Construction of regular 2n41 designs with general minimum lower-order confounding

Mixed-level designs, especially two- and four-level designs, are very useful in practice. In the last two decades, there are quite a few literatures investigating the selection of this kind of optimal designs. Recently, the general minimum lower-order confounding (GMC) criterion (Zhang et al., 2008 Zhang, R.C., Li, P.F., Zhao, S.L., Ai, M.Y. (2008). A general minimum lower order confounding criterion for two-level regular design. Stat. Sin. 18:1689–1705.[Web of Science ®] , [Google Scholar]) gave a new approach for choosing optimal factorials. It is proved that the GMC designs are more powerful than other criteria in the widely practical situations. In this paper, we extend the GMC theory to the mixed-level designs. Under the theory we establish a new criterion for choosing optimal regular two- and four-level designs. Further, a construction method is proposed to obtain all the 2ⁿ4¹ GMC designs with N/4 + 1 ? n + 2 ? 5N/16, where N is the number of runs and n is the number of two-level factors.     

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.     

Construction of blocked two-level regular designs with general minimum lower order confounding

Zhang et al. (2008) proposed a general minimum lower order confounding (GMC for short) criterion, which aims to select optimal factorial designs in a more elaborate and explicit manner. By extending the GMC criterion to the case of blocked designs, Wei et al. (submitted for publication) proposed a B¹-GMC criterion. The present paper gives a construction theory and obtains the B¹-GMC 2^n−m:2^r$2^{n-m}:2^r$ designs with n≥5N/16+1$n\ge 5N/16+1$, where 2^n−m:2^r$2^{n-m}:2^r$ denotes a two-level regular blocked design with N=2^n−m$N=2^{n-m}$ runs, n   treatment factors, and 2^r$2^r$ blocks. The construction result is simple. Up to isomorphism, the B¹-GMC 2^n−m:2^r$2^{n-m}:2^r$ designs can be constructed as follows: the n   treatment factors and the 2^r−1$2^r-1$ block effects are, respectively, assigned to the last n   columns and specific 2^r−1$2^r-1$ columns of the saturated 2^{(N−1)−(N−1−n+m)}$2^{(N-1)-(N-1-n+m)}$ design with Yates order. With such a simple structure, the B¹-GMC designs can be conveniently used in practice. Examples are included to illustrate the theory.     

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.     

Three‐level regular designs with general minimum lower‐order confounding

In this paper, we extend the general minimum lower‐order confounding (GMC) criterion to the case of three‐level designs. First, we review the relationship between GMC and other criteria. Then we introduce an aliased component‐number pattern (ACNP) and a three‐level GMC criterion via the consideration of component effects, and obtain some results on the new criterion. All the 27‐run GMC designs, 81‐run GMC designs with factor numbers $n=5,\ldots,20$ and 243‐run GMC designs with resolution $IV$ or higher are tabulated. The Canadian Journal of Statistics 41: 192–210; 2013 © 2012 Statistical Society of Canada     

Construction of minimum aberration blocked two-level regular factorial designs

ABSTRACT

In this article, we consider the construction of minimum aberration 2^{n ? k}: 2^p designs with respect to some existing combined wordlength patterns, where a 2^{n ? k}: 2^p design is a blocked two-level design with n treatment factors, 2^p blocks, and N = 2^q runs with q = n ? k. Two methods are proposed for two situations: n ? 2^{q ? p} ? 1 and n > N/2. These methods enable us to obtain some new minimum aberration 2^{n ? k}: 2^p designs from existing minimum aberration unblocked and blocked designs. Examples are included to illustrate the theory.     

Some results on two-level factorial designs with dependent observations

There are many situations in which observations in factorial experiments may be dependent. When this is so, run orders are needed that result in efficient estimates of contrasts. The Cheng and Steinberg reverse foldover algorithm, which gives a maximal number of level changes, is known to produce very efficient main-effects two-level designs using the D-criterion, but less is known about other designs, models and criteria. We present some further theoretical results, and give another statistic of importance in predicting efficiency under strong dependence. The theory is illustrated using some 16-run designs.     

