input variables on the response of the process. Missing observations can make the results of a response surface experiment quite misleading, especially in the case of one-off experiments or high cost experiments. Designs robust to missing observations can attract the user since they are comparatively more reliable. Subset designs are studied for their robustness to missing observations in different experimental regions. The robustness of subset designs is also improved for multiple levels by using the minimax loss criterion.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal 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 3×3 factorial and 3-ray designs for drug interaction experiments with dichotomous responses

The only information available to an investigator designing an initial combination drug study is for each drug used singly. The designs that we investigated are constructed using this information. Within the major body of the paper we consider experiments using nine points arrived at from 3x3 factorial and 3-ray design plans for which D-optimal solutions are obtained under the hypothesis of no interaction.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

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.     

E-optimal designs for regression models with quantitative factors— a reasonable choice?

For regression models with quantitative factors it is illustrated that the E-optimal design can be extremely inefficient in the sense that it degenerates to a design which takes all observations at only one point. This phenomenon is caused by the different size of the elements in the covariance matrix of the least-squares estimator for the unknown parameters. For these reasons we propose to replace the E-criterion by a corresponding standardized version. The advantage of this approach is demonstrated for the polynomial regression on a nonnegative interval, where the classical and standardized E-optimal designs can be found explicitly. The described phenomena are not restricted to the E-criterion but appear for nearly all optimality criteria proposed in the literature. Therefore standardization is recommended for optimal experimental design in regression models with quantitative factors. The optimal designs with respect to the new standardized criteria satisfy a similar invariance property as the famous D-optimal designs, which allows an easy calculation of standardized optimal designs on many linearly transformed design spaces.     

D-optimal designs with minimal and nearly minimal number of units

D-optimal designs are identified in classes of connected block designs with fixed block size when the number of experimental units is one or two more than the minimal number required for the design to be connected. An application of one of these results is made to identify D-optimal designs in a class of minimally connected row-column designs. Graph-theoretic methods are employed to arrive at the optimality results.     

D-optimal minimax regression designs on discrete design space

One classical design criterion is to minimize the determinant of the covariance matrix of the regression estimates, and the designs are called D-optimal designs. To reflect the nature that the proposed models are only approximately true, we propose a robust design criterion to study response surface designs. Both the variance and bias are considered in the criterion. In particular, D-optimal minimax designs are investigated and constructed. Examples are given to compare D-optimal minimax designs with classical D-optimal designs.