Comparison of Simplex Designs for Quadratic Mixture Models

Designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x ₁, x ₂, …, x _{q + 1}) of nonnegative components x _i with Σ ^{q + 1} ₁ x _i = 1. The designs that are optimum with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria. Computational routines for obtaining these designs are developed, and the geometry of optimum structures is discussed. Except when q = 2, the A-optimum design is supported by the vertices and midpoints of edges of the simplex, as is the case for the previously known D-optimum design. Although the E-optimum design requires more observation points, it is more robust in its efficiency, under variation of criterion: but all three designs perform reasonably well in this sense.     

Use of Strip-Strip-Block Design for Multi-Stage Processes to Reduce Costs of Experimentation

A strip-strip-block design is proposed for use in investigating the effects of factors, and their interactions, in three-stage industrial processes. Consideration is given, more generally, to the question of how to generate such designs, what theoretical properties they should have, and how to analyze the results of such designs. A case study is used as an illustration.     

Probit Analysis as a Technique for Estimating the Reliability of a Simple System

In this paper the problem of estimating the relation between reliability and stress is considered for a simple system composed of several, say m, identical but independent components. The parameters of the failure probability equation P = H[F _X (x)] are estimated where F _X (x) has the cumulative normal form. The method of analysis used is a modification of probit analysis.     

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.     

Construction of Optimal Block Designs by Computer

This article describes an effective algorithm for constructing optimal or near-optimal incomplete block designs with up to 100 treatments. The algorithm is found to perform well when evaluated against 874 optimal or near-optimal incomplete block designs in the literature. Examples that motivated the development of the algorithm are given.     

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.     

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.     

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.     

Sequential Application of Simplex Designs in Optimisation and Evolutionary Operation

A technique for empirical optimisation is presented in which a sequence of experimental designs each in the form of a regular or irregular simplex is used, each simplex having all vertices but one in common with the preceding simplex, and being completed by one new point. Reasons for the choice of design are outlined, and a formal procedure given. The performance of the technique in the presence and absence of error is studied and it is shown (a) that in the presence of error the rate of advance is inversely proportional to the error standard deviation, so that replication of observations is not beneficial, and (b) that the “efficiency” of the technique appears to increase in direct proportion to the number of factors investigated. It is also noted that, since the direction of movement from each simplex is dependent solely on the ranking of the observations, the technique may be used even in circumstances when a response cannot be quantitatively assessed. Attention is drawn to the ease with which second-order designs having the minimum number of experimental points may be derived from a regular simplex, and a fitting procedure which avoids a direct matrix inversion is suggested. In a brief appendix one or two new rotatable designs derivable from a simplex are noted.     

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.     

Applied Time Series Analysis II

The problem of identification and construction of optimal designs for comparing several test treatments with a control treatment is studied. Two-way heterogeneity in the experimental units is considered under the usual additive model. A general class of designs—named balanced treatment row-column designs—appears to be natural candidates for the A-optimality criterion. Catalogs of A-optimal balanced treatment row-column designs have been prepared and tabulated for a practical range of parameter values. The use of the catalogs is explained with examples.     

Strategies for sequential design of experiments and augmentation

The benefits of sequential design of experiments have long been described for both model-based and space-filling designs. However, in our experience, too few practitioners take advantage of the opportunity afforded by this approach to maximize the learning from their experimentation. By obtaining data sequentially, it is possible to learn from the early stages to inform subsequent data collection, minimize wasted resources, and provide answers for a series of objectives for the overall experiment. This paper provides methods and algorithms to create augmented distance-based space-filling designs, using both uniform and non-uniform space-filling strategies, that can be constructed at each stage based on information learned in earlier stages. We illustrate the methods with several examples that involve different initial data, types of space-filing designs and experimental goals.     

A New Class of Supersaturated Designs

Supersaturated designs are useful in situations in which the number of active factors is very small compared to the total number of factors being considered. In this article, a new class of supersaturated designs is constructed using half fractions of Hadamard matrices. When a Hadamard matrix of order N is used, such a design can investigate up to N – 2 factors in N/2 runs. Results are given for N ≤ 60. Extension to larger N is straightforward. These designs are superior to other existing supersaturated designs and are easy to construct. An example with real data is used to illustrate the ideas.     

