Three-stage group screening with unequal group sizes and with errors in observations

In this paper three-stage group screening in which group-factors contain differing number of factors is discussed. A procedure for grouping the factors in the absence of concrete prior information is described. Formulas for the expected number of runs and the expected number of incorrect decisions have also been obtained. These formulas are used to formulate criteria for optimal designs.     

The performance of multistage group screening designs

This article deals with multistage group screening in which group-factors contain the same number of factors. A usual assumption of this procedure is that the directions of possible effects are known. In practice, however, this assumption i s often unreasonable. This paper examines, in the case of no errors in observations, the performance of multistage group screening when this assumption is false . This enails consideration of cancellation effects within group-factors.     

Two-stage group-screening designs with equal prior probabilities and no errors in decisions

A discrete approach to group-screening designs in the sense of discontinuous variation in the sizes of group-factors is studied. The results obtained using the finite differences method are compared with Watson!s results obtained by assuming continuous variation. Conditions for two-stage designs to be more economic than the corresponding single-stage designs are also given.     

设计包含最多纯净两因子交互效应分量条件

对于任意的素数或者素数幂 ,本文通过分析 设计中不纯净两因子交互效应(Two-Factor Interaction, 2FI)分量的个数,推导出某些 设计包含最多纯净2FI分量的一个条件。该结论对构造包含最多纯净2FI分量的 设计具有重要意义。     

Response Surface Designs Where Levels of Some Factors are Difficult to Change

This article considers response surface designs in which the number of levels of some of the factors are constrained. Two general types of designs are examined: CUBE designs and STAR designs. The specific factor levels are chosen to give variance contours with a high level of sphericity, thus providing designs that are close to rotatable.     

Minimum Aberration Two-Level Fractional Factorial Split-Plot Designs with Hard to Change Factors

In this article we will consider industrial experiments in which some experimental factors have hard to change levels and others have levels which are easy to change. In such situations, fractional factorial split plot designs are often used where the hard to change factors are included as a subset of the whole plot factors and the easy to change factors make up the subplot factors. Here we consider the problem of finding two-level split plot designs which have minimum aberration among those designs which also minimize the number of level changes for the hard to change factors.     

Efficient circular neighbour designs for spatial interference model

We consider circular block designs for field-trials when there are two-sided spatial interference between neighbouring plots of the same blocks. The parameter of interest is total effects that is the sum of direct effect of treatment and neighbour effects, which correspond to the use of a single treatment in the whole field. We determine universally optimal approximate designs. When the number of blocks may be large, we propose efficient exact designs generated by a single sequence of treatment. We also give efficiency factors of the usual binary block neighbour balanced designs which can be used when the number of blocks is small.     

Analysis of a supersaturated design using Entropy Prior Complexity for binary responses via generalized linear models

A supersaturated design is a factorial design in which the number of effects to be estimated is greater than the available number of experimental runs. It is used in many experiments for screening purposes, i.e., for studying a large number of factors and then identifying the active ones. The goal with such a design is to identify just a few of the factors under consideration, that have dominant effects and to do this at minimum cost. While most of the literature on supersaturated designs has focused on the construction of designs and their optimality, the data analysis of such designs remains still at an early stage. In this paper, we incorporate the parameter model complexity into the supersaturated design analysis process, by assuming generalized linear models for a Bernoulli response, for analyzing main effects designs and discovering simultaneously the effects that are significant.     

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.     

Experimentation order in factorial designs: new findings

Under some very reasonable hypotheses, it becomes evident that randomizing the run order of a factorial experiment does not always neutralize the effect of undesirable factors. Yet, these factors do have an influence on the response, depending on the order in which the experiments are conducted. On the other hand, changing the factor levels is many times costly; therefore it is not reasonable to leave to chance the number of changes necessary. For this reason, run orders that offer the minimum number of factor level changes and at the same time minimize the possible influence of undesirable factors on the experimentation have been sought. Sequences which are known to produce the desired properties in designs with 8 and 16 experiments can be found in the literature. In this paper, we provide the best possible sequences for designs with 32 experiments, as well as sequences that offer excellent properties for designs with 64 and 128 experiments. The method used to find them is based on a mixture of algorithmic searches and an augmentation of smaller designs.     

Supersaturated design including an orthogonal base

Recently, many supersaturated designs have been proposed. A supersaturated design is a fractional factorial design in which the number of factors is greater than the number of experimental runs. The main thrust of the previous studies has been to generate more columns while avoiding large values of squared inner products among all design columns. These designs would be appropriate if the probability for each factor being active is uniformly distributed. When factors can be partitioned into two groups, namely, with high and low probabilities of each factor being active, it is desirable to maintain orthogonality among columns to be assigned to the factors in the high-probability group. We discuss a supersaturated design including an orthogonal base which is suitable for this common situation. Mathematical results on the existence of the supersaturated designs are shown, and the construction of supersaturated designs is presented. We next discuss some properties of the proposed supersaturated designs based on the squared inner products.     

A multi-objective coordinate-exchange two-phase local search algorithm for multi-stratum experiments

A multi-stratum design is a useful tool for industrial experimentation, where factors that have levels which are harder to set than others, due to time or cost constraints, are frequently included. The number of different levels of hardness to set defines the number of strata that should be used. The simplest case is the split-plot design, which includes two strata and two sets of factors defined by their level of hardness-to-set. In this paper, we propose a novel computational algorithm which can be used to construct optimal multi-stratum designs for any number of strata and up to six optimality criteria simultaneously. Our algorithm allows the study of the entire Pareto front of the optimization problem and the selection of the designs representing the desired trade-off between the competing objectives. We apply our algorithm to several real case scenarios and we show that the efficiencies of the designs obtained present experimenters with several good options according to their objectives.     

