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.     

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.     

Partial Duplication of Response Surface Designs

Continuing the author's earlier work [6] a method is described which requires that certain experimental runs of a central composite, second-order, response surface design be repeated, thereby providing a more general estimate of the experimental error, at the same time providing more reliable estimates of the effects.

The partial duplication of the factorial portion as well as the partial duplication of the star portion has been considered and described. The response surface designs with the star portion duplicated seem to have more potential than the designs with their factorial portions duplicated or partially duplicated.     

Graphical and computer-aided approaches to plan experiments

Design of experiments is a quality technology to achieve product excellence, that is to achieve high quality at low cost. It is a tool to optimize product and process designs, to accelerate the development cycle, to reduce development costs, to improve the transition of products from R & D to manufacturing and to troubleshoot manufacturing problems effectively. It has been successfully, but sporadically, used in the United States. More recently, it has been identified as a major technological reason for the success of Japan in producing high-quality products at low cost. In the United States, the need for increased competitiveness and the emphasis on quality improvement demands a widespread use of design of experiments by engineers, scientists and quality professionals. In the past, such widespread use has been hampered by a lack of proper training and a lack of availability of tools to easily implement design of experiments in industry. Three steps are essential, and are being taken, to change this situation dramatically. First, simple graphical methods, to design and analyse experiments, need to be developed, particularly when the necessary microcomputer resources are not available. Secondly, engineers, scientists and quality professionals must have access to microcomputer-based software for design and analysis of experiments.¹ Availability of such software would allow users to concentrate on the important scientific and engineering aspects of the problem by computerizing the necessary statistical expertise. Finally, since a majority of the current workforce is expected to be working in the year 2000, a massive training effort, based upon simple graphical methods and appropriate computer software, is necessary.² The purpose of this paper is to describe a methodology based upon a new graphical method called interaction graphs and other previously known techniques, to simplify the correct design of practically important fractional factorial experiments. The essential problem in designing a fractional factorial experiment is first stated. The interaction graph for a 16-trial fractional factorial design is given to illustrate how the graphical procedure can be easily used to design a two-level fractional factorial experiment. Other previously known techniques are described to easily modify the two-level fractional factorial designs to create mixed multi-level designs. Interaction graphs for other practically useful fractional factorial designs are provided. A computer package called CADE (computer aided design of experiments), which automatically generates the appropriate fractional factorial designs based upon user specifications of factors, levels and interactions and conducts complete analyses of the designed experiments is briefly described.¹ Finally, the graphical method is compared with other available methods for designing fractional factorial experiments.     

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.     

Constructing General Orthogonal Fractional Factorial Split-Plot Designs

While the orthogonal design of split-plot fractional factorial experiments has received much attention already, there are still major voids in the literature. First, designs with one or more factors acting at more than two levels have not yet been considered. Second, published work on nonregular fractional factorial split-plot designs was either based only on Plackett–Burman designs, or on small nonregular designs with limited numbers of factors. In this article, we present a novel approach to designing general orthogonal fractional factorial split-plot designs. One key feature of our approach is that it can be used to construct two-level designs as well as designs involving one or more factors with more than two levels. Moreover, the approach can be used to create two-level designs that match or outperform alternative designs in the literature, and to create two-level designs that cannot be constructed using existing methodology. Our new approach involves the use of integer linear programming and mixed integer linear programming, and, for large design problems, it combines integer linear programming with variable neighborhood search. We demonstrate the usefulness of our approach by constructing two-level split-plot designs of 16–96 runs, an 81-run three-level split-plot design, and a 48-run mixed-level split-plot design. Supplementary materials for this article are available online.     

A Minimum Cost Approach for Finding Fractional Factorials

A procedure is given for finding fractional factorial designs which permit the estimation of specified main and interaction effects. This procedure directly accounts for the actual costs of experimentation. Given the cost data for the factor level combinations and a rank ordering of the importance to the experimenters for obtaining estimates of the effects, a sequence of sets of words eligible to appear in defining relations is constructed and cost-optimal fractional factorial designs are found over such sets. The procedure gives experimenters information regarding which effects can be estimated and the costs required to obtain such estimates. This information along with budgetary constraints can then be used to best allocate experimental resources. An example is given to illustrate the method.     

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.     

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.     

