Minimum Cost Linear Trend Free 2n-(n-k) Designs of Resolution IV

This article extends the resolution of time trend free designs for sequential 2^n-p experiments from III into IV and minimizes the number of factor level changes between runs (i.e., cost) by constructing a catalog of (2^k?2 ?1) minimum cost linear trend free resolution IV 2^n?(n?k) designs (2^k?2 ≤ n ≤ 2^k?1?2) from the full 2^k factorial experiment using the interactions-main effects assignment technique. Each systematic 2^n?(n?k) design in the catalog is economic in minimum number of factor level changes and allows for the estimation of all n main effects unbiased by either the linear time trend (which may be present in the 2^n?(n?k) sequentially generated responses) or the non negligible two-factor interactions. This article provides for each 2^n?(n?k) design: (1) the defining relation or the alias structure; (2) the k independent generators for sequencing the 2^n?(n?k) runs by the generalized foldover scheme; and (3) the minimum cost represented by the total number of factor level changes between the 2^n?(n?k) runs. All k main effects of the 2^k experiment are excluded from the selection assignment process due to their nonlinear time trend resistance as well as excluding a total of (2^k?1 –k +1) interactions violating the resolution IV requirement.     

Some new factor screening designs using the search linear model

Factor screening designs for searching two and three effective factors using the search linear model are discussed. The construction of such factor screening designs involved finding a fraction with small number of treatments of a 2^m factorial experiment having the property P_2t (no 2t columns are linearly dependent) for t=2 and 3. A ‘Packing Problem’ is introduced in this connection. A complete solution of the problem in one case and partial solutions for the other cases are presented. Many practically useful new designs are listed.     

A series of single array 2m factorial search designs for even m

By means of a search design one is able to search for and estimate a small set of non‐zero elements from the set of higher order factorial interactions in addition to estimating the lower order factorial effects. One may be interested in estimating the general mean and main effects, in addition to searching for and estimating a non‐negligible effect in the set of 2‐ and 3‐factor interactions, assuming 4‐ and higher‐order interactions are all zero. Such a search design is called a ‘main effect plus one plan’ and is denoted by MEP.1. Construction of such a plan, for 2^m factorial experiments, has been considered and developed by several authors and leads to MEP.1 plans for an odd number m of factors. These designs are generally determined by two arrays, one specifying a main effect plan and the other specifying a follow‐up. In this paper we develop the construction of search designs for an even number of factors m, m≠6. The new series of MEP.1 plans is a set of single array designs with a well structured form. Such a structure allows for flexibility in arriving at an appropriate design with optimum properties for search and estimation.     

Optimal orthogonal-array-based latin hypercubes

The use of optimal orthogonal array latin hypercube designs is proposed. Orthogonal arrays were proposed for constructing latin hypercube designs by Tang (1993). Such designs generally have better space filling properties than random latin hypercube designs. Even so, these designs do not necessarily fill the space particularly well. As a result, we consider orthogonal-array-based latin hypercube designs that try to achieve optimality in some sense. Optimization is performed by adapting strategies found in Morris & Mitchell (1995) and Ye et al. (2000). The strategies here search only orthogonal-array-based latin hypercube designs and, as a result, optimal designs are found in a more efficient fashion. The designs found are in general agreement with existing optimal designs reported elsewhere.     

Comparisons of search designs using search probabilities

Search designs are considered for searching and estimating one nonzero interaction from the two and three factor interactions under the search linear model. We compare three 12-run search designs D1, D2, and D3, and three 11-run search designs D4, D5, and D6, for a 2⁴ factorial experiment. Designs D2 and D3 are orthogonal arrays of strength 2, D1 and D4 are balanced arrays of full strength, D5 is a balanced array of strength 2, and D6 is obtained from D3 by deleting the duplicate run. Designs D4 and D5 are also obtained by deleting a run from D1 and D2, respectively. Balanced arrays and orthogonal arrays are commonly used factorial designs in scientific experiments. “Search probabilities” are calculated for the comparison of search designs. Three criteria based on search probabilities are presented to determine the design which is most likely to identify the nonzero interaction. The calculation of these search probabilities depends on an unknown parameter ρ which has a signal-to-noise ratio form. For a given value of ρ, Criteria I and II are newly proposed in this paper and Criteria III is given in Shirakura et al. (Ann. Statist. 24 (6) (1996) 2560). We generalize Criteria I–III for all values of ρ so that the comparison of search designs can be made without requiring a specific value of ρ. We have developed simplified methods for comparing designs under these three criteria for all values of ρ. We demonstrate, under all three criteria, that the balanced array D1 is more likely to identify the nonzero interaction than the orthogonal arrays D2 and D3, and the design D4 is more likely to identify the nonzero interaction than the designs D5 and D6.The methods of comparing designs developed in this paper are applicable to other factorial experiments for searching one nonzero interaction of any order.     

