Angela M. Dean

k-Circulant Supersaturated Designs

A class of supersaturated designs called k-circulant designs is explored. These designs are constructed from cyclic generators by cycling k elements at a time. The class of designs includes many Es2-optimal designs, some of which are already known and some of which are more efficient than known designs for model estimation under factor sparsity. Generators for the most efficient designs are listed, and projection properties of some of the designs are explored. We also illustrate that some k-circulant supersaturated designs can be augmented with interaction columns to produce efficient designs for a larger number of factors or for estimating interactions.

Detection of interactions in experiments on large numbers of factors

One of the main advantages of factorial experiments is the information that they can offer on interactions. When there are many factors to be studied, some or all of this information is often sacrificed to keep the size of an experiment economically feasible. Two strategies for group screening are presented for a large number of factors, over two stages of experimentation, with particular emphasis on the detection of interactions. One approach estimates only main effects at the first stage (classical group screening), whereas the other new method (interaction group screening) estimates both main effects and key two-factor interactions at the first stage. Three criteria are used to guide the choice of screening technique, and also the size of the groups of factors for study in the first-stage experiment. The criteria seek to minimize the expected total number of observations in the experiment, the probability that the size of the experiment exceeds a prespecified target and the proportion of active individual factorial effects which are not detected. To implement these criteria, results are derived on the relationship between the grouped and individual factorial effects, and the probability distributions of the numbers of grouped factors whose main effects or interactions are declared active at the first stage. Examples are used to illustrate the methodology, and some issues and open questions for the practical implementation of the results are discussed.

Mixture Designs for Four Components in Orthogonal Blocks

The problem of partitioning the blends (runs) of four mixture components into two orthogonal blocks when a quadratic model is fitted is considered. This is motivated by an industrial investigation of bread-making flours carried out at Spillers Milling Limited, a member of the Dalgety group of companies in the United Kingdom. The design solution proposed by John and described by Cornell is discussed and extended. Study of the characteristics of Latin squares of side 4 leads to reliable rules for quickly obtaining designs of specified kinds. One such design was selected for the experiment at Spillers Milling. Mixture-component values that cause singularity in the new designs are identified, and values that provide designs with highest D-criterion values are obtained for the class of designs discussed. Conveniently rounded, near-optimal mixture component values were chosen for the Spillers Milling experiment, and the analysis led to the prediction of an optimal flour mixture.

Optimal designs for diallel cross experiments

Using nested balanced incomplete block designs, new families of optimal block designs for a certain type of dialled cross experiments are obtained. It is further shown that triangular partially balanced incomplete block designs satisfying a certain parametric condition also lead to optimal designs for diallel crosses. These results unify and extend some of the earlier results on optimality of block designs for diallel crosses.

Mixture experiments: ILL-conditioning and quadratic model specification

Well-conditioned models are important, particularly for practitioners who work with regression models for mixture experiments where parameter estimates are individually meaningful. In this article we investigate conditioning in second-order mixture models, using variance inflation factors, maximum and minimum eigenvalues of the information matrix and condition numbers to assess conditioning. A range of equivalent mixture models that lie “between” the Scheffé model (S-model) and the Kronecker model (K-model) is examined, and pseudocomponent transformations for lower bounds (L-pseudocomponents) and upper bounds (U-pseudocomponents) are also discussed. We prove that the maximum eigenvalue for the information matrix for the K-model is always smaller than that for any other model in the above range. We recommend in practice the use of the K-model, to reduce ill-conditioning, and the appropriate use of pseudocomponents.

Noncollapsing Space-Filling Designs for Bounded Nonrectangular Regions.

Many researchers use computer simulators as experimental tools, especially when physical experiments are infeasible. When computer codes are computationally intensive, nonparametric predictors can be fitted to training data for detailed exploration of the input–output relationship. The accuracy of such flexible predictors is enhanced by taking training inputs to be “space-filling.” If there are inputs that have little or no effect on the response, it is desirable that the design be “noncollapsing” in the sense of having space-filling lower dimensional projections. This article describes an algorithm for constructing noncollapsing space-filling designs for bounded input regions that are of possibly high dimension. Online supplementary materials provide the code for the algorithm, examples of its use, and show its performance in multiple settings.

Mixture designs for five components in orthogonal blocks

The methods developed by John and Draper et al. of partitioning the blends (runs) of four mixture components into two or more orthogonal blocks when fitting quadratic models are extended to mixtures of five components. The characteristics of Latin squares of side five are used to derive rules for reliably and quickly obtaining designs with specific properties. The designs also produce orthogonal blocks when higher order models are fitted.

Complete Block Designs

In the presence of controllable nuisance factors, the device of blocking divides the experimental material into homogeneous blocks in such a way that treatments can be compared under similar conditions. This chapter describes complete block designs (including randomized block designs), together with block design models, model assumption checks, multiple comparisons, sample size calculations, and analysis of variance. The analyses of complete block designs are illustrated through two real experiments, one having factorial treatment combinations. The use of R and SAS software is described.

Response Surface Methodology

Experiments for fitting a predictive model involving several continuous variables are known as response surface experiments. The objectives of response surface methodology include the determination of variable settings for which the mean response is optimized and the estimation of the response surface in the vicinity of this good location. The first part this chapter discusses first-order designs and first-order models, including lack of fit and the path of steepest ascent to locate the optimum. The second part of the chapter introduces second-order designs and models for exploring the vicinity of the optimum location. The application of response surface methodology is demonstrated through a real experiment. The concepts introduced in this chapter are illustrated through the use of SAS and R software.

Two-Stage Sensitivity-Based Group Screening in Computer Experiments.

Sophisticated computer codes that implement mathematical models of physical processes can involve large numbers of inputs, and screening to determine the most active inputs is critical for understanding the input-output relationship. This article presents a new two-stage group screening methodology for identifying active inputs. In Stage 1, groups of inputs showing low activity are screened out; in Stage 2, individual inputs from the active groups are identified. Inputs are evaluated through their estimated total (effect) sensitivity indices (TSIs), which are compared with a benchmark null TSI distribution created from added low noise inputs. Examples show that, compared with other procedures, the proposed method provides more consistent and accurate results for high-dimensional screening. Additional details and computer code are provided in supplementary materials available online.