Constructing optimal projection designs

ABSTRACT

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.     

Analysis of supersaturated designs via the Dantzig selector

A supersaturated design is a design whose run size is not enough for estimating all the main effects. It is commonly used in screening experiments, where the goals are to identify sparse and dominant active factors with low cost. In this paper, we study a variable selection method via the Dantzig selector, proposed by Candes and Tao [2007. The Dantzig selector: statistical estimation when p$p$ is much larger than n$n$. Annals of Statistics 35, 2313–2351], to screen important effects. A graphical procedure and an automated procedure are suggested to accompany with the method. Simulation shows that this method performs well compared to existing methods in the literature and is more efficient at estimating the model size.     

The convolution theorem for estimating linear functionals in indirect nonparametric regression models

Nonparametric regression—directly or indirectly observed—is one of the important statistical models. On one hand it contains two infinite dimensional parameters (the regression function and the error density), and on the other it is of rather simple structure. Therefore, it may serve as an interesting paradigm for illustrating or developing abstract statistical theory for non-Euclidean parameters. In this paper estimation of a linear functional of the indirectly observed regression function is considered, when a deterministic design is used. It should be noted that any Fourier coefficient of an expansion of the regression function in an orthonormal basis is such a functional. Because the design is deterministic the observables are independent but not identically distributed. Local asymptotic normality is established and applied to prove Hájek's convolution theorem for this functional. Pertinent references are Beran [1977. Robust location estimates. Ann. Statist. 5, 431–444] and McNeney and Wellner [2000. Application of convolution theorems in semiparametric models with non-i.i.d. data. J. Statist. Plann. Inference 91, 441–480]. For purposes explained above, however, the paper is kept self-contained and full proofs are provided.     

A method for screening active effects in supersaturated designs

A supersaturated design (SSD) is a design whose run size is not enough for estimating all the main effects. The goal in conducting such a design is to identify, presumably only a few, relatively dominant active effects with a cost as low as possible. However, data analysis of such designs remains primitive: traditional approaches are not appropriate in such a situation and several methods which were proposed in the literature in recent years are effective when used to analyze two-level SSDs. In this paper, we introduce a variable selection procedure, called the PLSVS method, to screen active effects in mixed-level SSDs based on the variable importance in projection which is an important concept in the partial least-squares regression. Simulation studies show that this procedure is effective.     

Split-plot designs for robust product experimentation

Genichi Taguchi has emphasized the use of designed experiments in several novel and important applications. In this paper we focus on the use of statistical experimental designs in designingproducts to be robust to environmental conditions. The engineering concept of robust product design is very important because it is frequently impossible or prohibitively expensive to control or eliminate variation resulting from environmental conditions. Robust product design enablesthe experimenter to discover how to modify the design of the product to minimize the effect dueto variation from environmental sources. In experiments of this kind, Taguchi's total experimental arrangement consists of a cross-product of two experimental designs:an inner array containing the design factors and an outer array containing the environmental factors. Except in situations where both these arrays are small, this arrangement may involve a prohibitively large amount of experimental work. One of the objectives of this paper is to show how this amount of work can be reduced. In this paper we investigate the applicability of split-plot designs for thisparticular experimental situation. Consideration of the efficiency of split-plot designs and anexamination of several variants of split-plot designs indicates that experiments conductedin a split-plot mode can be of tremendous value in robust product design since they not only enable the contrasts of interest to be estimated efficiently but also the experiments can be considerably easier to conduct than the designs proposed by Taguchi.     

Bounds on the Maximum Number of Factors in Designs with Two Distinct Groups of Factors

This article focuses on designs involving two distinct groups of factors. In particular, we assume that between-group interactions are more important than within-group interactions. Under this assumption, a new word-length pattern is proposed to characterize the aliasing severity of a design, and the concepts of resolution and aberration are defined accordingly. Furthermore, we have obtained various bounds on the maximum number of factors that a design with given resolution can accommodate.     

Identification of factors influencing dispersion in split-plot experiments

As split-plot designs are commonly used in robust design it is important to identify factors in these designs that influence the dispersion of the response variable. In this article, the Bergman-Hynén method, developed for identification of dispersion effects in unreplicated experiments, is modified to be used in the context of split-plot experiments. The modification of the Bergman-Hynén method enables identification of factors that influence specific variance components in unreplicated two-level fractional factorial splitplot experiments. An industrial example is used to illustrate the proposed method.     

Nested Latin Hypercube Designs with Sliced Structures

In computer experiments, space-filling designs with a sliced structure or nested structure have received much recent interest and been studied separately. However, it is likely that designs with both structures are needed in some situations, but there are no suitable designs so far. In this paper, we construct a special class of nested Latin hypercube designs with sliced structures, in such a design, a small sliced Latin hypercube design is nested within a large one. The construction method is easy to implement and the number of factors is flexible. Numerical simulations show the usefulness of the newly proposed designs.     

Difference variance dispersion graphs for comparing response surface designs with applications in food technology

Variance dispersion graphs have become a popular tool in aiding the choice of a response surface design. Often differences in response from some particular point, such as the expected position of the optimum or standard operating conditions, are more important than the response itself. We describe two examples from food technology. In the first, an experiment was conducted to find the levels of three factors which optimized the yield of valuable products enzymatically synthesized from sugars and to discover how the yield changed as the levels of the factors were changed from the optimum. In the second example, an experiment was conducted on a mixing process for pastry dough to discover how three factors affected a number of properties of the pastry, with a view to using these factors to control the process. We introduce the difference variance dispersion graph (DVDG) to help in the choice of a design in these circumstances. The DVDG for blocked designs is developed and the examples are used to show how the DVDG can be used in practice. In both examples a design was chosen by using the DVDG, as well as other properties, and the experiments were conducted and produced results that were useful to the experimenters. In both cases the conclusions were drawn partly by comparing responses at different points on the response surface.     

