Tables for Normal Sampling with Unknown Variance-the Student Distribution and Economically Optimal Sampling Plans

Industrial scientists and engineers often use experimental designs in which all degrees of freedom are used to estimate effects and consequently no classical estimate of the error is possible. Robust scale estimates provide an alternative measure of the error. In this study, several such scale estimators are evaluated based on the power or related significance tests. The pseudo standard error method of Lenth provides the best overall performance. Lenth's t approximation for critical values was found to be inaccurate, however, so new tables are provided. Additional recommendations are made according to the experimenter's prior belief in the number of likely important factors.     

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.     

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.     

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.     

Orthogonal Main-Effect Plans Permitting Estimation of All Two-Factor Interactions for the 2n3m Factorial Series of Designs

If we assume no higher order interactions for the 2ⁿ3^m factorial series of designs, then relaxing the restrictions concerning equal frequency for the factors and complete orthogonality for each estimate permits considerable savings in the number of runs required to estimate all the main effects and two-factor interactions. Three construction techniques are discussed which yield designs providing orthogonal estimates of all the main effects and allowing estimation of all the two-factor interactions. These techniques are: (i) collapsing of factors in symmetrical fractionated 3^m–p designs, (ii) conjoining fractionated designs, and (iii) combinations of (i) and (ii). Collapsing factors in a design either maintains or increases the resolution of the original design, but does not decrease it. Plans are presented for certain values of (n, m) as examples of the construction techniques. Systematic methods of analysis are also discussed.     

Bayesian Inference: Parameter Estimation and Decisions

Xu has cataloged 165 minimum aberration (MA) regular fractional factorial (FF) designs with 2-levels and large run sizes N=128 (m=8–64 factors), N=256 (m=9–28, 109–119), N=512 (m=10–25, 238–246), N=1024 (m=11–24, 488–501), N=2048 (m=12–23), and N=4096 (m=13–24). Such an extensive catalog was produced because of an improved algorithm. We extend the catalog by 36 MA, 2-level regular FF designs: N=256 (m=29–36, 100–108), N=512 (m=26–29), N=1024 (m=25–28), N=2048 (m=24–32), and N=4096 (m=25–26). Although such enumeration problems are notoriously difficult with increased N and/or m, we brought the newly solved problems within computational reach by changing the vital isomorphism check component of Xu’s algorithm. Here we present a new, compact graph for solving regular design isomorphism problems and use the nauty program to solve the corresponding graph isomorphism problems. Supplemental appendices are available online.     

Constructing Tables of Minimum Aberration Pn–m Designs

Fries and Hunter (1980) presented a practical algorithm for selecting standard 2 ^n–m fractional factorial designs based on a criterion they called “minimum aberration.” In this article some simple results are presented that enable the Fries–Hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p ≥ 2 is prime. Examples of minimum aberration 2 ^n–m designs with resolution R ≥ 4 are given for m, n – m < 9. A matrix is given for generating 3 ^n–m designs with m, n – m ≤ 6, which have, or nearly have, minimum aberration.     

Orthogonal Main-Effect 2 n 3 m Designs and Two-Factor Interaction Aliasing

Methods are presented for the determination of the alias matrix of two-factor interactions for the orthogonal main-effect 2 ⁿ 3 ^m plans catalogued by Addelman and Kempthorne. This catalogue includes Placket-Burman designs and designs obtained by replacement in 2 ^n–p plans or collapsing in 3 ^m–m plans. Systematic methods are included to facilitate the data computations. For standard r ^n–p factorial designs, techniques are given to determine a set of live factors, a generating set of linear sum congruences and the alias matrix. Additional orthogonal main-effect 2 ⁿ 3 ^m designs are constructed to supplement the Addelman-Kempthorne catalogue of 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.     

Constructing Two-Level Designs by Concatenation of Strength-3 Orthogonal Arrays

Two-level orthogonal arrays of N runs, k factors, and a strength of 3 provide suitable fractional factorial designs in situations where many of the main effects are expected to be active, as well as some two-factor interactions. If they consist of N/2 mirror image pairs, these designs are fold-over designs. They are called even and provide at most N/2 ? 1 degrees of freedom to estimate interactions. For k < N/3 factors, there exist strength-3 designs that are not fold-over designs. They are called even-odd designs and they provide many more degrees of freedom to estimate interactions. For N ? 48, attractive even-odd designs can be extracted from complete catalogs of strength-3 orthogonal arrays. However, for larger run sizes, no complete catalogs exist. To construct even-odd designs with N > 48, we develop an algorithm for an optimal concatenation of strength-3 designs involving N/2 runs. Our approach involves column permutations of one of the concatenated designs, as well as sign switches of the elements of one or more columns of that design. We illustrate the potential of the algorithm by generating two-level even-odd designs with 64 and 128 runs involving up to 33 factors, because this allows a comparison with benchmark designs from the literature. With a few exceptions, our even-odd designs outperform or are competitive with the benchmark designs in terms of the aliasing of two-factor interactions and in terms of the available degrees of freedom to estimate two-factor interactions. Supplementary materials for the article are available online.     

