Small asymmetric fractional factorial plans for main effects and specified two-factor interactions

Fractional factorial plans represented by orthogonal arrays of strength two are known to be optimal in a very strong sense under a model that includes the mean and all the main effects, when all interactions are assumed to be absent. When a fractional factorial plan given by an orthogonal array of strength two is not saturated, one might think of entertaining some two-factor interactions also in the model. In such a situation, it is of interest to examine which of the two-factor interactions can be estimated via a plan represented by an orthogonal array, as also to study the overall efficiency of the plan when some interactions are in the model alongwith the mean and all main effects. In this paper, an attempt has been made to examine these issues by considering some practically useful plans for asymmetric (mixed level) factorials with small number of runs.     

On a property of orthogonal arrays and optimal blocking of fractional factorial plans

It is shown that fractional factorial plans represented by orthogonal arrays of strength three are universally optimal under a model that includes the mean, all main effects and all two-factor interactions between a specified factor and each of the other factors. Thus, such plans exhibit a kind of model robustness in being universally optimal under two different models. Procedures for obtaining universally optimal block designs for fractional factorial plans represented by orthogonal arrays are also discussed. Acknowledgements. The authors wish to thank the referees for making several useful comments on a previous version.     

A note on the projection properties of Hadamard matrices obtained by the first Payley construction

We show that the projections on four factors of an arbitrary orthogonal array of strength 2 allow the estimation of main effects and two-factor interactions when all other effects are assumed to be zero, if those projections satisfy the bounds given by Weils theorem. The only exceptions are the Hadamard matrices of orders 16 and 24. A consequence is again the estimability of main effects and two-factor interactions for the projections on four factors of the first Payley construction for arbitrary run size.     

U-type and column-orthogonal designs for computer experiments

U-type designs and orthogonal Latin hypercube designs (OLHDs) have been used extensively for performing computer experiments. Both have good spaced filling properties in one-dimension. U-type designs may not have low correlations among the main effects, quadratic effects and two-factor interactions. On the other hand, OLHDs are hard to be found due to their large number of levels for each factor. Recently, alternative classes of U-type designs with zero or low correlations among the effects of interest appear in the literature. In this paper, we present new classes of U-type or quantitative \(3\) -orthogonal designs for computer experiments. The proposed designs are constructed by combining known combinatorial structures and they have their main effects pairwise orthogonal, orthogonal to the mean effect, and orthogonal to both quadratic effects and two-factor interactions.     

Projection properties of certain three level orthogonal arrays

Two orthogonal arrays based on 3 symbols are said to be isomorphic or combinatorially equivalent if one can be obtained from the other by a sequence of row permutations, column permutations and permutations of symbols in each column. Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a ``projection design'. In this paper we have identified all the inequivalent projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k=3,4 and 5 factors. It is shown that the generalized wordlength pattern criterion proposed by Ma and Fang [23] can distinguish between most, but not all, inequivalent classes. We propose an extension of the Es² criterion (which is commonly used for measuring efficiency of 2-level designs) to distinguish further between the non-isomorphic classes and to measure the efficiency of the designs in these classes. Some concepts on generalized resolution are also discussed.     

Orthogonal main effect plans for two and three factors in small blocks

Orthogonal main-effect plans for two and three factors in small blocks are obtained from the dual of adjusted orthogonal row-column designs. The method for constructing efficient plans is presented, and a relationship between the average efficiency factors of the row-column design and the corresponding main effects is given for the two-factor case. Orthogonal Main Effect.     

A new approach in constructing orthogonal and nearly orthogonal arrays

Orthogonal arrays have been constructed by a number of mathematical tools such as orthogonal Latin squares, Hadamard matrices, group theory and finite fields. Wang and Wu (1992) proposed the concept of a nearly orthogonal array and found a number of such arrays with high efficiency. In this paper we propose some criteria for non-orthogonality and two algorithms for the construction of orthogonal and nearly orthogonal arrays evincing higher efficiency than that obtained by Wang and Wu. Received: September 1999     

An effective step-down algorithm for the construction and the identification of nonisomorphic orthogonal arrays

In this paper we present an effective algorithm for the construction and the identification of two-level nonisomorphic orthogonal arrays. Using this algorithm, we identify and list a full catalogue of nonisomorphic orthogonal arrays with parameters OA(24,7,2,t), OA(28,6,2,t) and OA(32,6,2,t), t ≥ 2. Some statistical properties of these designs are also considered.     

18-run nonisomorphic three level orthogonal arrays

In this paper we construct all possible orthogonal arrays OA(18,q, 3,2) with 18 runs and 3 ≤ q ≤ 7 columns and present those that are nonisomorphic. A discussion on the novelty and the superiority of many of the designs found in terms of isomorphism and generalized minimum aberration has been made.      

D-optimal weighing designs for six objects

For all integers m≥6, we determine the maximum value of det X ^T X, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix X ^T X is discussed. Received: May 2000     

Barycentric algorithm for computing D-optimal size- and cost-constrained designs of experiments

In this paper, we study the problem of D-optimal experimental design under two linear constraints, which can be interpreted as simultaneous restrictions on the size and on the cost of the experiment. For computing a size- and cost-constrained approximate D-optimal design, we propose a specification of the “barycentric” multiplicative algorithm with sequential removal of redundant design points. We analytically prove convergence results for the proposed algorithm and numerically demonstrate its favorable properties compared to competing methods.     

