Optimal foldover plans for mixed‐level fractional factorial designs

Reversing plus and minus signs of one or more factors is the traditional method to fold over two‐level fractional factorial designs. However, when factors in the original design have more than two levels, the method of ‘reversing signs’ loses its efficacy. This article develops a mechanism to foldover designs involving factors with different numbers of levels, say mixed‐level designs. By exhaustive search we identify the optimal foldover plans. The criterion used is the general balance metric, which can reveal the aberration properties of the combined designs (original design plus foldover). The optimal foldovers for some efficient mixed‐level fractional factorial designs are provided for practical use. Copyright © 2008 John Wiley & Sons, Ltd.     

A comparison of designs for one‐step screening and response surface estimation

The sequential design approach to response surface exploration is often viewed as advantageous as it provides the opportunity to learn from each successive experiment with the ultimate goal of determining optimum operating conditions for the system or process under study. Recent literature has explored factor screening and response surface optimization using only one three‐level design to handle situations where conducting multiple experiments is prohibitive. The most straightforward and accessible analysis strategy for such designs is to first perform a main‐effects only analysis to screen important factors before projecting the design onto these factors to conduct response surface exploration. This article proposes the use of optimal designs with minimal aliasing (MA designs) and demonstrates that they are more effective at screening important factors than the existing designs recommended for single‐design response surface exploration. For comparison purposes, we construct 27‐run MA designs with up to 13 factors and demonstrate their utility using established design criterion and a simulation study. Copyright 2011 © John Wiley & Sons, Ltd.     

Semifold plans for mixed‐level designs

This article presents a more efficient method for sequential augmentation of mixed‐level designs. The proposed approach reduces the optimal foldover plan of a mixed‐level design to a semifold plan by selecting half of the treatment combinations of the foldover fraction using exhaustive search and the criterion of general balance metric. The resulting design is a more economic run size augmented fraction that possesses good balance and orthogonality properties for main effects and two‐factor interactions. Three efficient arrays consisting of 20, 24 and 30 runs were selected for the analysis. Efficient arrays composed of a higher number of runs can be semifolded in a similar manner. Copyright © 2011 John Wiley & Sons, Ltd.     

The general balance metric for mixed‐level fractional factorial designs

Mixed‐level designs are employed when factors with different numbers of levels are involved. Practitioners use mixed‐level fractional factorial designs as the total number of runs of the full factorial increases rapidly as the number of factors and/or the number of factor levels increases. One important decision is to determine which fractional designs should be chosen. A new criterion, the general balance metric (GBM), is proposed to evaluate and compare mixed‐level fractional factorial designs. The GBM measures the degree of balance for both main effects and interaction effects. This criterion is tied to, and dominates orthogonality criteria as well as traditional minimum aberration criteria. Furthermore, the proposal is easy to use and has practical interpretations. As part of the GBM, the concept of resolution is generalized and the confounding structure of mixed‐level fractional factorial designs is also revealed. Moreover, the metric can also be used for the purpose of design augmentation. Examples are provided to compare this approach with existing criteria. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimation of missing observations in two‐level split‐plot designs

Inserting estimates for the missing observations from split‐plot designs restores their balanced or orthogonal structure and alleviates the difficulties in the statistical analysis. In this article, we extend a method due to Draper and Stoneman to estimate the missing observations from unreplicated two‐level factorial and fractional factorial split‐plot (FSP and FFSP) designs. The missing observations, which can either be from the same whole plot, from different whole plots, or comprise entire whole plots, are estimated by equating to zero a number of specific contrast columns equal to the number of the missing observations. These estimates are inserted into the design table and the estimates for the remaining effects (or alias chains of effects as the case with FFSP designs) are plotted on two half‐normal plots: one for the whole‐plot effects and the other for the subplot effects. If the smaller effects do not point at the origin, then different contrast columns to some or all of the initial ones should be discarded and the plots re‐examined for bias. Using examples, we show how the method provides estimates for the missing observations that are very close to their actual values. Copyright © 2007 John Wiley & Sons, Ltd.     

Minimum Setup Minimum Aberration Two‐level Split‐plot Type Designs for Physical Prototype Testing

Although new technologies allow for less effort in prototyping, physical testing still remains an important step in the product development cycle. Well‐planned experiments are useful to guide the decision‐making process. During the design of an experiment, one of the challenges is to balance limited resources and system constraints to obtain useful information. It is common that prototypes are composed of several parts, with some parts more difficult to assemble than others. And, usually, there is only one piece available of each part type and a large number of different setups. Under these conditions, designs with randomization restrictions become attractive approaches. Considering this scenario, a new and additional criterion, minimum setup, to construct split‐plot type designs is presented. Designs with the minimum number of setups of the more difficult parts, which are especially useful for screening purposes in physical prototype testing, are discussed. The use of the proposed criterion combined with minimum aberration for selecting a regular design is shown through a real application in testing car prototypes. As a tool to practitioners, catalogs of selected 32‐run minimum setup minimum aberration split‐split‐plot and split‐split‐split‐plot designs are presented. More complete catalogs are available as Supporting information. Copyright © 2015 John Wiley & Sons, Ltd.     

