Partial Duplication of Factorial Experiments

Two level factorial and fractional factorial designs are described in which a subset of the treatment combinations are duplicated. The duplicate runs provide an unbiased estimate of the experimental error variance and more reliable estimates of the effects. Several analysis procedures are described and a numerical example is given. Designs for partially duplicated two level factorial and fractional factorial designs are proposed for as many as eleven factors.     

An Approximation to Student's t

This paper describes two level fractional factorial designs in which some of the treatment combinations are duplicated. As is well known, the duplicated runs provide an unbiased estimate of error variance and more precise estimates of the effects. An example is given with the analysis procedure. Block designs for the corresponding fractional factorial designs are also given.     

Saturated Fractions of 2 n and 3 n Factorial Designs

Saturated fractions of 2 ⁿ and 3 ⁿ factorial designs which permit the estimation of both main effects and first order interactions are described. A simple method of generating these particular designs is given. In addition to presenting the specific designs for n = 3, …, 10, tables of variances and relative efficiencies are included to assist the potential user in assessing the suitability of the described designs.     

Experimental Design in Dissolution Testing

Abstract

Comparison of dissolution profiles may be facilitated by blocking the individual units of a given batch, thus greatly reducing the possibility of error from variation between experimental runs. Experimental designs are described which allow valid comparisons to be made between batches, as well as allowing the between run variation to he assessed and identifying any systematic errors resulting from differences between vessels. The number of tests reguired may freguently be reduced, and the need for replicate testing eliminated. The limitation of 6 vessels per run imposes certain restrictions in the experimental designs possible. Applications of these experimental designs in characterisation of dosage forms by their pH-dissolution topography and their use in factorial formulation experiments are described.     

Adapting second‐order response surface designs to specific needs

Experimental design strategies most often involve an initial choice of a classic factorial or response surface design and adapt that design to meet restrictions or unique requirements of the system under study. One such experience is described here, in which the objective was to develop an efficient experimental design strategy that would facilitate building second‐order response models with excellent prediction capabilities. In development, careful consideration was paid to the desirable properties of response surface designs. Once developed, the proposed design was evaluated using Monte Carlo simulation to prove the concept, a pilot implementation of the design carried out to evaluate the accuracy of the response models, and a set of validation runs enacted to look for potential weaknesses in the approach. The purpose of the exercise was to develop a procedure to efficiently and effectively calibrate strain‐gauge balances to be used in wind tunnel testing. The current calibration testing procedure is based on a time‐intensive one‐factor‐at‐a‐time method. In this study, response surface methods were used to reduce the number of calibration runs required during the labor‐intensive heavy load calibration, to leverage the prediction capabilities of response surface designs, and to provide an estimate of uncertainty for the calibration models. Results of the three‐phased approach for design evaluation are presented. The new calibration process will require significantly fewer tests to achieve the same or improved levels of precision in balance calibration. Copyright © 2008 John Wiley & Sons, Ltd.     

Analyzing unreplicated 2k factorial designs by examining their projections into k‐1 factors

Unreplicated factorial designs are widely used as experimental designs because of the economy they offer in run size. However, they are difficult to analyze because there are no degrees of freedom left to estimate the experimental error. Many methods have been proposed for the analysis of such designs with Lenth's (Technometrics 1989; 31 :469–473) and Dong's (Statist. Sinica 1993; 3 :209–217) being the most popular. In this paper we take advantage of the projective property of factorial designs and we propose a simple yet effective method for analyzing unreplicated factorial designs. The results are compared through a simulation study with Lenth's and Dong's methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Past Editors and Editorial Collaborators 1976

D-optimal fractions of three-level factorial designs for p factors are constructed for factorial effects models (2 ≤ p ≤ 4) and quadratic response surface models (2 ≤ p ≤ 5). These designs are generated using an exchange algorithm for maximizing |X′X| and an algorithm which produces D-optimal balanced array designs. The design properties for the DETMAX designs and the balanced array designs are tabulated. An example is given to illustrate the use of such designs.     

Stu Hunter's Contributions to Experimental Design and Quality Engineering

ABSTRACT

Professor J. Stuart Hunter has long been one of the leaders of our field. He has made many pioneering contributions to experimental design and the general field of quality engineering, including response surface methodology, fractional factorial designs, and the use of experimental design in product and process design including the robust design problem. This article highlights some of his key technical contributions and identifies some additional work that has been inspired by his research.     

