Partially A-optimal blocked multi-factor designs

In this paper, we propose a partially A-optimal criterion for block designs where multiple factors are arranged. The number of levels of each factor is assumed to be arbitrary and unequal block sizes are allowed. A sufficient condition is derived for a design to be partially A-optimal among all feasible designs. Then the properties of the selected design and its relation with orthogonal arrays are studied. Methods of constructing designs satisfying the sufficient condition are also given.     

Projection estimation capacity of Hadamard designs

In a screening design, often only a few factors among a large number of potential factors are significantly important. Usually, it is not known which factors will be important ones. Thus, it is of practical interest to know if each projection of a design onto a small subset of factors is able to entertain and estimate all two-factor-interactions along with its main effects, assuming higher order interactions are negligible. In this paper, we investigate the estimation capacity of projections of Hadamard designs with run size up to 60. Possible applications of our results to robust parameter designs are also discussed.     

A unified theory for generalized cyclic designs

Properties of abelian groups are used to unify the construction of cyclic and generalized cyclic designs. Necessary and sufficient conditions are obtained for the generation of fractional sets of blocks, and also for the generation of resolvable designs. It is shown how to apply these results in conjunction with efficiency arguments to obtain useful designs.     

On the enumeration of some D-optimal designs

Two matrices with elements taken from the set {-1,1}$\{-1,1\}$ are Hadamard equivalent if one can be converted into the other by a sequence of permutations of rows and columns, and negations of rows and columns. In this paper we summarize what is known about the number of equivalence classes of matrices having maximal determinant. We establish that there are 7 equivalence classes for matrices of order 21 and that there are at least 9884 equivalence classes for matrices of order 26. The latter result is obtained primarily using a switching technique for producing new designs from old.     

Optimal saturated block designs when observations are correlated

A- and MV-optimal block designs are identified in the class of minimally connected designs when the observations within blocks are spatially correlated. All connected designs are shown to be D-equal regardless of the correlation structure, and a sufficient condition for E-optimality is presented. Earlier results for the uncorrelated case are strengthened.     

Φp-optimal second order designs for symmetric regions

Designs for quadratic regression on a cube, on a cube with truncated vertices and on a ball are studied in terms of a family of criteria, introduced by Kiefer (1974, 1975), that includes A-, D- and E-optimality. Both theoretical and numerical results on structure and performance are presented. In particular, D- and E-optimal designs are described and a procedure of construction of nearly robust (under variation of criterion) integer designs is suggested. Some examples are given for dimensions 4, 5 and 6.     

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only few experiments in the preliminary stages of experimentation. This paper explores how to construct E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs using k-cyclic generators. The necessary and sufficient conditions for the existence of mixed-level k-circulant SSDs with the equal occurrence property are provided. Properties of the mixed-level k  -circulant SSDs are investigated, in particular, the sufficient condition under which the generator vector produces an E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal SSD is obtained. Moreover, many new E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean [2004. k$k$-circulant supersaturated designs. Technometrics 46, 32–43] for two-level SSDs and the one due to Georgiou and Koukouvinos [2006. Multi-level k-circulant supersaturated designs. Metrika 64, 209–220] for the multi-level case.     

Uniform coverage designs for mixture experiments

We investigate an optimization problem for mixture experiments. We consider the case when a large number of ingredients are available but mixtures can contain only a few number of ingredients. These conditions are held in experiments for self assembling molecular systems. First, we introduce a concept of uniform coverage design specialized for the situation. Next, we propose to use the stepwise technique for estimating coefficients of third-order Scheffe model which describes a response surface. After that, we propose a method of adding new mixtures for a movement to an extremum region. By this method, additional mixtures of experiments are extremum points of current estimated model and points which lead to more accurate estimation of current model prediction. This methodology is studied numerically for a model constructed from real data.     

Efficient experimental designs for sigmoidal growth models

For the Weibull- and Richards-regression model robust designs are determined by maximizing a minimum of D  - or _D1$D_1$-efficiencies, taken over a certain range of the non-linear parameters. It is demonstrated that the derived designs yield a satisfactory solution of the optimal design problem for this type of model in the sense that these designs are efficient and robust with respect to misspecification of the unknown parameters. Moreover, the designs can also be used for testing the postulated form of the regression model against a simplified sub-model.     