Replicating Walrasian equilibria using markets for membership in labor-managed firms

In standard models of the labor-managed firm, labor is misallocated in the short run. This problem can be eliminated by introducing a market for membership in each firm. These markets substitute for the capitalist labor market and support equilibria isomorphic to Walrasian equilibria. Such LMFs resemble familiar Ward-Domar-Vanek LMFs by ensuring that firm sizes are optimal for ex post members, but they also have the property that firm sizes are optimal for workers endowed with ex ante membership rights.     

A recursive method for orthogonal designs

We introduce a class of arrays which can be used with amicable sets of matrices. Using a recursive method we introduce a class of amicable matrices which can be used in the new arrays to generate a large class of full orthogonal designs. The work was completed while the author was on study leave from University of Lethbridge, visiting IPM in Tehran. Hospitality and research grant provided by IPM is greatly appreciated. Supported by an NSERC Discovery grant     

Een criterium voor het kiezen van een proefopzet

A characterization of D-optimality is given together with several examples where D-optimal designs are computed.     

Optimality results on orthogonal arrays plus <Emphasis Type="Italic">p</Emphasis> runs for <Emphasis Type="Italic">s</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript> factorial experiments

The optimality of designs obtained by adding p runs to an orthogonal array is studied for experiments involving m factors each at s levels. The optimality criterion used here, is the Type 1 criterion due to Cheng (1978) which is an extension of Kiefer (1975) universal optimality criterion. Unlike what happens with orthogonal array plus one run designs, the behavior of designs obtained via augmentation of an orthogonal array by p runs depends on the particular runs added.     

Some orthogonal arrays with 32 runs and their projection properties

Screening designs are useful for situations where a large number of factors are examined but only a few, k, of them are expected to be important. Traditionally orthogonal arrays such as Hadamard matrices and Plackett Burman designs have been studied for this purpose. It is therefore of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties. In this paper we present 15 inequivalent Hadamard matrices of order n=32 constructed from circulant cores. We study their projection properties using several well-known statistical criteria and we provide minimum generalized aberration 2 level designs with 32 runs and up to seven factors that are embedded into these Hadamard matrices. A concept of generalized projectivity and design selection of such designs is also discussed.AMS Subject Classification: Primary 62K15, Secondary 05B20     

Exact and approximate D-optimal designs in polynomial regression

If the sample sizen is large enough, then the exact polynomial regression designs obtained by rounding the weights of the approximate D-optimal design to integral multiples of 1/n are D-optimal. This was shown by alaevskiî (1966) and Gaffke (1987). In this note, an efficient algorithm to determine the minimum sample sizen _d for a polynomial model of degreed is derived from a condition given by Huang (1987). Under an additional assumption we show that the conditions of Gaffke and Huang are equivalent; we verify the additional assumption for polynomial degreed40.     

D-optimal chemical balance weighing designs with autoregressive errors

In this paper, we consider the estimation problem of individual weights of three objects. For the estimation we use the chemical balance weighing design and the criterion of D-optimality. We assume that the error terms ${\varepsilon_{i},\ i=1,2,\dots,n,}$ are a first-order autoregressive process. This assumption implies that the covariance matrix of errors depends on the known parameter ρ. We present the chemical balance weighing design matrix ${\widetilde{\bf X}}$ and we prove that this design is D-optimal in certain classes of designs for ${\rho\in[0,1)}$ and it is also D-optimal in the class of designs with the design matrix ${{\bf X} \in M_{n\times 3}(\pm 1)}$ for some ρ ≥ 0. We prove also the necessary and sufficient conditions under which the design is D-optimal in the class of designs ${M_{n\times 3}(\pm 1)}$ , if ${\rho\in[0,1/(n-2))}$ . We present also the matrix of the D-optimal factorial design with 3 two-level factors.     

Bootstrapping sequential change-point tests for linear regression

Bootstrap methods for sequential change-point detection procedures in linear regression models are proposed. The corresponding monitoring procedures are designed to control the overall significance level. The bootstrap critical values are updated constantly by including new observations obtained from the monitoring. The theoretical properties of these sequential bootstrap procedures are investigated, showing their asymptotic validity. Bootstrap and asymptotic methods are compared in a simulation study, showing that the studentized bootstrap tests hold the overall level better especially for small historic sample sizes while having a comparable power and run length.     

Optimal designs for generalized linear models with biased response

This paper is concerned with the search for locally optimal designs when the observations of the response variable arise from a weighted distribution in the exponential family. Locally optimal designs are derived for regression models in which the response follows a weighted version of Normal, Gamma, Inverse Gaussian, Poisson or Binomial distributions. Some conditions are given under which the optimal designs for the weighted and original (non-weighted) distributions are the same. An efficiency study is performed to find out the behavior of the D-optimal designs for the original distribution when they are used to estimate models with weighted distributions.