Reverse Foldovers for Blocked 2 m−k Fractional Factorial Designs

Two-level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. When the experimental material to be used in the experiment is heterogenous or the experiment must be performed over several well-defined time periods, blocking is often used as a means to improve experimental efficiency by removing the possible effects of heterogenous experimental material or possible time period effects. In a recent article, Li and Jacroux (2007 Li , F. , Jacroux , M. (2007). Optimal foldover plans for blocked 2 ^m?k fractional factorial designs. J. Statsist. Plann. Infer 137:2434–2452. [Google Scholar]) suggested a strategy for constructing optimal follow-up designs for blocked fractional factorial designs using the well-known foldover technique in conjunction with several optimality criteria. In this article, we consider the reverse foldover problem for blocked fractional factorial designs. In particular, given a 2^(m+p)?(p+k) blocked fractional factorial design D, we derive simple sufficient conditions which can be used to determine if there exists a 2^{(m+p?1)?(p?1+k+1)} initial fractional factorial design d which yields D as a foldover combined design as well how to generate all such d. Such information is useful in developing an overall experimental strategy in situations where an experimenter wants an overall blocked fractional factorial design with “desirable” properties but also wants the option of analyzing the observed data at the halfway mark to determine if the significant experimental variables are obvious (and the experiment can be terminated) or if a different path of experimentation should be taken from that initially planned.     

A new look on optimal foldover plans in terms of uniformity criteria

In this paper, we develop a new mechanism for finding the optimal foldover plans (OFPs) which is based on the uniformity criteria measured by Lee discrepancy, wrap-around L₂-discrepancy, and centered L₂-discrepancy. For three-level fractional factorials as the original designs, general foldover plans and combined designs are defined, and lower bounds of these three discrepancies of combined designs under general foldover plans are also obtained, which can be used as benchmarks for searching OFPs. Illustrative examples with a comparison study between the foldover plans under these discrepancies are provided. Our results provide a theoretical justification for OFPs of three-level designs in terms of uniformity criteria.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

A bidimensional class of optimality criteria involving φp and characteristic criteria

A new biparametric class of criteria for Optimal Experimental Design generalizing the families of φ_p and Characteristic Criteria is presented. Some properties of the Characteristic Criteria are provided: in particular, differentiability, monotonicity and convexity. A statistical interpretation is also offered. Optimal designs with respect to Characteristic Criteria are obtained for polynomial and compartmental models and an extension of the Michaelis-Menten model. A thorough discussion for the trigonometric model is given. The computed optimal designs are shown to be quite efficient for A-, D- and D_s -optimality. Thus, some of them appear as a natural compromise between different optimality criteria. A simulation in multiple linear regression confirms the quality of Characteristic designs for discovering significant differences between parameters.     

On constructing general minimum lower order confounding two-level block designs

In practice, to reduce systematic variation and increase precision of effect estimation, a practical design strategy is then to partition the experimental units into homogeneous groups, known as blocks. It is an important issue to study the optimal way on blocking the experimental units. Blocked general minimum lower order confounding (B¹-GMC) is a new criterion for selecting optimal block designs. The paper considers the construction of optimal two-level block designs with respect to the B¹-GMC criterion. By utilizing doubling theory and MaxC2 design, some optimal block designs with respect to the B¹-GMC criterion are obtained.     

D‐optimal minimax fractional factorial designs

The D‐optimal minimax criterion is proposed to construct fractional factorial designs. The resulting designs are very efficient, and robust against misspecification of the effects in the linear model. The criterion was first proposed by Wilmut & Zhou (2011); their work is limited to two‐level factorial designs, however. In this paper we extend this criterion to designs with factors having any levels (including mixed levels) and explore several important properties of this criterion. Theoretical results are obtained for construction of fractional factorial designs in general. This minimax criterion is not only scale invariant, but also invariant under level permutations. Moreover, it can be applied to any run size. This is an advantage over some other existing criteria. The Canadian Journal of Statistics 41: 325–340; 2013 © 2013 Statistical Society of Canada     

On the E-optimality of blocked main effects plans when n ≡ 1(mod4)

ABSTRACT

In this article, we consider experimental situations in which m two-lever factors are to be studied using a two-level main effects plan involving n runs which are partitioned into b blocks of size k =?n/b. For the casen ≡ 1(mod?4) and b???k, we derive some new methods of constructing E-optimal designs which tend to be highly efficient under other optimality criteria as well.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

A Class of Experimental Designs Balanced for First Residuals

In this paper a new class of designs involving sequences of treatments balanced for first residual effects has been introduced. These designs require only t experimental units for 2t periods, t being the number of treatments to be tested. A unified method of constructing these designs for all values of t (≥2) along with an appropriate method of analysis is presented. Besides, their efficiency relative to some well known designs is investigated.     