The existence of the strong combined-optimal design

The technique of fold-over is useful for conducting follow-up experiments. Based on the minimum aberration criterion, Li and Lin (2003) developed an algorithm and used computer to search the corresponding optimal foldover designs for 16 and 32 runs in the 2 ^k-p design. In their study, they found that the 2¹⁰⁻⁶ design is the only one that is not a strong combined-optimal design among all the designs. However, they did not interpret the reason causing the phenomenon. This article will explore under what kind of conditions, that the strong combined-optimal design will exist, and the solutions of the related problems.     

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

We consider a search design for the 2^m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2^m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2^m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).     

Factorial experimentation in second-order Latin cubes

A Second-order Latin cube of size n x n x n can be used as the design for an experiment in three space dimensions, with the three sets of layers used as three sets of blocks. The n2 treatments are then orthogonal to the main effects X, Y and Z of the blocking systems. Particular interest attaches to second-order Latin cubes whose treatments are an n x n factorial set, with the main effects A and B of the treatment factors orthogonal to the interactions XY, XZ and YZbetween pairs of blocking systems. This note describes such designs where components of the interaction AB are each totally confounded with one of XY, XZ and YZ. Cubes with n = 4 are then described where components of A, B and AB are each partially confounded. Finally, a defective design with n = 4 is described, to illustrate the need for care in composing designs for three dimensions.     

On the number of 2n-p fractional factorials of resolution III

An algorithm is specified and demonstrated which will compute the total number of ways a 2ⁿ factorial design may be partitioned into 2^p mutually exclusive 2^n-p fractional factorial designs, each having resolution III. The results of its application to all designs possessing resolution III fractions for n=5,…,20 are also given.     

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we obtain balanced resolution V plans for 2^m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal balanced resolution V designs obtained earlier in the work of Srivastava and Chopra.     

Comparing search and estimation performances of designs based on a compound criterion

Since the introduction of the search design by Srivastava [Designs for searching non-negligible effects. In: Srivastava, editor. A survey of statistical design and linear models. Amsterdam: North-Holland, Elsevier; 1975. p. 507–519], construction of such designs has been considered by many researchers. The efficient performances of constructed search designs in terms of parameter estimation and search ability of parameters have also been investigated by several authors. They have proposed suitable optimality measures such as DD- and AD-optimality for estimation in the early stage of search design construction. Moreover, since 1990s, some criteria have been developed to evaluate search performance of a design. Although these criteria are useful none of them is able to evaluate both estimation and search efficiency of a design simultaneously. In this paper, we propose dual-task criteria to deal with searching and estimating performances of search designs. These compound criteria are weighted multiplication of estimation and search suitable criteria. They will be used for design comparison and the results will be presented.     

A closer look at de-aliasing effects using an efficient foldover technique

Foldover techniques are used to reduce the confounding when some important effects (usually lower order effects) cannot be estimated independently. This article develops an efficient foldover mechanism for symmetric or asymmetric designs, whether regular or nonregular. In this paper, we take the uniformity criteria (UC) as the optimality measures to construct the optimal combined designs (initial design plus its corresponding foldover design) which have better capability of estimating lower order effects. The relationship between any initial design and its combined design is studied. A comparison study between the combined designs via different UC is provided. Equivalence between any combined design and its complementary combined design is investigated, which is a very useful constraint that reduce the search space. Using our results as benchmarks, we can implement a powerful algorithm for constructing optimal combined designs. Our work covers as well as gives results better than recent works of about 20 articles in the last few years as special cases. So this article is a good reference for constructing effective designs.     

Robust extrapolation designs and weights for biased regression models with heteroscedastic errors

We consider the construction of designs for the extrapolation of regression responses, allowing both for possible heteroscedasticity in the errors and for imprecision in the specification of the response function. We find minimax designs and correspondingly optimal estimation weights in the context of the following problems: (1) for ordinary least squares estimation, determine a design to minimize the maximum value of the integrated mean squared prediction error (IMSPE), with the maximum being evaluated over both types of departure; (2) for weighted least squares estimation, determine both weights and a design to minimize the maximum IMSPE; (3) choose weights and design points to minimize the maximum IMSPE, subject to a side condition of unbiasedness. Solutions to (1) and (2) are given for multiple linear regression with no interactions, a spherical design space and an annular extrapolation space. For (3) the solution is given in complete generality; as one example we consider polynomial regression. Applications to a dose-response problem for bioassays are discussed. Numerical comparisons, including a simulation study, indicate that, as well as being easily implemented, the designs and weights for (3) perform as well as those for (1) and (2) and outperform some common competitors for moderate but undetectable amounts of model bias.     

