Addition of runs to a two-level supersaturated design

The purpose of this article is to introduce a new class of extended E(s²)-optimal two level supersaturated designs obtained by adding runs to an existing E(s²)-optimal two level supersaturated design. The extended design is a union of two optimal SSDs belonging to different classes. New lower bound to E(s²) has been obtained for the extended supersaturated designs. Some examples and a small catalogue of E(s²)-optimal SSDs are also included.     

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.     

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.     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

D-optimal block designs with at most six varieties

For given positive integers v, b, and k (all of them ≥2) a block design is a k × b array of the variety labels 1,…,v with blocks as columns. For the usual one-way heterogeneity model in standard form the problem is studied of finding a D-optimal block design for estimating the variety contrasts, when no balanced block design (BBD) exists. The paper presents solutions to this problem for v≤6. The results on D-optimality are derived from a graph-theoretic context. Block designs can be considered as multigraphs, and a block design is D-optimal iff its multigraph has greatest complexity (=number of spanning trees).     

Optimal foldover designs

An obvious strategy for obtaining a Doptimal foldover design for p factors at two levels each in 2N runs is to fold a Doptimal main effects plan. We show that this strategy works except when N = 4t + 2 and s is even In that case there are two different classes of D-optimal main effects plans with N runs that have the same determinant. However folding them gives two different values foi the D-optimality criteiion One set of designs is D-optimal The other is not.     

Saturated designs for multivariate cubic regression

This paper deals with the problem of finding saturated designs for multivariate cubic regression on a cube which are nearly D-optimal. A finite class of designs is presented for the k dimensional cube having the property that the sequence of the best designs in this class for each k is asymptotically efficient as k increases. A method for constructing good designs in this class is discussed and the construction is carried out for 1?k?8. These numerical results are presented in the last section of the paper.     

On E(s)-optimal supersaturated designs

A popular measure to assess 2-level supersaturated designs is the E(s²)$E(s^2)$ criterion. In this paper, improved lower bounds on E(s²)$E(s^2)$ are obtained. The same improvement has recently been established by Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. However, our analysis provides more details on precisely when an improvement is possible, which is lacking in Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. The equivalence of the bounds obtained by Butler et al. [2001. A general method of constructing E(s²)$E(s^2)$-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632] (in the cases where their result applies) and those obtained by Bulutoglu and Cheng [2004. Construction of E(s²)$E(s^2)$-optimal supersaturated designs. Ann. Statist. 32, 1662–1678] is established. We also give two simple methods of constructing E(s²)$E(s^2)$-optimal designs.     

Optimal designs for statistical analysis with Zernike polynomials

The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and Φ _p -optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849–879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's Φ _p -criteria (p>?∞) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.     

Construction of Efficient Multi-Level k-Circulant Supersaturated Designs

This article proposes an algorithm to construct efficient balanced multi-level k-circulant supersaturated designs with m factors and n runs. The algorithm generates efficient balanced multi-level k-circulant supersaturated designs very fast. Using the proposed algorithm many balanced multi-level supersaturated designs are constructed and cataloged. A list of many optimal and near optimal, multi-level supersaturated designs is also provided for m ≤ 60 and number of levels (q) ≤10. The algorithm can be used to generate two-level k-circulant supersaturated designs also and some large optimal two-level supersaturated designs are presented. An upper bound to the number of factors in a balanced multi-level supersaturated design such that no two columns are fully aliased is also provided.     

An algorithm for calculating D-optimal designs for polynomial regression through a fixed point

Optimal designs are required to make efficient statistical experiments. By using canonical moments, in 1980, Studden found D_s-optimal designs for polynomial regression models. On the other hand, integrable systems are dynamical systems whose solutions can be written down concretely. In this paper, polynomial regression models through a fixed point are discussed. In order to calculate D-optimal designs for these models, a useful relationship between canonical moments and discrete integrable systems is introduced. By using canonical moments and discrete integrable systems, a new algorithm for calculating D-optimal designs for these models is proposed.     

One-factor-at-a-time designs of resolution V

Two classes of designs of resolution V are constructed using one-factor-at-a-time techniques. They facilitate sequential learning and are more economical in run size than regular 2_V^k−p designs. Comparisons of D efficiency with other designs are given to assess the suitability of the proposed designs. Their D_s efficiencies can be dramatically improved with the addition of a few runs.     

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.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

E-optimal incomplete block designs with two distinct block sizes

Sufficient conditions are derived for the determination of E-optimal designs in the class D(v,b₁,b₂,k₁,k₂) of incomplete block designs for v treatments in b₁ blocks of size k₁ each and b₂ blocks of size k₂ each. Some constructions for E-optimal designs that satisfy the sufficient conditions obtained here are given. In particular, it is shown that E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by augmenting b₂ blocks, with k₂ − k₁ extra plots each, of a BIBD(v,b = b₁ + b₂,k₁,λ) and GDD(v,b = b₁ + b₂,k₁,λ₁,λ₂). It is also shown that equireplicate E-optimal designs in D(v,b₁,b₂,k₁,k₂) can be constructed by combining disjoint blocks of BIBD(v,b,k₁,λ) and GDD(v,b,k₁,λ₁,λ₂) into larger blocks. As applications of the construction techniques, several infinite series of E-optimal designs with small block sizes differing by at most two are given. Lower bounds for the A-efficiency are derived and it is found that A-efficiency exceeds 99% for v ⩾ 10, and at least 97.5% for 5 ⩽v < 10.     

D s -optimal designs for Kozak's tree taper model

In this work, we study D _s -optimal design for Kozak's tree taper model. The approximate D _s -optimal designs are found invariant to tree size and hence create a ground to construct a general replication-free D _s -optimal design. Even though the designs are found not to be dependent on the parameter value p of the Kozak's model, they are sensitive to the s×1 subset parameter vector values of the model. The 12 points replication-free design (with 91% efficiency) suggested in this study is believed to reduce cost and time for data collection and more importantly to precisely estimate the subset parameters of interest.     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

On the use of partial confounding for the construction of alternative regular two-level blocked fractional factorial designs

In this paper, we consider experimental situations in which a regular fractional factorial design is to be used to study the effects of m two-level factors using n=2^m−k experimental units arranged in 2^p blocks of size 2^m−k−p. In such situations, two-factor interactions are often confounded with blocks and complete information is lost on these two-factor interactions. Here we consider the use of the foldover technique in conjunction with combining designs having different blocking schemes to produce alternative partially confounded blocked fractional factorial designs that have more estimable two-factor interactions or a higher estimation capacity or both than their traditional counterparts.     

Optimal regular graph designs

A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). To explore the landscape of designs with a large number of blocks, we introduce extensions of regular graph designs. These are constructed by adding the blocks of a balanced incomplete block design repeatedly to the original design. We present the results of an exact computer search for the best regular graph designs and the best extended regular graph designs with up to 20 treatments v, block size \(k \le 10\) and replication r \(\le 10\) and \(r(k-1)-(v-1)\lfloor r(k-1)/(v-1)\rfloor \le 9\).     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.