A Bayesian criterion for selecting super saturated screening designs

Super-saturated designs in which the number of factors under investigation exceeds the number of experimental runs have been suggested for screening experiments initiated to identify important factors for future study. Most of the designs suggested in the literature are based on natural but ad hoc criteria. The “average s²” criteria introduced by Booth and Cox (Technometrics 4 (1962) 489) is a popular choice. Here, a decision theoretic approach is pursued leading to an optimality criterion based on misclassification probabilities in a Bayesian model. In certain cases, designs optimal under the average s² criterion are also optimal for the new criterion. Necessary conditions for this to occur are presented. In addition, the new criterion often provides a strict preference between designs tied under the average s² criterion, which is advantageous in numerical search as it reduces the number of local minima.     

Split-plot designs for multistage experimentation

Most of today’s complex systems and processes involve several stages through which input or the raw material has to go before the final product is obtained. Also in many cases factors at different stages interact. Therefore, a holistic approach for experimentation that considers all stages at the same time will be more efficient. However, there have been only a few attempts in the literature to provide an adequate and easy-to-use approach for this problem. In this paper, we present a novel methodology for constructing two-level split-plot and multistage experiments. The methodology is based on the Kronecker product representation of orthogonal designs and can be used for any number of stages, for various numbers of subplots and for different number of subplots for each stage. The procedure is demonstrated on both regular and nonregular designs and provides the maximum number of factors that can be accommodated in each stage. Furthermore, split-plot designs for multistage experiments with good projective properties are also provided.     

D-Optimal Two-Level Fractional Factorial Designs of Resolution III with Two Dispersion Factors

The problem of finding D-optimal designs, with two dispersion factors, for the estimation of all location main effects is investigated in the class of regular unreplicated two-level fractional factorial designs of resolution III. Designs having length three words involving both of the dispersion factors in the defining relation are shown to be inferior in terms of D-optimality. Tables of factors that are named as the two dispersion factors so that the resulting design is either D-optimal or has the largest determinant of the information matrix are provided. Rank-order of designs is studied when the number of length three words involving either one of the dispersion factors and the number of length four words involving both of the dispersion factors are fixed. Rank-order of designs when the numbers of aforementioned words are less than or equal to ten is given.     

Construction and analysis of edge designs from skew-symmetric supplementary difference sets

The purpose of screening experiments is to identify the dominant variables from a set of many potentially active variables which may affect some characteristic y. Edge designs were recently introduced in the literature and are constructed by using conferences matrices and were proved to be robust. We introduce a new class of edge designs which are constructed from skew-symmetric supplementary difference sets. These designs are particularly useful since they can be applied for experiments with an even number of factors and they may exist for orders where conference matrices do not exist. Using this methodology, examples of new edge designs for 6, 14, 22, 26, 38, 42, 46, 58, and 62 factors are constructed. Of special interest are the new edge designs for studying 22 and 58 factors because edge designs with these parameters have not been constructed in the literature since conference matrices of the corresponding order do not exist. The suggested new edge designs achieve the same model-robustness as the traditional edge designs. We also suggest the use of a mirror edge method as a test for the linearity of the true underlying model. We give the details of the methodology and provide some illustrating examples for this new approach. We also show that the new designs have good D-efficiencies when applied to first order models.     

NESTING OPTIMAL MAIN EFFECTS PLANS AND OPTIMAL FOLDOVER DESIGNS

ABSTRACT

Optimal main effects plans (MEPs) and optimal foldover designs can often be performed as a series of nested optimal designs. Then, if the experiment cannot be completed due to time or budget constraints, the fraction already performed may still be an optimal design. We show that the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEP for 4t factors in 4t + 2 points nested within it. In general, the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEPs for 4t factors in 4t + 1, 4t + 2, or 4t + 3 points and the optimal MEP for 4t + 1 factors in 4t + 4 points does not contain the optimal MEPs for 4t + 1 factors in 4t + 2 or 4t + 3 points. We also show that the runs in an orthogonal design for 4t factors in 4t + 4 points, and the optimal foldover designs obtained by folding, should be performed in a certain sequence in order to avoid the possibility of a singular X'X matrix.     

A catalogue of designs for partial profiles in paired comparison experiments with three groups of factors

A common strategy for avoiding information overload in multi-factor paired comparison experiments is to employ pairs of options which have different levels for only some of the factors in a study. For the practically important case where the factors fall into three groups such that all factors within a group have the same number of levels and where one is only interested in estimating the main effects, a comprehensive catalogue of D-optimal approximate designs is presented. These optimal designs use at most three different types of pairs and have a block diagonal information matrix.     

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 designs for main effects in linear paired comparison models

In psychology, marketing research and sensory analysis paired comparisons which demand judges to evaluate the trade-off between two alternatives constitute a popular method of data collection. For this situation we present optimal designs in a discrete setting when the alternatives are specified by an analysis of variance model with main effects only. We employ combinatorial tools to achieve optimal designs which have sufficiently small sample sizes. Moreover, optimal designs are constructed when the number of factors presented is restricted for each pair of alternatives.