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.     

An optimality criterion for supersaturated designs with quantitative factors

A supersaturated design (SSD) is a factorial design in which the degrees of freedom for all its main effects exceed the total number of distinct factorial level-combinations (runs) of the design. Designs with quantitative factors, in which level permutation within one or more factors could result in different geometrical structures, are very different from designs with nominal ones which have been treated as traditional designs. In this paper, a new criterion is proposed for SSDs with quantitative factors. Comparison and analysis for this new criterion are made. It is shown that the proposed criterion has a high efficiency in discriminating geometrically nonisomorphic designs and an advantage in computation.     

Optimal Latin hypercube designs for the Kullback–Leibler criterion

Space-filling designs are commonly used for selecting the input values of time-consuming computer codes. Computer experiment context implies two constraints on the design. First, the design points should be evenly spread throughout the experimental region. A space-filling criterion (for instance, the maximin distance) is used to build optimal designs. Second, the design should avoid replication when projecting the points onto a subset of input variables (non-collapsing). The Latin hypercube structure is often enforced to ensure good projective properties. In this paper, a space-filling criterion based on the Kullback–Leibler information is used to build a new class of Latin hypercube designs. The new designs are compared with several traditional optimal Latin hypercube designs and appear to perform well.     

Model-Robust Design of Conjoint Choice Experiments

Within the context of choice experimental designs, most authors have proposed designs for the multinomial logit model under the assumption that only the main effects matter. Very little attention has been paid to designs for attribute interaction models. In this article, three types of Bayesian D-optimal designs for the multinomial logit model are studied: main-effects designs, interaction-effects designs, and composite designs. Simulation studies are used to show that in situations where a researcher is not sure whether or not attribute interaction effects are present, it is best to take into account interactions in the design stage. In particular, it is shown that a composite design constructed by including an interaction-effects model and a main-effects model in the design criterion is most robust against misspecification of the underlying model when it comes to making precise predictions.     

Efficient maximin distance designs for experiments in mixtures

In this paper, different dissimilarity measures are investigated to construct maximin designs for compositional data. Specifically, the effect of different dissimilarity measures on the maximin design criterion for two case studies is presented. Design evaluation criteria are proposed to distinguish between the maximin designs generated. An optimization algorithm is also presented. Divergence is found to be the best dissimilarity measure to use in combination with the maximin design criterion for creating space-filling designs for mixture variables.     

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.     

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.     

On minimax designs when there are two candidate models

This work is motivated by the need to find experimental designs which are robust under different model assumptions. We measure robustness by calculating a measure of design efficiency with respect to a design optimality criterion and say that a design is robust if it is reasonably efficient under different model scenarios. We discuss two design criteria and an algorithm which can be used to obtain robust designs. The first criterion employs a Bayesian-type approach by putting a prior or weight on each candidate model and possibly priors on the corresponding model parameters. We define the first criterion as the expected value of the design efficiency over the priors. The second design criterion we study is the minimax design which minimizes the worst value of a design criterion over all candidate models. We establish conditions when these two criteria are equivalent when there are two candidate models. We apply our findings to the area of accelerated life testing and perform sensitivity analysis of designs with respect to priors and misspecification of planning values.     

A maximin criterion for the logistic random intercept model with covariates

A maximin criterion is used to find optimal designs for the logistic random intercept model with dichotomous independent variables. The dichotomous independent variables can be subdivided into variables for which the distribution is specified prior to data sampling, called variates, and into variables for which the distribution is not specified prior to data sampling, but is obtained from data sampling, called covariates. The proposed maximin criterion maximizes the smallest possible relative efficiency not only with respect to all possible values of the model parameters, but also with respect to the joint distribution of the covariates. We have shown that, under certain conditions, the maximin design is balanced with respect to the joint distribution of the variates. The proposed method will be used to plan a (stratified) clinical trial where variates and covariates are involved.     