Spatial sampling design for parameter estimation of the covariance function

We study the spatial optimal sampling design for covariance parameter estimation. The spatial process is modeled as a Gaussian random field and maximum likelihood (ML) is used to estimate the covariance parameters. We use the log determinant of the inverse Fisher information matrix as the design criterion and run simulations to investigate the relationship between the inverse Fisher information matrix and the covariance matrix of the ML estimates. A simulated annealing algorithm is developed to search for an optimal design among all possible designs on a fine grid. Since the design criterion depends on the unknown parameters, we define relative efficiency of a design and consider minimax and Bayesian criteria to find designs that are robust for a range of parameter values. Simulation results are presented for the Matérn class of covariance functions.     

Optimal multi-criteria designs for Fourier regression models

Riccomagno, Schwabe and Wynn (RSW) (1997) have given a necessary and sufficient condition for obtaining a complete Fourier regression model with a design based on lattice points that is D-optimal. However, in practice, the number of factors to be considered may be large, or the experimental data may be restricted or not homogeneous. To address these difficulties we extend the results of RSW to obtain a sufficient condition for an incomplete interaction Fourier model design based on lattice points that is D-, A-, E- and G-optimal. We also propose an algorithm for finding such optimal designs that requires fewer design points than those obtained using RSW's generators when the underlying model is a complete interaction model.     

2 m 4 n designs with resolution III or IV containing clear two-factor interaction components

The orthogonal arrays with mixed levels have become widely used in fractional factorial designs. It is highly desirable to know when such designs with resolution III or IV have clear two-factor interaction components (2fic’s). In this paper, we give a complete classification of the existence of clear 2fic’s in regular 2 ^m 4 ⁿ designs with resolution III or IV. The necessary and sufficient conditions for a 2 ^m 4 ⁿ design to have clear 2fic’s are given. Also, 2 ^m 4 ⁿ designs of 32 runs with the most clear 2fic’s are given for n = 1,2.      

Designs for two-colour microarray experiments

Summary.  Designs for two-colour microarray experiments can be viewed as block designs with two treatments per block. Explicit formulae for the A- and D-criteria are given for the case that the number of blocks is equal to the number of treatments. These show that the A- and D-optimality criteria conflict badly if there are 10 or more treatments. A similar analysis shows that designs with one or two extra blocks perform very much better, but again there is a conflict between the two optimality criteria for moderately large numbers of treatments. It is shown that this problem can be avoided by slightly increasing the number of blocks. The two colours that are used in each block effectively turn the block design into a row–column design. There is no need to use a design in which every treatment has each colour equally often: rather, an efficient row–column design should be used. For odd replication, it is recommended that the row–column design should be based on a bipartite graph, and it is proved that the optimal such design corresponds to an optimal block design for half the number of treatments. Efficient row–column designs are given for replications 3–6. It is shown how to adapt them for experiments in which some treatments have replication only 2.     

Comparison of Four New General Classes of Search Designs

A factor screening experiment identifies a few important factors from a large list of factors that potentially influence the response. If a list consists of m factors each at three levels, a design is a subset of all possible 3 ^m runs. This paper considers the problem of finding designs with small numbers of runs, using the search linear model introduced in Srivastava (1975). The paper presents four new general classes of these 'search designs', each with 2 m −1 runs, which permit, at most, two important factors out of m factors to be searched for and identified. The paper compares the designs for 4 ≤ m ≤ 10, using arithmetic and geometric means of the determinants, traces and maximum characteristic roots of particular matrices. Two of the designs are found to be superior in all six criteria studied. The four designs are identical for m = 3 and this design is an optimal design in the class of all search designs under the six criteria. The four designs are also identical for m = 4 under some row and column permutations.     

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Some remarks about the gasser-sroka-jennen-steinmetz variance estimator

This article proposes some simplifications of the residual variance estimator of Gasset, Sroka, and Jeneen-Steinmetz (GSJ, 1986) which is often used in conjunction with non parametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, the estimate is distributed as a weighted sum of x² variables. By matching the first two moments, the distribution can be approximated by a x² with degrees of freedom determined by the coefficients of the. quadratic form. Computation of the estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.