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.     

Designing Experiments for Tolerancing Assembled Products

This article provides an outline of theory and methods for the experimental determination of tolerance limits for mating components of assembled products. The emphasis is on novel combinatorial problems of pre- and post-fracfionafion of certain products of two-level factorial designs. The cost of experimentation is discussed and used as a guide to allocating experimental runs. Several design examples are provided. The article also includes a comprehensive example of the experimental determination of tolerances for the components of a throttle handle for small outboard motors.     

Comparison of the Ewald and Wolf methods for modeling electrostatic interactions in nanowires

Ionic compounds pose extra challenges with the appropriate modeling of long‐range coulombic interactions. Here, we study the mechanical properties of zinc oxide (ZnO) nanowires using molecular dynamic simulations with Buckingham potential and determine the suitability of the Ewald (Ann. Phys. 1921; 19) and Wolf (J. Chem. Phys. 1999; 110 (17):8254–8282) summation methods to account for the long‐range Coulombic forces. A comparative study shows that both the summation methods are suitable for modeling bulk structures with periodic boundary conditions imposed on all sides; however, significant differences are observed when nanowires with free surfaces are modeled. As opposed to Wolf's prediction of a linear stress–strain response in the elastic regime, Ewald's method predicts an erroneous behavior. This is attributed to the Ewald method's inability to account for surface effects properly. Additionally, Wolf's method offers highly improved computational performance as the model size is increased. This gain in computational time allows for modeling realistic nanowires, which can be directly compared with the existing experimental results. We conclude that the Wolf summation is a superior technique when modeling non‐periodic structures in terms of both accuracy of the results and computational performance. Copyright © 2010 John Wiley & Sons, Ltd.     

A New Balanced Design for Eighteen Varieties

A new balanced incomplete block design for v = |8, b = 5|, r = 17, k = 6. λ = 5 is presented. The design is resolvable and can be split into useful partially balanced suhdesigns. When these designs are used for the 2 × 3² factorial experiment they all have factorial structure.     

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.     

Near Minimal Resolution IV Designs for the sn Factorial Experiment

A fractional factorial design is of resolution IV if all main effects are estimable in the presence of two-factor interactions. For the sⁿ factorial experiment such a design requires at least N = s[(s – I)n – (s – 2)] runs. In this paper a series of resolution IV designs are given for the s” factorial, s = p ^α where p is prime, in N = s(s – I)n runs. A special case of the construction method produces a series of generalized foldover designs for the sⁿ experiment, s ≥ 3 and n ≥ 3, in N = s(s – I)n + s runs. These foldover designs permit estimation of the general mean in addition to all main effects and provide s degrees of freedom for estimating error. A section on analysis is included.     

Classical design structure of orthogonal designs with six to eight factors and sixteen runs

Most two‐level fractional factorial designs used in practice involve independent or fully confounded effects (so‐called regular designs). For example, for 16 runs and 6 factors, the classical resolution IV design with defining relation I = ABCE = BCDF = ADEF has become the de facto gold standard. Recent work has indicated that non‐regular orthogonal designs could be preferable in some circumstances. Inhibiting a wider usage of these non‐regular designs seems to be a combination of inertia/status quo and perhaps the general resistance and suspicion to designs that are computer generated to achieve ‘X–Y–Z’ optimality. In this paper each of the orthogonal non‐isomorphic two‐level, 16 run designs with 6, 7, or 8 factors (both regular and non‐regular) are shown to have a classical‐type construction with a 2⁴ or a replicated 2³ starting point. Additional factor columns are defined either using the familiar one‐term column generators or generators using weighted sums of effects. Copyright © 2010 John Wiley & Sons, Ltd.     

A METHOD TO IDENTIFY DISPERSION EFFECTS FROM UNREPLICATED MULTILEVEL EXPERIMENTS

The use of experimental design to identify dispersion effects is an effective tool for improving quality. In this article the problem of identifying such effects from unreplicated experiments is studied. We focus on a method developed for two-level 2^k-p experiments and discuss how the method can be generalized to a wider range of experimental designs. In particular, we show how dispersion effects can be identified from non-geometric Plackett–Burman designs and from experiments with more than two factor levels. In contrast to related methods our approach provides a test statistic with a well-known reference distribution. Three actual experiments are used to illustrate the method and to make comparisons to related methods. © 1997 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.     

Reliability Engineering Bible

In unreplicated 2^k?p designs, the assumption of constant variance is commonly made. When the variance of the response differs between the two levels of a column in the effect matrix, that column produces a dispersion effect. In this article we show that two active dispersion effects may create a spurious dispersion effect in their interaction column. Most existing methods for dispersion-effect testing in unreplicated fractional factorial designs are subject to these spurious effects. We propose a method of dispersion-effect testing based on geometric means of residual sample variances. We show through examples from the literature and simulations that the proposed test has many desirable properties that are lacking in other tests.