Steven G. Gilmour

A general method of constructing E(s2)-optimal supersaturated designs

There has been much recent interest in supersaturated designs and their application in factor screening experiments. Supersaturated designs have mainly been constructed by using the E(s2)‐optimality criterion originally proposed by Booth and Cox in 1962. However, until now E(s2)‐optimal designs have only been established with certainty for n experimental runs when the number of factors m is a multiple of n‐1, and in adjacent cases where m=q( n‐1) +r (|r| 2, q an integer). A method of constructing E(s2)‐optimal designs is presented which allows a reasonably complete solution to be found for various numbers of runs n including n,=8 12, 16, 20, 24, 32, 40, 48, 64.

Multistratum Response Surface Designs

Response surface designs are usually described as if the treatments have been completely randomized to the experimental units. However, in practice there is often a structure to the units, implying the need for blocking. If, in addition, some factors are more difficult to vary between units than others, a multistratum structure arises naturally. We present a general strategy for constructing response surface designs in multistratum unit structures. Designs are constructed stratum by stratum, starting in the highest stratum. In each stratum a prespecified treatment set for the factors applied in that stratum is arranged to be nearly orthogonal to the units in the higher strata, allowing for all the effects that have to be estimated. Three examples are given to show the applicability of the method and are also used to check the relationship of the final design to the choice of treatment set. Finally, some practical considerations in randomization are discussed.

Characterisation of colloidal gas aphrons for subsequent use for protein recovery

Abstract Colloidal gas aphrons (CGAs) were first reported by Sebba (J. Colloid Interface Sci., 35 (4) (1971) 643) as micro bubbles (10-100 μm), composed of a gaseous inner core surrounded by a thin surfactant film, which are created by intense stirring of a surfactant solution. Since then, these colloidal dispersions have been used for diverse applications (clarification of suspensions, removal of sulphur crystals, separation of organic dyes from wastewater, etc.). However, there have been no reports, as yet, of their direct application for protein recovery. In this study, CGAs are created from an anionic surfactant (AOT) and are characterised in terms of stability and gas hold-up for a range of process parameters relevant to their proposed use for protein recovery, at a later stage. A statistical experimental design was developed in order to study the effect of different factors (surfactant concentration, salt concentration, pH, time of stirring and temperature) on the stability and gas hold-up of CGAs. The analysis of results from the experimental design provides predictive statistical models. Stability was found to depend mainly on salt and surfactant concentration. Several interactions are shown to be significant including the time-temperature interaction. Gas hold-up was found to depend mainly on salt and surfactant concentration and time of stirring. Also, results from power measurements are presented and the minimum energy for the formation of CGAs, for one set of solution properties, is determined.

The interpretation of Mallows's C p -statistic

When selecting variables in multiple-regression studies, the model with the lowest value of Mallowss C p -statistic is often chosen. It is shown here that when the estimate of σ 2 comes from the full model an adjusted C p , V p , has the property that E(C p ) = p. It is suggested that a procedure be adopted which involves testing whether the model with minimum C p is really better than a simpler model. Tables approximating the null distribution of the test statistics are given.

Factor Screening via Supersaturated Designs

Supersaturated designs are fractional factorial designs that have too few runs to allow the estimation of the main effects of all the factors in the experiment. There has been a great deal of interest in the development of these designs for factor screening in recent years. A review of this work is presented, including criteria for design selection, in particular the popular E(s 2) criterion, and methods for constructing supersaturated designs, both combinatorial and computational. Various methods, both classical and partially Bayesian, have been suggested for the analysis of data from supersaturated designs and these are critically reviewed and illustrated. Recommendations are made about the use of supersaturated designs in practice and suggestions for future research are given.

An algorithm for arranging response surface designs in small blocks

We consider the problem of blocking response surface designs when the block sizes are prespecified to control variation efficiently and the treatment set is chosen independently of the block structure. We show how the loss of information due to blocking is related to scores defined by Mead and present an interchange algorithm based on scores to improve a given blocked design. Examples illustrating the performance of the algorithm are given and some comparisons with other designs are made.

Design of microarray experiments for genetical genomics studies.

Microarray experiments have been used recently in genetical genomics studies, as an additional tool to understand the genetic mechanisms governing variation in complex traits, such as for estimating heritabilities of mRNA transcript abundances, for mapping expression quantitative trait loci, and for inferring regulatory networks controlling gene expression. Several articles on the design of microarray experiments discuss situations in which treatment effects are assumed fixed and without any structure. In the case of two-color microarray platforms, several authors have studied reference and circular designs. Here, we discuss the optimal design of microarray experiments whose goals refer to specific genetic questions. Some examples are used to illustrate the choice of a design for comparing fixed, structured treatments, such as genotypic groups. Experiments targeting single genes or chromosomic regions (such as with transgene research) or multiple epistatic loci (such as within a selective phenotyping context) are discussed. In addition, microarray experiments in which treatments refer to families or to subjects (within family structures or complex pedigrees) are presented. In these cases treatments are more appropriately considered to be random effects, with specific covariance structures, in which the genetic goals relate to the estimation of genetic variances and the heritability of transcriptional abundances.

Efficient and accurate experimental design for enzyme kinetics: Bayesian studies reveal a systematic approach

In areas such as drug development, clinical diagnosis and biotechnology research, acquiring details about the kinetic parameters of enzymes is crucial. The correct design of an experiment is critical to collecting data suitable for analysis, modelling and deriving the correct information. As classical design methods are not targeted to the more complex kinetics being frequently studied, attention is needed to estimate parameters of such models with low variance. We demonstrate that a Bayesian approach (the use of prior knowledge) can produce major gains quantifiable in terms of information, productivity and accuracy of each experiment. Developing the use of Bayesian Utility functions, we have used a systematic method to identify the optimum experimental designs for a number of kinetic model data sets. This has enabled the identification of trends between kinetic model types, sets of design rules and the key conclusion that such designs should be based on some prior knowledge of K(M) and/or the kinetic model. We suggest an optimal and iterative method for selecting features of the design such as the substrate range, number of measurements and choice of intermediate points. The final design collects data suitable for accurate modelling and analysis and minimises the error in the parameters estimated.

Some practical advice on polynomial regression analysis from blocked response surface designs

It is often necessary to run response surface designs in blocks. In this paper the analysis of data from such experiments, using polynomial regression models, is discussed. The definition and estimation of pure error in blocked designs are considered. It is recommended that pure error is estimated by assuming additive block and treatment effects, as this is more consistent with designs without blocking. The recovery of inter-block information using REML analysis is discussed, although it is shown that it has very little impact if the design is nearly orthogonally blocked. Finally prediction from blocked designs is considered and it is shown that prediction of many quantities of interest is much simpler than prediction of the response itself.

Planning incomplete block experiments when treatments are genetically related.

Selection trials in plant and animal breeding, in incomplete blocks, are described by linear models with random effect parameters associated with treatments with known genetic covariance structure. It is now well known that the information on relatives can improve the analysis and many extensions of this model have been proposed, but no studies have been done on the consequences of this genetical relatedness among treatments for the optimality of block designs. Using a suitable optimality criterion, we show that the knowledge on relatedness may imply that the optimal design is not in the class of designs which are optimal for unrelated treatments. Implications for practical applications are discussed.