All‐subsets regression under effect heredity restrictions for experimental designs with complex aliasing

We consider an all‐subsets regression method for models under effect heredity restrictions for experimental designs with complex aliasing, whose number of potential main effects and two‐factor interactions exceed the number of runs. In this paper, we present an algorithm that systematically attempts to fit all such models. We illustrate the algorithm with two published experiments. Copyright © 2009 John Wiley & Sons, Ltd.     

Cost‐constrained G‐efficient Response Surface Designs for Cuboidal Regions

In many industrial experiments there are restrictions on the resource (or cost) required for performing the runs in a response surface design. This will require practitioners to choose some subset of the candidate set of experimental runs. The appropriate selection of design points under resource constraints is an important aspect of multi‐factor experimentation. A well‐planned experiment should consist of factor‐level combinations selected such that the resulting design will have desirable statistical properties but the resource constraints should not be violated or the experimental cost should be minimized. The resulting designs are referred to as cost‐efficient designs. We use a genetic algorithm for constructing cost‐constrained G‐efficient second‐order response surface designs over cuboidal regions when an experimental cost at a certain factor level is high and a resource constraint exists. Consideration of practical resource (or cost) restrictions and different cost structures will provide valuable information for planning effective and economical experiments when optimizing statistical design properties. Copyright © 2005 John Wiley & Sons, Ltd.     

On the Analysis of Balanced Two‐Level Factorial Whole‐Plot Saturated Split‐Plot Designs

This paper considers an experimentation strategy when resource constraints permit only a single design replicate per time interval and one or more design variables are hard to change. The experimental designs considered are two‐level full‐factorial or fractional‐factorial designs run as balanced split plots. These designs are common in practice and appropriate for fitting a main‐effects‐plus‐interactions model, while minimizing the number of times the whole‐plot treatment combination is changed. Depending on the postulated model, single replicates of these designs can result in the inability to estimate error at the whole‐plot level, suggesting that formal statistical hypothesis testing on the whole‐plot effects is not possible. We refer to these designs as balanced two‐level whole‐plot saturated split‐plot designs. In this paper, we show that, for these designs, it is appropriate to use ordinary least squares to analyze the subplot factor effects at the ‘intermittent’ stage of the experiments (i.e., after a single design replicate is run); however, formal inference on the whole‐plot effects may or may not be possible at this point. We exploit the sensitivity of ordinary least squares in detecting whole‐plot effects in a split‐plot design and propose a data‐based strategy for determining whether to run an additional replicate following the intermittent analysis or whether to simply reduce the model at the whole‐plot level to facilitate testing. The performance of the proposed strategy is assessed using Monte Carlo simulation. The method is then illustrated using wind tunnel test data obtained from a NASCAR Winston Cup Chevrolet Monte Carlo stock car. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Correlation‐Based r‐plot for Evaluating Supersaturated Designs

Orthogonality or near‐orthogonality is an important property in the design of experiments. Supersaturated designs are natural when we wish to investigate the main effects for a large number of factors but are restricted to a small number of runs. These supersaturated designs, by definition, cannot satisfy pairwise orthogonality of all the factor columns in the design matrix. Hence, we need a means to evaluate the degree of near‐orthogonality of different alternative supersaturated designs. It is usual to use numerical measures that condense the rich information from many pairwise column measures to assess the degree of orthogonality of given supersaturated designs, but we propose using graphical methods to better understand patterns between sets of columns and evaluate the degree of near‐orthogonality to compare and select between alternative supersaturated designs. The methods are illustrated with a number of diverse examples to illustrate the information that can be extracted from the summary. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust split‐plot designs

In many experimental situations, practitioners are confronted with costly, time consuming, or hard‐to‐change (HTC) factors. These practical or economic restrictions on randomization can be accommodated with a split‐plot design structure that minimizes the manipulation of the HTC factors. Selecting a good design is a challenging task and requires knowledge of the opportunities and restrictions imposed by the experimental apparatus and an evaluation of statistical performance among competing designs. Building on the well‐established evaluation criteria for the completely randomized context, we emphasize the unique qualitative and quantitative evaluation criteria for split‐plot designs. An example from hypersonic propulsion research is used to demonstrate the consideration of multiple design evaluation criteria. Published in 2007 by John Wiley & Sons, Ltd.     

Partition experimental designs for sequential processes: Part I—First‐order models

The output quality or performance characteristics of a product often depend not only on the effect of the factors in the current process but on the effect of factors from preceding processes. Statistically‐designed experiments provide a systematic approach to study the effects of multiple factors on process performance by offering a structured set of analyses of data collected through a design matrix. One important limitation of experimental design methods is that they have not often been applied to multiple sequential processes. The objective is to create a first‐order experimental design for multiple sequential processes that possess several factors and multiple responses. The first‐order design expands the current experimental designs to incorporate two processes into one partitioned design. The designs are evaluated on the complexity of the alias structure and their orthogonality characteristics. The advantages include a decrease in the number of experimental design runs, a reduction in experiment execution time, and a better understanding of the overall process variables and their influence on each of the responses. Copyright © 2001 John Wiley & Sons, Ltd.     