Minimal circular balanced repeated measurement designs in unequal period sizes

Abstract

Repeated measurement designs (RMDs) are widely used in medicine, pharmacology, animal sciences and psychology. In these fields, there are several situations where these designs should be used in periods of different sizes. With the use of RMD, residual effects or carry over effects may arise and balanced RMDs are solution to this problem. In this article, therefore, some infinite series are developed through method of cyclic shifts to obtain circular balanced repeated measurements designs in periods of two different sizes.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

Approximate E-optimal designs for the model of spring balance weighing with a constant bias

The present paper analyzes the linear regression model with a nonzero intercept term on the vertices of a d-dimensional unit cube. This setting may be interpreted as a model of weighing d objects on a spring balance with a constant bias. We give analytic formulas for E-optimal designs, as well as their minimal efficiencies under the class of all orthogonally invariant optimality criteria, proving the criterion-robustness of the E-optimal designs. We also discuss the D- and A-optimal designs for this model.     

Optimum Designs for Parameter Estimation in Linear Mixture Models with Synergistic Effects

In this article, we derive optimum designs for parameter estimation in a mixture experiment when the response function is linear in the mixing components with some synergistic effects. The D- and A-optimality criteria have been used for the purpose. The Equivalence Theorem has been used to check for the optimality of the proposed designs.     

Two-Level Designs in Blocks of Size Two in Which All Main Effects and Two-Factor Interactions Are Estimable

There are many problems in the real world for which it is necessary to perform two points in one block. In this situation, information about certain treatments may be indistinguishable from, or confounded with, blocks. If the experimenter wants to estimate all main effects and two-factor interactions in which two points are in one block, then how to perform the blocking schemes is an important topic to study. In this article, we combine several 2 ^k?p fractions from same or different family to obtain better combined designs requiring fewer runs than those appearing in the literature.     

Using orthogonal array for constructing three-level search designs

We consider the problem of constructing search designs for 3^m factorial designs. By using projection properties of some three-level orthogonal arrays, some search designs are obtained for 3 ? m ? 11. The new obtained orthogonal search designs are capable of searching and identifying up to four two-factor interactions and estimating them along with the general mean and main effects. The resulted designs have very high searching probabilities; it means that besides the well-known orthogonal structure, they have high ability in searching the true effects.     

On the use of partial confounding for the construction of alternative regular two-level blocked fractional factorial designs

In this paper, we consider experimental situations in which a regular fractional factorial design is to be used to study the effects of m two-level factors using n=2^m−k experimental units arranged in 2^p blocks of size 2^m−k−p. In such situations, two-factor interactions are often confounded with blocks and complete information is lost on these two-factor interactions. Here we consider the use of the foldover technique in conjunction with combining designs having different blocking schemes to produce alternative partially confounded blocked fractional factorial designs that have more estimable two-factor interactions or a higher estimation capacity or both than their traditional counterparts.     

Covariance Analysis for Split-Plot and Split-Block Designs

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

Proper methodology for the analysis of covariance for experiments designed in a split-plot or split-block design is not found in the statistical literature. Analyses for these designs are often performed incompletely or even incorrectly. This is especially true when popular statistical computer software packages are used for the analysis of these designs. This article provides several appropriate models, ANOVA tables, and standard errors for comparisons from experiments arranged in a standard split-plot, split–split-plot, or split-block design where a covariate has been measured on the smallest size experimental unit.     

Construction of Efficient Multi-Level k-Circulant Supersaturated Designs

This article proposes an algorithm to construct efficient balanced multi-level k-circulant supersaturated designs with m factors and n runs. The algorithm generates efficient balanced multi-level k-circulant supersaturated designs very fast. Using the proposed algorithm many balanced multi-level supersaturated designs are constructed and cataloged. A list of many optimal and near optimal, multi-level supersaturated designs is also provided for m ≤ 60 and number of levels (q) ≤10. The algorithm can be used to generate two-level k-circulant supersaturated designs also and some large optimal two-level supersaturated designs are presented. An upper bound to the number of factors in a balanced multi-level supersaturated design such that no two columns are fully aliased is also provided.     

IN JUSTIFICATION OF WALD'S CRITERION OF EFFICIENCY OF EXPERIMENTAL DESIGNS IN RELATION TO THE WEIGHING PROBLEM1

Wald (1943), among others, examined the question of efficiency of experimental designs in general and suggested the criterion of maximization of the det. [X'X] for the purpose, where X denotes the design matrix. In the same context, Ehrenfeld (1955) recommended the maximization of the minimum eigenvalue of [X'X] as the desideratum. It has been shown in this note that for weighing designs in general maximization of det. [X'X] may be preferred to the latter criterion.     

Some results on constructing two-level block designs with general minimum lower order confounding

Block designs are widely used in experimental situations where the experimental units are heterogeneous. The blocked general minimum lower order confounding (B-GMC) criterion is suitable for selecting optimal block designs when the experimenters have some prior information on the importance of ordering of the treatment factors. This paper constructs B-GMC 2^{n ? m}: 2^r designs with 5 × 2^l/16 + 1 ? n ? (N ? 2^l) < 2^{l ? 1} for l(r + 1 ? l ? n ? m ? 1), where 2^{n ? m}: 2^r denotes a two-level regular block design with N = 2^{n ? m} runs, n treatment factors, and 2^r blocks. With suitable choice of the blocking factors, each B-GMC block design has a common specific structure. Some examples illustrate the simple and effective construction method.