Sequentially Generated Second Order Quasi D-Optimal Designs for Experiments With Mixture and Process Variables

Second order designs for experiments with mixture and process variables are proposed. They are constructed on the basis of continuous D-optimal designs by use of a three-stage procedure for sequentially generating optimal designs. The determinants of the information matrices of the designs obtained are very near to those of continuous D-optimal designs. Tables of discrete quasi D-optimal designs for q + r ≤ 7 are given, where q is the number of mixture components and r is the number of process variables. The experimenter can choose the number of trials N within the interval k ≤ N ≤ min(2k, k + 20), where k is the number of model coefficients. An application of the proposed designs in an investigation of truck tire properties is given.     

Another Look at First-Order Saturated Designs: The p-efficient Designs

The problem of constructing first-order saturated designs that are optimal in some sense has received a great deal of attention in the literature. Since these saturated designs are frequently used in screening situations, the focus will be on the potential projective models rather than the full model. This article discusses some practical concerns in choosing a design and presents some first-order saturated designs having two desirable properties, (near-) equal occurrence and (near-) orthogonality. These saturated designs are shown to be reasonably efficient for estimating the parameters of projective submodels and thus are called p-efficient designs. Comparisons with the efficiency of D-optimal designs are given for designs for all n from 3 to 30.     

Benefits and Fast Construction of Efficient Two-Level Foldover Designs

Recent work in two-level screening experiments has demonstrated the advantages of using small foldover designs, even when such designs are not orthogonal for the estimation of main effects (MEs). In this article, we provide further support for this argument and develop a fast algorithm for constructing efficient two-level foldover (EFD) designs. We show that these designs have equal or greater efficiency for estimating the ME model versus competitive designs in the literature and that our algorithmic approach allows the fast construction of designs with many more factors and/or runs. Our compromise algorithm allows the practitioner to choose among many designs making a trade-off between efficiency of the main effect estimates and correlation of the two-factor interactions (2FIs). Using our compromise approach, practitioners can decide just how much efficiency they are willing to sacrifice to avoid confounded 2FIs as well as lowering an omnibus measure of correlation among the 2FIs.     

Design of experiments to investigate main effects and all two-factor interactions when one of the factors has more than two levels—application to nuclear waste cementation

When both main effects and first-order interactions are looked for, experimental designs usually require a high number of experiments as soon as one of the investigated factors has more than two levels of variation. A new strategy, based on the use of a series of two-level saturated designs, allowed to reduce the number of trials without compromising the quality of the achieved results. This approach revealed very fruitful in a practical case aiming at selecting a proper binder for nuclear waste encapsulation, and at investigating factors which influence the stability of the resulting solidified waste forms.     

Split-Plot Experiments with Unusual Numbers of Subplot Runs

In many experimental situations, it may not be feasible or even possible to run experiments in a completely randomized fashion as usually recommended. Under these circumstances, split-plot experiments in which certain factors are changed less frequently than the others are often used. Most of the literature on split-plot designs is based on 2-level factorials. For those designs, the number of subplots is a power of 2. There may however be some situations where for cost purposes or physical constraints, we may need to have unusual number of subplots such as 3, 5, 6, etc. In this article, we explore this issue and provide some examples based on the Plackett and Burman designs. Also algorithmically constructed D-optimal split-plot designs are compared to those based on Plackett and Burman designs.     

Blocking versus robustness in industrial contexts

The paper discusses the similarities and differences between blocking factors (blocked designs) and noise factors (robust designs) in industrial two‐level factorial experiments. The discussion covers from the objectives of both design types and the nature of blocking and noise factors to the types of designs and the assumptions needed in each case. The conclusions are as follows: the nature and characteristics of noise and blocking factors are equal or very similar; the designs used in both situations are also similar; and the main differences lie in the assumptions and the objectives. The paper argues that the objectives are not in conflict and can easily be harmonized. In consequence, we argue in favor of a unified approach that would clarify the issue, especially for students and practitioners.     

Aliased informed model selection strategies for six-factor no-confounding designs

Nonregular designs are a preferable alternative to regular resolution IV designs because they avoid confounding two-factor interactions. As a result nonregular designs can estimate and identify a few active two-factor interactions. However, due to the sometimes complex alias structure of nonregular designs, standard screening strategies can fail to identify all active effects. In this paper, we explore a specific no-confounding six-factor 16-run nonregular design with orthogonal main effects. By utilizing our knowledge of the alias structure, we can inform the model selection process. Our aliased informed model selection (AIMS) strategy is a design-specific approach that we compare to three generic model selection methods; stepwise regression, Lasso, and the Dantzig selector. The AIMS approach substantially increases the power to detect active main effects and two-factor interactions versus the aforementioned generic methodologies.