Interaction balance for symmetrical factorial designs with generalized minimum aberration

There are two different systems of contrast parameterization when analyzing the interaction effects among the factors with more than two levels, i.e., linear-quadratic system and orthogonal components system. Based on the former system and an ANOVA model, Xu and Wu (2001) introduced the generalized wordlength pattern for general factorial designs. This paper shows that the generalized wordlength pattern exactly measures the balance pattern of interaction columns of a symmetrical design ground on the orthogonal components system, and thus an alternative angle to look at the generalized minimum aberration criterion is given. This work is partially supported by NNSF of China grant No. 10231030.     

Optimal discrimination designs for exponential regression models

We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw [1995. Robust sampling designs for compartmental models under large prior eigenvalue uncertainties. Math. Comput. Biomed. Appl. 181–187] or Gibaldi and Perrier [1982. Pharmacokinetics. Marcel Dekker, New York]. We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory's Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach.     

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.     

Universal admissibility in second order asymptotics

Some sufficient conditions for an estimator to be universally second order admissible are derived. Those sufficient conditions consist of the elementary integrals with respect to the Fisher information and the limits of some functions characterized by the dealt statistical model, and thus can be checked with comparative ease. In location model and scale model, the sufficient condition for the linear estimator with respect to the maximum likelihood estimator (MLE) to be universally second order admissible is given. Furthermore, a guide for classifying any estimator into either the universal admissibility or the non-universal admissibility is proposed.     

Characterization of minimum aberration mixed factorials in terms of consulting designs

In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain combinatorial identities that relate the wordlength pattern of a regular mixed factorial design to that of its consulting design. According to these identities, we further-more establish the general and unified rules for identifying minimum aberration mixed factorial designs through their consulting designs. It is an improvement and generalization of the results in Mukerjee and Wu (2001). This paper is supported by NNSF of China grant No. 10171051 and RFDP grant No. 1999005512.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

Minimal point second order designs

This paper deals with the problem of finding nearly D-optimal designs for multivariate quadratic regression on a cube which take as few observations as possible and still allow estimation of all parameters. It is shown that among the class of all such designs taking as many observations as possible on the corners of the cube there is one which is asymptotically efficient as the dimension of the cube increases. Methods for constructing designs in this class, using balanced arrays, are given. It is shown that the designs so constructed for dimensions ≤6 compare well with existing computer generated designs, and in dimensions 5 and 6 are better than those in literature prior to 1978.     

Optimal crossover designs when carryover effects are proportional to direct effects

Crossover designs are used for a variety of different applications. While these designs have a number of attractive features, they also induce a number of special problems and concerns. One of these is the possible presence of carryover effects. Even with the use of washout periods, which are for many applications widely accepted as an indispensable component, the effect of a treatment from a previous period may not be completely eliminated. A model that has recently received renewed attention in the literature is the model in which first-order carryover effects are assumed to be proportional to direct treatment effects. Under this model, assuming that the constant of proportionality is known, we identify optimal and efficient designs for the direct effects for different values of the constant of proportionality. We also consider the implication of these results for the case that the constant of proportionality is not known.     

Robust designs for one-point extrapolation

We consider the construction of designs for the extrapolation of a regression response to one point outside of the design space. The response function is an only approximately known function of a specified linear function. As well, we allow for variance heterogeneity. We find minimax designs and corresponding optimal regression weights in the context of the following problems: (P1) for nonlinear least squares estimation with homoscedasticity, determine a design to minimize the maximum value of the mean squared extrapolation error (MSEE), with the maximum being evaluated over the possible departures from the response function; (P2) for nonlinear least squares estimation with heteroscedasticity, determine a design to minimize the maximum value of MSEE, with the maximum being evaluated over both types of departures; (P3) for nonlinear weighted least squares estimation, determine both weights and a design to minimize the maximum MSEE; (P4) choose weights and design points to minimize the maximum MSEE, subject to a side condition of unbiasedness. Solutions to (P1)–(P4) are given in complete generality. Numerical comparisons indicate that our designs and weights perform well in combining robustness and efficiency. Applications to accelerated life testing are highlighted.     

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.     

Robust estimators and designs for field experiments

We consider the construction of designs for test-control field experiments, with particular attention being paid to the effects of spatial correlation between adjoining plots. In contrast to previous approaches, in which very specific correlation structures were modelled, we explicitly allow a degree of uncertainty on the part of the experimenter. While fitting a particular correlation structure—and variance structure and regression response—the experimenter is thought to be seeking protection against other possible structures in full neighbourhoods of these particular choices. Robustness, in a minimax sense, is obtained through a modification of the kriging estimation procedure, and through the assignment of treatments to field plots.