Experimental Design in Dissolution Testing

Comparison of dissolution profiles may be facilitated by blocking the individual units of a given batch, thus greatly reducing the possibility of error from variation between experimental runs. Experimental designs are described which allow valid comparisons to be made between batches, as well as allowing the between run variation to he assessed and identifying any systematic errors resulting from differences between vessels. The number of tests reguired may freguently be reduced, and the need for replicate testing eliminated. The limitation of 6 vessels per run imposes certain restrictions in the experimental designs possible. Applications of these experimental designs in characterisation of dosage forms by their pH-dissolution topography and their use in factorial formulation experiments are described.     

Discussion: Posterior Probabilities and Experimental Designs for Discriminating Between Regression Models

The treatment-design portion of fractionated two-level split-plot designs is associated with a subset of the 2 ^n–k fractional factorial designs. The concept of aberration is then extended to these splitplot designs to compare designs. Two methods are presented for constructing two-level minimumaberration split-plot designs, along with examples. An extensive catalog of such designs is tabulated. Extensions to prime-level designs and relations to inner-outer arrays are also presented.     

Experimental Designs for Constrained Regions

The familiar factorial, fractional factorial, and response surface designs are designs for regularly-shaped regions of interest, typically cuboidal regions and spherical regions. An irregularly shaped region of experimentation arises in situations where there are constraints on the factor level combinations that can be run or restrictions on portions of the region of exploration. Computer-generated designs based on some optimality criterion are a logical alternative for these problems. We give a brief tutorial on design optimality criteria and show how one of these, the D-optimality criteria, can lead to very reasonable designs for constrained regions of interest. We show through a simulation study that D-optimal designs perform very well with respect to the capability of selecting the correct model and accurately estimating the design factor levels that result in the optimal response.     

Factorial Designs,the |X′X| Criterion,and Some Related Matters

Two of the basic approaches to choosing an n-point experimental design in many industrial situations are (i) to set down a simple factorial or fractional factorial design in the factors being studied, or (ii) to choose a design based on the well-known |X′X| criterion. Experimenters often prefer (i) due to its simplicity; our viewpoint here is that (ii) is much better. We first indicate some situations for which (when all the factors are restricted to a cuboidal region) the factorial approach is optimal, as judged by the |X′X| criterion, but the assumed models are often not sensible ones in practical work. We then examine what (similarly restricted) designs are optimal under the |X′X| criterion for the standard linear models of first and second order; because of the very rapid increase in computational difficulties, we consider only “cube plus star” type designs for k ≥ 3 (except for k = 3, n = 10). In spite of computational requirements, we recommend use of the |X′X| criterion in general rather than the indiscriminate use of factorials and we briefly discuss the reasons why, both for linear and nonlinear model situations.     

Experimental Design: Review and Comment

We review major developments in the design of experiments, offer our thoughts on important directions for the future, and make specific recommendations for experimenters and statisticians who are students and teachers of experimental design, practitioners of experimental design, and researchers jointly exploring new frontiers. Specific topics covered are optimal design, computer-aided design, robust design, response surface design, mixture design, factorial design, block design, and designs for nonlinear models.     

No‐confounding designs with 20 runs—Alternatives to resolution IV screening designs

When experimental resources are significantly constrained, resolution V fractional factorial designs are often prohibitively large for experiments with 6 or more factors. Resolution IV designs may also be cost prohibitive, as additional experimentation may be required to de‐alias active 2‐factor interactions (2FI). This paper introduces 20‐run no‐confounding screening designs for 6 to 12 factors as alternatives to resolution IV designs. No‐confounding designs have orthogonal main effects, and since no 2FI is completely confounded with another main effects or 2FI, the experimental results can be analyzed without follow‐on experimentation. The paper concludes with the results of a Monte Carlo simulation used to assess the model‐fitting accuracy of the recommended designs.     

Construction of mixed-level supersaturated designs and comparison of their performance: Application to a gas chromatographic method

A supersaturated design is a design for which there are fewer runs than effects to be estimated. Although two-level supersaturated designs are becoming increasingly popular, mixed-level designs are scarcely used. Mixed-level designs are needed when the response is based on a polynomial response surface model or in situations where factors are nominal variables (with more than two modalities). The aim of this study is to explore the construction of mixed-level supersaturated designs and to evaluate their performance from the analysis of peppermint oil using a gas chromatographic method as application. This experimental setup requires the study of seven factors at two levels and five factors at three levels. Different building methods are tested from asymmetric or symmetric supersaturated designs. The mixed-level supersaturated designs obtained are compared from the point of view of a priori criteria with the aim of evaluating which criteria are better suited to judge the quality and fitness for purpose of these experimental designs. Finally, the results of the supersaturated designs are compared to the complete classical design.     