Robust two-level regular fractional factorial designs

Abstract

In this paper, we introduce the concept of model quality for two-level regular fractional factorial designs. Under the effect hierarchy principle, this paper raises the definition of model quality and introduces robust model-number pattern (RP) to choose the optimal robust design. Some theoretical results on this optimality and comparisons with GMC and MEC criterion are given.     

Generalized minimum aberration two-level split-plot designs

Split-plot experiments may arise when it is impractical to completely randomize the treatment combinations of a designed experiment. To provide more flexible design choices in the nonregular split-plot setting, we describe an approach for constructing minimum aberration orthogonal two-level split-plot designs having 12, 16, 20 and 24 runs. We consider five design scenarios that may be of importance to practitioners, and then propose an approach for assigning word lengths under these five scenarios. We then use the extended word length patterns to rank both regular and nonregular orthogonal split-plot designs. While most existing papers concerning orthogonal split-plot designs focus on regular orthogonal designs, we find that many minimum aberration split-plot designs are nonregular orthogonal designs.     

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.     

Graph based isomorph-free generation of two-level regular fractional factorial designs

We provide a new necessary and sufficient check for testing the isomorphism of two 2-level regular fractional factorial designs. The approach is based on modeling fractional factorial designs as bipartite graphs. We employ an efficient canonical graph labeling approach to compare two designs for isomorphism. We then improve upon the existing non-isomorphic fractional factorial design generation algorithm by reducing the number of candidate designs from which isomorphs need to be removed. Not only does our method generate non-isomorphic designs much faster, it is also able to generate designs with run sizes of 2048 and 4096 runs, which were not generated by the existing methods.     

General minimum lower order confounding designs: An overview and a construction theory

For fractional factorial (FF) designs, Zhang et al. (2008) introduced a new pattern for assessing regular designs, called aliased effect-number pattern (AENP), and based on the AENP, proposed a general minimum lower order confounding (denoted by GMC for short) criterion for selecting design. In this paper, we first have an overview of the existing optimality criteria of FF designs, and then propose a construction theory for 2^n−m$2^{n-m}$ GMC designs with 33N/128≤n≤5N/16$33N/128\le n\le 5N/16$, where N=2^n−m$N=2^{n-m}$ is the run size and n is the number of factors, for all N's and n  's, via the doubling theory and SOS resolution IV designs. The doubling theory is extended with a new approach. By introducing a notion of rechanged (RC) Yates order for the regular saturated design, the construction result turns out to be quite transparent: every GMC 2^n−m$2^{n-m}$ design simply consists of the last n columns of the saturated design with a specific RC Yates order. This can be very conveniently applied in practice.     

Some results on coverings of galois spaces with ovoids and related bib designs

Some different coverings of Galois planes with ovoids are obtained and a three-dimensional example is also introduced. The regularity of these structures is put in evidence by constructing related BIB designs.     

Some results on optimality in models with heteroscedastic errors from partial optimum designs

This paper studies optimum designs for linear models when the errors are heteroscedastic. Sufficient conditions are given in order to obtainD-, A- andE-optimum designs for a complete regression model from partial optimum designs for some sub-parameters. A result about optimality for a complete model from the optimality for the submodels is included. Supported by Junta de Andalucía, research group FQM244.     

Fisher''s inequality for designs on regular graphs

Let G=(V,E) be a regular graph of valency d. A (v,k,λ,μ)-design over G is a pair , where is a family of k-subsets of V (blocks) such that for any distinct vertices x and y, the number of blocks containing {x,y} is equal to λ if {x,y} is an edge and is equal to μ if {x,y} is not an edge. We will prove that the number of vertices does not exceed the number of blocks (Fisher's Inequality) under the following condition: (r−μ)/(μ−λ) is not a multiple eigenvalue of the adjacency matrix of the graph (r is the replication number of the design). We also give examples showing that this restriction is essential.