Fast flexible space‐filling designs with nominal factors for nonrectangular regions

Space‐filling designs allow for exploration of responses when each continuous input factor is set to many different values over its range. While typical space‐filling designs treat all the input factors as continuous, some problems necessitate the use of nominal input factors. In such cases, it is desirable that the design for the continuous inputs only be space‐filling, as well as the subdesign of the continuous factors for each level of a nominal input. In this article, we present a solution that takes both aspects into account and allows the experimenter to balance between the space‐filling properties of the design ignoring the nominal inputs versus subdesigns on each level of nominal inputs. Space‐filling designs commonly employ rectangular design spaces. However, it is sometimes necessary to place constraints on the design region where some regions of the design space are impossible or undesirable to run. In addition to incorporating nominal factors, the methods presented in this article generate space‐filling designs that have the flexibility to accommodate nonrectangular design regions.     

Model‐robust choice experiments: discussion and case study

Choice experiments are an effective way of obtaining objective information regarding the voice of the customer. They can be used to obtain the relevant customer attributes and importance rankings used in the first step of quality function deployment. They are also used extensively in marketing research. Optimal designs for choice experiments have been discussed in the literature. However, optimal designs are only optimal for a particular model. In this article we borrow ideas from quality engineering and industrial experimentation to develop designs for choice experiments that are model‐robust (in the sense that they are efficient for fitting a model involving main effects plus a few interactions that need not be specified in advance). A case study is presented to illustrate the use of a model‐robust design for a choice experiment. Two unsuspected interactions were discovered in the case study, and this discovery led to added insight regarding customer preferences and importance rankings of product attributes. These insights would not have been possible if an optimal design for the main effects model had been used. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Cost‐penalized estimation and prediction evaluation for split‐plot designs

Comparisons between different designs have traditionally focused on balancing the quality of estimation or prediction relative to the overall size of the design. For split‐plot designs with two levels of randomization, the total number of observations may not accurately summarize the true cost of the experiment, because different costs are likely associated with setting up the whole and subplot levels. In this paper, we present several flexible measures for design assessment based on D‐, G‐ and V‐optimality criteria that take into account potentially different cost structures for the split‐plot designs. The new measures are illustrated with two examples: a 2³ factorial experiment for first‐order models, where all possible designs are considered, and selective designs for a three‐factor second‐order model. Copyright © 2006 John Wiley & Sons, Ltd.     

Partition experimental designs for sequential processes: Part II—second‐order models

Second‐order experimental designs are employed when an experimenter wishes to fit a second‐order model to account for response curvature over the region of interest. Partition designs are utilized when the output quality or performance characteristics of a product depend not only on the effect of the factors in the current process, but the effects of factors from preceding processes. Standard experimental design methods are often difficult to apply to several sequential processes. We present an approach to building second‐order response models for sequential processes with several design factors and multiple responses. The proposed design expands current experimental designs to incorporate two processes into one partitioned design. Potential advantages include a reduction in the time required to execute the experiment, a decrease in the number of experimental runs, and improved understanding of the process variables and their influence on the responses. Copyright © 2002 John Wiley & Sons, Ltd.     

Fast Flexible Space‐Filling Designs for Nonrectangular Regions

Space‐filling designs allow for exploration of responses with many different settings for each input factor. While much research has been done using rectangular design spaces, it is not uncommon to have constraints on the design region where some combinations are impossible or undesirable to run. In this article, we present an intuitive method for quickly generating space‐filling designs that have the flexibility to accommodate nonrectangular design regions. We also show that these designs perform favorably compared with other standard designs with respect to the average distance of an arbitrary point in space to the closest design point. This property holds even when the design region is rectangular. Copyright © 2014 John Wiley & Sons, Ltd.     

Three‐level BDDC in two dimensions

Balancing Domain Decomposition by Constraints (BDDC) methods are non‐overlapping iterative substructuring domain decomposition methods for the solution of large sparse linear algebraic systems arising from discretization of elliptic boundary value problems. They are similar to the balancing Neumann–Neumann algorithm. However, in BDDC methods, a small number of continuity constraints are enforced across the interface, and these constraints form a new coarse, global component. An important advantage of using such constraints is that the Schur complements that arise in the computation will all be strictly positive definite. The matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation. However, this problem can ultimately become a bottleneck, if the number of subdomains is very large. In this paper, two three‐level BDDC methods are introduced for solving the coarse problem approximately in two‐dimensional space, while still maintaining a good convergence rate. Estimates of the condition numbers are provided for the two three‐level BDDC methods and numerical experiments are also discussed. Copyright © 2006 John Wiley & Sons, Ltd.