A Bayesian Variable-Selection Approach for Analyzing Designed Experiments With Complex Aliasing

Experiments using designs with complex aliasing patterns are often performed—for example, twolevel nongeometric Plackett-Burman designs, multilevel and mixed-level fractional factorial designs, two-level fractional factorial designs with hard-to-control factors, and supersaturated designs. Hamada and Wu proposed an iterative guided stepwise regression strategy for analyzing the data from such designs that allows entertainment of interactions. Their strategy provides a restricted search in a rather large model space, however. This article provides an efficient methodology based on a Bayesian variable-selection algorithm for searching the model space more thoroughly. We show how the use of hierarchical priors provides a flexible and powerful way to focus the search on a reasonable class of models. The proposed methodology is demonstrated with four examples, three of which come from actual industrial experiments.     

Reduced Designs of Resolution Five

The two level fractional factorial designs of resolution five enable the experimenter to estimate independently all main effects and two-factor interactions under the assumptions that higher order interaction effects are negligible. By relaxing, very slightly, the requirement that all two-factor interactions be estimable, or that all estimated effects be orthogonal, the number of runs required for many resolution five designs can be greatly reduced.     

Statistical Programs for High Speed Computers

Eight-run two level factorial and fractional factorial designs are examined from two points of view: (1) the number of levels which must be changed in performing the design, (2) the effect of a first order time trend on the main effects. It is found that only a few run orders are desirable from these viewpoints and thus randomization of the runs is likely, in general, to lead to an unfavorable sequence.     

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.     

A comparison of two-level designs to estimate all main effects and two-factor interactions

We compare cost-efficient alternatives for the full factorial 2⁴ design, the regular 2^5-1 fractional factorial design, and the regular 2^6-1 fractional factorial design that can fit the model consisting of all the main effects as well as all the two-factor interactions. For 4 and 5 factors we examine orthogonal arrays with 12 and 20 runs, respectively. For 6 factors we consider orthogonal arrays with 24 as well as 28 runs. We consult complete catalogs of two-level orthogonal arrays to find the ones that provide the most efficient estimation of all the effects in the model. We compare these arrays with D-optimal designs found using a coordinate exchange algorithm. The D-optimal designs are always preferable to the most efficient orthogonal arrays for fitting the full model in all the factors.     

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.     

Elementary Statistics

Fractional two-level factorial designs are often used in the early stages of an investigation to screen for important factors. Traditionally, 2 ^n-k fractional factorial designs of resolution III, IV, or V have been used for this purpose. When the investigator is able to specify the set of nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design with fewer runs than a standard textbook design by searching within a wider class of designs called parallel-flats designs. The run sizes in this class of designs do not necessarily need to be powers of 2. We discuss an algorithm for constructing orthogonal parallel-flats designs to meet user specifications. Several examples illustrate the use of the algorithm.     

Foldover Plans for Resolution IV Designs with 11 to 16 Factors

When there is a concern about the best way to apply the foldover technique to a Resolution IV fraction, it is often helpful to consult the literature available. Unfortunately, foldover plans for Resolution IV designs with 11 ≤ k factors are not provided. This article shows a methodology for the selection and classification of foldover plans for two‐level fractional factorial designs with 11 to 16 factors based on an exhaustive search and computer programming. The recommended foldovers are presented. This research extends the current foldover plans available for Resolution IV designs. Copyright © 2015 John Wiley & Sons, Ltd.     

Adaptive One-Factor-at-a-Time Experimentation and Expected Value of Improvement

This article concerns adaptive experimentation as a means for making improvements in design of engineering systems. A simple method for experimentation, called “adaptive one-factor-at-a-time,” is described. A mathematical model is proposed and theorems are proven concerning the expected value of the improvement provided and the probability that factor effects will be exploited. It is shown that adaptive one-factor-at-a-time provides a large fraction of the potential improvements if experimental error is not large compared with the main effects and that this degree of improvement is more than that provided by resolution III fractional factorial designs if interactions are not small compared with main effects. The theorems also establish that the method exploits two-factor interactions when they are large and exploits main effects if interactions are small. A case study on design of electric-powered aircraft supports these results.     

Some New Three Level Designs for the Study of Quantitative Variables

A class of incomplete three level factorial designs useful for estimating the coefficients in a second degree graduating polynomial are described. The designs either meet, or approximately meet, the criterion of rotatability and for the most part can be orthogonally blocked. A fully worked example is included.     

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.