Comparison of Four New General Classes of Search Designs

A factor screening experiment identifies a few important factors from a large list of factors that potentially influence the response. If a list consists of m factors each at three levels, a design is a subset of all possible 3 ^m runs. This paper considers the problem of finding designs with small numbers of runs, using the search linear model introduced in Srivastava (1975). The paper presents four new general classes of these 'search designs', each with 2 m −1 runs, which permit, at most, two important factors out of m factors to be searched for and identified. The paper compares the designs for 4 ≤ m ≤ 10, using arithmetic and geometric means of the determinants, traces and maximum characteristic roots of particular matrices. Two of the designs are found to be superior in all six criteria studied. The four designs are identical for m = 3 and this design is an optimal design in the class of all search designs under the six criteria. The four designs are also identical for m = 4 under some row and column permutations.     

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.     

The Construction of Efficient Two-replicate Row–Column Designs for Use in Field Trials

Two-replicate row–column designs are often used for field trials in multisite tree or plant breeding programmes. With only two replicates for each trial, it is important to use designs with optimal or near optimal efficiency factors. This paper presents an algorithm for generating such designs. The method extends the contraction approach of Bailey and Patterson to any set of parameters and uses the factorial design construction algorithm of Williams and John to generate designs. Our experience with the algorithm is that it produces designs that are at least as good as, and often much better and more quickly generated than, those obtained by other recent computer algorithms.     

On Kronecker block designs for factorial experiments

In this paper the use of Kronecker designs for factorial experiments is considered. The two-factor Kronecker design is considered in some detail and the efficiency factors of the main effects and interaction in such a design are derived. It is shown that the efficiency factor of the interaction is at least as large as the product of the efficiency factors of the two main effects and when both the component designs are totally balanced then its efficiency factor will be higher than the efficiency factor of either of the two main effects. If the component designs are nearly balanced then its efficiency factor will be approximately at least as large as the efficiency factor of either of the two main effects. It is argued that these designs are particularly useful for factorial experiments.Extensions to the multi-factor design are given and it is proved that the two-factor Kronecker design will be connected if the component designs are connected.     

The construction of multi-factor crossover designs in animal husbandry studies

For ethical reasons it is important to try to obtain as much useful information as possible from an animal experiment whilst minimizing the number of animals used. Crossover designs, where applicable, provide an ideal framework for achieving this. If two or more treatment factors are included in the crossover design then the reduction in total animal usage can be considerable. In this paper we consider such designs, defined as multi-factor crossover designs. The designs are applicable when there are several different treatment factors, each at t levels, to be applied to the experimental units. The motivation for investigating these designs was a study conducted at GlaxoSmithKline to determine the preference of male and female dogs for t=5 different types of bed and t=5 different bedding conditions. A construction method is given for forming universally optimal designs for t not too large. Also given is an example for the special case where the number of treatment levels t=6.     

Review of random and group-screening designs

Experiments with real systems, but especially with simulated systems, may involve hundreds of factors. However, only a few factors are really important. Detecting these Important factors requires special designs such as random and group-screening de-signs. Random designs are simple, but they yield biased esti-mators. Group-screening is based on aggregation. The assumptions of group-screening are discussed in detail, and seem not very restrictive. References to applications of both design types are given.     

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

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 article, we have identified all the geometric non isomorphic 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 when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.     

Marginal Regression for Binary Longitudinal Data in Adaptive Clinical Trials

Abstract.  In an adaptive clinical trial research, it is common to use certain data-dependent design weights to assign individuals to treatments so that more study subjects are assigned to the better treatment. These design weights must also be used for consistent estimation of the treatment effects as well as the effects of the other prognostic factors. In practice, there are however situations where it may be necessary to collect binary responses repeatedly from an individual over a period of time and to obtain consistent estimates for the treatment effect as well as the effects of the other covariates in such a binary longitudinal set up. In this paper, we introduce a binary response-based longitudinal adaptive design for the allocation of individuals to a better treatment and propose a weighted generalized quasi-likelihood approach for the consistent and efficient estimation of the regression parameters including the treatment effects.     

On the minimum bias response surface designs of Box and Draper

The minimum bias design criterion is an important part of response surface methodology. In this paper, necessary and sufficient conditions for minimum bias designs are derived in terms of the design moments. It is shown that the sufficient conditions of Box and Draper (J. Amer. Statist. Assoc. 54 (1959) 622) are sometimes, but not always, necessary depending on the order of the polynomial model and the number of factors.     

Analyzing supersaturated designs with entropic measures

A supersaturated design is a design for which there are fewer runs than effects to be estimated. In this paper, we propose a method for screening out the important factors from a large set of potentially active variables, based on an information theoretical approach. Three entropy measures: Rényi entropy, Tsallis entropy and Havrda–Charvát entropy, have been associated with the measure of information gain, in order to identify the significant factors using data and assuming generalized linear models. The investigation of the proposed method performance and the comparison of each entropic measure application have been accomplished through simulation experiments. A noteworthy advantage of this paper is the use of generalized linear models for analyzing data from supersaturated designs, a fact that, to the best of our knowledge, has not yet been studied.     

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.