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.     

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.     

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.     

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.     

Screening Designs for Poly-Factor Experimentation

A survey is given of the following types of screening designs: Incomplete 2 ^k designs, srlpersaturated and grollp-screening designs. These designs are compared with each other. Some new results for group-screening are derived.     

Slope-Rotatable Central Composite Designs

An analogue of the Box-Hunter rotatability property for second order response surface designs in k independent variables is presented. When such designs are used to estimate the first derivatives with respect to each independent variable the variance of the estimated derivative is a function of the coordinates of the point at which the derivative is evaluated and is also a function of the design. By choice of design it is possible to make this variance constant for all points equidistant from the design origin. This property is called slope-rotatability by analogy with the corresponding property for the variance of the estimated response, ?.

For central composite designs slope-rotatability can be achieved simply by adjusting the axial point distances (α), so that the variance of the pure quadratic coefficients is one-fourth the variance of the mixed second order coefficients. Tables giving appropriate values of α have been constructed for 2 ≤ k ≤ 8. For 5 ≤ k ≤ 8 central composite designs involving fractional factorials are used. It is also shown that appreciable advantage is gained by replicating axial points rather than confining replication to the center point only.     

Bridge Designs for Modeling Systems With Low Noise

For deterministic computer simulations, Gaussian process models are a standard procedure for fitting data. These models can be used only when the study design avoids having replicated points. This characteristic is also desirable for one-dimensional projections of the design, since it may happen that one of the design factors has a strongly nonlinear effect on the response. Latin hypercube designs have uniform one-dimensional projections, but are not efficient for fitting low-order polynomials when there is a small error variance. D-optimal designs are very efficient for polynomial fitting but have substantial replication in projections. We propose a new class of designs that bridge the gap between D-optimal designs and D-optimal Latin hypercube designs. These designs guarantee a minimum distance between points in any one-dimensional projection allowing for the fit of either polynomial or Gaussian process models. Subject to this constraint they are D-optimal for a prespecified model.     

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.     

On Minimum-Point Second-Order Designs

In this note, we discuss k-factor, second order designs with minimum number of points ½(k + l)(k + 2), in particular, those which are extensions of designs that give minimum generalized variance for k = 2 and 3. The experimental region is the unit cuboid. Minimum point designs of this type are unknown for k ≥ 4, and these designs are the best found to date except for k = 4, where a better design is known. Kiefer has shown that these designs cannot be the best for k ≥ 7, via an existence result but, even here, specific better designs are not known and appear difficult to obtain. We also discuss some difficulties of using, in practice, designs that, are D-optimal (that is give minimum generalized variance when the number of points is not restricted).     

Two-Level Multifactor Designs for Detecting the Presence of Interactions

A design optimality criterion, tr (L)-optimality, is applied to the problem of designing two-level multifactor experiments to detect the presence of interactions among the controlled variables. We give rules for constructing tr (L)-optimal foldover designs and tr (L)-optimal fractional factorial designs. Some results are given on the power of these designs for testing the hypothesis that there are no two-factor interactions. Augmentation of the tr (L)-optimal designs produces designs that achieve a compromise between the criteria of D-optimality (for parameter estimation in a first-order model) and tr (L)-optimality (for detecting lack of fit). We give an example to demonstrate an application to the sensitivity analysis of a computer model.     

Three-Factor Additive Designs More General Than the Latin Square

A class of designs are described for the case of three factors at levels m, n and p respectively for which the total number of experiments required is a fraction of mnp.     

Robust designs with high projection efficiency

Alphabetic optimality criteria, such as the D, A, and I criteria, require specifying a model to select optimal designs. They are not model‐free, and the designs obtained by them may not be robust. Recently, many extensions of the D and A criteria have been proposed for selecting robust designs with high estimation efficiency. However, approaches for finding robust designs with high prediction efficiency are rarely studied in the literature. In this paper, we propose a compound criterion and apply the coordinate‐exchange 2‐phase local search algorithm to generate robust designs with high estimation, high prediction, or balanced estimation and prediction efficiency for projective submodels. Examples demonstrate that the designs obtained by our method have better projection efficiency than many existing designs.     

Blocking of 3(2 n–k ) Designs

3(2 ^n–2) designs may be divided into two blocks, one of size 2 ^n–1 and the other of size 2 ^n–2 by blocking on one of the defining contrasts and into three blocks of size 2 ^n–2 by blocking on all three defining contrasts. Blocking on an effect which is not a defining contrast gives two blocks of 3(2 ^n–3) runs each. In this paper these methods are applied to 3(2 ^n–k ) designs with twelve or twenty-four points. The designs considered are the 3(2^4–2) and 3(2^5–2) designs with all main effects and all two factor interactions estimable (assuming that higher order interactions are negligible), and saturated main effect plans with twelve points and up to eleven factors.     

Fast Computation of Exact G-Optimal Designs Via Iλ-Optimality

Exact G-optimal designs have rarely, if ever, been employed in practical applications. One reason for this is that, due to the computational difficulties involved, no statistical software system currently provides capabilities for constructing them. Two algorithms for exact G-optimal design construction of small designs involving one to three factors have been discussed in the literature: one employing a genetic algorithm and one employing a coordinate-exchange algorithm. However, these algorithms are extremely computer intensive in small experiments and do not scale beyond two or three factors. In this article, we develop a new method for constructing exact G-optimal designs using the integrated variance criterion, I_λ-optimality. We show that with careful selection of the weight function, a difficult exact G-optimal design construction problem can be converted to an equivalent exact I_λ-optimal design problem, which is easily and quickly solved. We illustrate the use of the algorithm for full quadratic models in one to five factors. The MATLAB codes used to implement our algorithm and the exact G-optimal designs produced by the algorithm for each test case are available online as supplementary material.     

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.     

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.     

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.     

Letter to the Editor: A List of New Group Divisible Designs

Group divisible designs are the most important class of partially balanced incomplete block (PBIB) designs. A list of new group divisible designs with r, k ⩽ 10 is provided.     

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.     

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.