Balancing mixed-model assembly lines with adjacent task duplication

When a mixed-model assembly line (MAL) is balanced, it is generally assumed that the variants of a task over different models should be assigned to an identical station. In this study, this restriction is relaxed and the variants of a task over different models can be duplicated on two adjacent stations (referred to as adjacent task duplication) to improve the MAL’s efficiency. The adjacent task duplication incurs few additional training and tool duplication costs as each task is duplicated on at most two stations. Moreover, for each task, the assembly part storage is not duplicated as it can be shared by the two adjacent stations. The mathematical model of this problem is formulated and some important properties are characterised. A branch, bound and remember algorithm is then developed to solve the problem. The performance of the proposed algorithm is tested on 900 representative instances, of which 889 instances are optimally solved. The experimental results show that the use of the adjacent task duplication policy effectively reduces the number of stations, especially when the WEST ratios are small.     

Analysis and efficient 2k − 1 designs for experiments in blocks of size two

Two‐level factorial designs in blocks of size two are useful in a variety of experimental settings, including microarray experiments. Replication is typically used to allow estimation of the relevant effects, but when the number of factors is large this common practice can result in designs with a prohibitively large number of runs. One alternative is to use a design with fewer runs that allows estimation of both main effects and two‐factor interactions. Such designs are available in full factorial experiments, though they may still require a great many runs. In this article, we develop fractional factorial design in blocks of size two when the number of factors is less than nine, using just half of the runs needed for the designs given by Kerr (J Qual. Tech. 2006; 38 :309–318). Two approaches, the orthogonal array approach and the generator approach, are utilized to construct our designs. Analysis of the resulting experimental data from the suggested design is also given. Copyright © 2011 John Wiley & Sons, Ltd.     

Partial confounding and projective properties of Plackett–Burman designs

Screening experiments are typically used when attempting to identify a few active factors in a larger pool of potentially significant factors. In general, two‐level regular factorial designs are used, but Plackett–Burman (PB) designs provide a useful alternative. Although PB designs are run‐efficient, they confound the main effects with fractions of strings of two‐factor interactions, making the analysis difficult. However, recent discoveries regarding the projective properties of PB designs suggest that if only a few factors are active, the original design can be reduced to a full factorial, with additional trials frequently forming attractive patterns. In this paper, we show that there is a close relationship between the partial confounding in certain PB designs and their projective properties. With the aid of examples, we demonstrate how this relationship may help experimenters better appreciate the use of PB designs. Copyright © 2006 John Wiley & Sons, Ltd.     

Response surface optimization for a nonlinearly constrained irregular experimental design space

Finding optimum conditions for process factors in an engineering optimization problem with response surface functions requires structured data collection using experimental design. When the experimental design space is constrained owing to external factors, its design space may form an asymmetrical and irregular shape and thus standard experimental design methods become ineffective. Computer-generated optimal designs, such as D-optimal designs, provide alternatives. While several iterative exchange algorithms for D-optimal designs are available for a linearly constrained irregular design space, it has not been clearly understood how D-optimal design points need to be generated when the design space is nonlinearly constrained and how response surface models are optimized. This article proposes an algorithm for generating the D-optimal design points that satisfy both feasibility and optimality conditions by using piecewise linear functions on the design space. The D-optimality-based response surface design models are proposed and optimization procedures are then analysed.     

Low‐cost response surface methods from simulation optimization

We propose ‘low‐cost response surface methods’ (LCRSMs) that typically require half the experimental runs of standard response surface methods based on central composite and Box Behnken designs, but yield comparable or lower modeling errors under realistic assumptions. In addition, the LCRSMs have substantially lower modeling errors and greater expected savings compared with alternatives with comparable numbers of runs, including small composite designs and computer‐generated designs based on popular criteria such as D‐optimality. The LCRSM procedures appear to be the first experimental design methods derived as the solution to a simulation optimization problem. Together with modern computers, simulation optimization offers unprecedented opportunities for applying clear, realistic multicriterion objectives and assumptions to produce useful experimental design methods. We compare the proposed LCRSMs with alternatives based on six criteria. We conclude that the proposed methods offer attractive alternatives when the experimenter is considering dropping factors to use standard response surface methods or would like to perform relatively few runs and stop with a second‐order model. Copyright © 2002 John Wiley & Sons, Ltd.