Maximin efficient design of experiment for exponential regression models

In this paper robust and efficient designs are derived for several exponential decay models. These models are widely used in chemistry, pharmacokinetics or microbiology. We propose a maximin approach, which determines the optimal design such that a minimum of the D-efficiencies (taken over a certain range) becomes maximal. Analytic solutions are derived if optimization is performed in the class of minimal supported designs. In general the optimal designs with respect to the maximin criterion have to be determined numerically and some properties of these designs are also studied. We also illustrate the benefits of our approach by reanalysing a data example from the literature.     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

Efficient Use of Two Decks of Cards in Randomized Response Sampling

This paper extends the work of Russell (1976). Proof Is given that, for many parameter sets, all O:XB designs belonging to the set are connected. It is shown how an (M,S)-optinal design nay be selected from the M-optimal designs of a given parameter set. The efficiencies of these (M,S)-optimal designs are displayed, and it is concluded that the (M,S)-optimality criterion is useful for selecting designs of high efficiency.     

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.     

Slope-rotatable designs with equal maximum directional variance for second order response surface models

The concept of sloperotaiability with equal maximum directional vari ance for second order response surface models is introduced as a new design property. This requires that the maximum variance of the estimated slope over all possible directions be only a function of p, which is the distance from the design originif is shown that a rotatable design satisfies this property Also, minimization of tiie maximum variance of the estimated slope over all possible directions is proposed as a new design optirnality criterion, and op¬timal designs are called slope-directional minirnax designs. For the class of cquiradial designs, the slope-directional minirnax designs are compared with D— optimal designs.     

Weighted A-optimality for fractional 2m factorial designs of resolution V

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.     

A semi-infinite programming based algorithm for finding minimax optimal designs for nonlinear models

Minimax optimal experimental designs are notoriously difficult to study largely because the optimality criterion is not differentiable and there is no effective algorithm for generating them. We apply semi-infinite programming (SIP) to solve minimax design problems for nonlinear models in a systematic way using a discretization based strategy and solvers from the General Algebraic Modeling System (GAMS). Using popular models from the biological sciences, we show our approach produces minimax optimal designs that coincide with the few theoretical and numerical optimal designs in the literature. We also show our method can be readily modified to find standardized maximin optimal designs and minimax optimal designs for more complicated problems, such as when the ranges of plausible values for the model parameters are dependent and we want to find a design to minimize the maximal inefficiency of estimates for the model parameters.     

Catalogue of small runs nonregular designs from hadamard matrices with generalized minimum aberration

Fries and Hunter ( 1980 ) proposed the Minimum Aberration criterion (MA) for selecting regular designs. The regular designs with MA are msot commonly used because they are considered as the best designs. How ever, as pointed out by Chen, Sun and Wu ( 1993 ), there are situations that other designs may better meet the design need. Therefore, they catalogued some two-level and three-level fractional factorial regular designs with small (16,27,32,64) runs. For nonregular designs, such as the ones taken from Hadamard matrices, the MA criterion is not appUcable. Deng and Tang ( 1999 ) introduced Generalized Minimum Aberration Criterion (GMA) as a natural extension to the MA criterion. Similar to the case in the regular designs, other designs may better meet practical need, In this paper, we use the GMA criterion to give a catalogue of nonregular designs with smaU (16,20,24) runs.     

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.     

Defining equations for two-level factorial designs

Defining equations are introduced in the context of two-level factorial designs and they are shown to provide a concise specification of both regular and nonregular designs. The equations are used to find orthogonal arrays of high strength and some optimal designs. The latter optimal designs are formed in a new way by augmenting notional orthogonal arrays which are allowed to have some runs with a negative number of replicates before augmentation. Defining equations are also shown to be useful when the factorial design is blocked.     

Maximum expected entropy transformed Latin hypercube designs

Existing projection designs (e.g. maximum projection designs) attempt to achieve good space-filling properties in all projections. However, when using a Gaussian process (GP), model-based design criteria such as the entropy criterion is more appropriate. We employ the entropy criterion averaged over a set of projections, called expected entropy criterion (EEC), to generate projection designs. We show that maximum EEC designs are invariant to monotonic transformations of the response, i.e. they are optimal for a wide class of stochastic process models. We also demonstrate that transformation of each column of a Latin hypercube design (LHD) based on a monotonic function can substantially improve the EEC. Two types of input transformations are considered: a quantile function of a symmetric Beta distribution chosen to optimize the EEC, and a nonparametric transformation corresponding to the quantile function of a symmetric density chosen to optimize the EEC. Numerical studies show that the proposed transformations of the LHD are efficient and effective for building robust maximum EEC designs. These designs give projections with markedly higher entropies and lower maximum prediction variances (MPV''s) at the cost of small increases in average prediction variances (APV''s) compared to state-of-the-art space-filling designs over wide ranges of covariance parameter values.