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.     

Robust prediction and extrapolation designs for censored data

In this paper we present the construction of robust designs for a possibly misspecified generalized linear regression model when the data are censored. The minimax designs and unbiased designs are found for maximum likelihood estimation in the context of both prediction and extrapolation problems. This paper extends preceding work of robust designs for complete data by incorporating censoring and maximum likelihood estimation. It also broadens former work of robust designs for censored data from others by considering both nonlinearity and much more arbitrary uncertainty in the fitted regression response and by dropping all restrictions on the structure of the regressors. Solutions are derived by a nonsmooth optimization technique analytically and given in full generality. A typical example in accelerated life testing is also demonstrated. We also investigate implementation schemes which are utilized to approximate a robust design having a density. Some exact designs are obtained using an optimal implementation scheme.     

Robust prediction and extrapolation designs for misspecified generalized linear regression models

We study minimax robust designs for response prediction and extrapolation in biased linear regression models. We extend previous work of others by considering a nonlinear fitted regression response, by taking a rather general extrapolation space and, most significantly, by dropping all restrictions on the structure of the regressors. Several examples are discussed.     

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.     

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.     

Optimality of random allocation design for the control of accidental bias in sequential experiments

In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. Known or unknown to the experimenter there may be nuisance factors systematically affecting the subjects. Accidental bias is a measure of the influence of these factors in the analysis of data. We show in this paper that the random allocation design minimizes the accidental bias among all designs that allocate n, out of 2n, subjects to each treatment and do not prefer either treatment in the assignment. When the final imbalance is allowed to be nonzero, optimal and efficient designs are given. In particular the random allocation design is shown to be very efficient in this broader setup.     

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.     

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.     

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.     

Construction of some classes of balanced cross-over designs from terraces

5 and 6 have recently introduced power-sequence terraces. In this paper we have used these terraces for the construction of some new families of balanced cross-over designs of first and second order which are variance-balanced. We have also used them for the construction of some new families of balanced ternary cross-over designs.     

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.     

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.     

Optimality of circular neighbor-balanced designs for total effects with autoregressive correlated observations

This paper studies the optimality of circular neighbor-balanced designs (CNBDs) for total effects when the one-sided or two-sided neighbor effects are present in the model and the observation errors are correlated according to a first-order circular autoregressive (AR(1,C$C$)) process. Some optimality results under some specified conditions are provided and the efficiency of a CNBD relative to the optimal block design is investigated. In order to discuss the efficiency of a CNBD among all possible block designs with the same size, the optimal equivalence classes of sequences under the one-sided neighbor effects model are characterized and the efficiencies of CNBDs with blocks of small size are illustrated.     

Approximate and exact optimal designs for paired comparisons of partial profiles when there are two groups of factors

For paired comparison experiments involving pairs of multifactor options differing in a specified number of factors the problem of finding optimal designs is considered, when only main effects are to be estimated. It is presumed that the set of factors can be partitioned into two groups such that the number of levels is constant within each group. The optimal designs for this frequently encountered case are also optimal for the corresponding choice experiments under the hypothesis that the parameters in the multinomial logit model are equal to zero.     

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.     

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.     

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.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

Robust designs for misspecified logistic models

We develop criteria that generate robust designs and use such criteria for the construction of designs that insure against possible misspecifications in logistic regression models. The design criteria we propose are different from the classical in that we do not focus on sampling error alone. Instead we use design criteria that account as well for error due to bias engendered by the model misspecification. Our robust designs optimize the average of a function of the sampling error and bias error over a specified misspecification neighbourhood. Examples of robust designs for logistic models are presented, including a case study implementing the methodologies using beetle mortality data.     

Optimality in presence of damaged observations

In this paper we study a situation where, instead of observing $\text{y}\text{(N×1)}$, we observe $\text{y}\text{+}\text{d}$; where the random variables in the vector d are called damage components. It is shown that A- and E-optimum designs minimize the effect of d in the mean square error (m.s.e.) of estimable linear functions of parameters under the ordinary linear model.