General Minimum Lower-order Confounding Research Articles

Construction of Split-Plot Designs with General Minimum Lower Order Confounding

Fractional factorial split-plot design has been widely used in many fields due to its advantage of saving experimental cost. The general minimum lower order confounding criterion is usually used as one of the attractive design criterion for selecting fractional factorial split-plot design. In this paper, we are interested in the theoretical construction methods of the optimal fractional factorial split-plot designs under the general minimum lower order confounding criterion. We present the theoretical construction methods of optimal fractional factorial split-plot designs under general minimum lower order confounding criterion under several conditions.

Results on constructing [formula omitted] regular designs with general minimum lower-order confounding

For s-level regular designs, an aliased component effect-number pattern (AENP) can be used to investigate all of confounding information among factorial component effects. Based on the AENP, we introduce general minimum lower-order confounding (GMC) criterion to choose optimal sn−m designs, where s is a prime or prime power. We first have an overview of GMC theory and introduce s-level GMC criterion. Then, some properties are provided to study s-level GMC designs in terms of their complementary sets. Further, we obtain the lower-order confounding expressions of the constructed GMC designs. Some examples are given to illustrate these results.

General Minimum Lower-Order Confounding Designs with Multi-Block Variables

Blocking the inhomogeneous units of experiments into groups is an efficient way to reduce the influence of systematic sources on the estimations of treatment effects. In practice, there are two types of blocking problems. One considers only a single block variable and the other considers multi-block variables. The present paper considers the blocking problem of multi-block variables. Theoretical results and systematical construction methods of optimal blocked 2 n − m designs with N / 4 + 1 ≤ n ≤ 5 N / 16 are developed under the prevalent general minimum lower-order confounding (GMC) criterion, where N = 2 n − m .

Construction of some [formula omitted]-level regular designs with general minimum lower-order confounding

Based on an aliased component-number pattern (ACNP), a general minimum lower-order confounding (GMC) criterion has been proposed to choose the optimal regular designs, which minimize the confounding among lower-order effects. This paper is ready to study the properties of GMC s-level designs in terms of complementary sets. It is proved that an sn−m design has GMC only if its complementary set is contained in a flat. Then some GMC sn−m designs are constructed when n=(N−sr)∕(s−1)+t and 0≤t≤(sr−sr−1)∕(s−1), where N=sn−m and r<n−m. These results are further illustrated with some examples.

Some properties of general minimum lower-order confounding designs

The general minimum lower order confounding (GMC) criterion was proposed to select factorial designs, called GMC designs. The theory of constructing GMC 2n–m designs with was studied by Li, Zhao, and Zhang (2011), where 2n–m denotes a two-level design with n factors and runs. In this article, we propose a method to study the properties of aliasing relations of GMC 2n–m designs with , which make the aliasing information of these designs obtained theoretically. We illustrate how our method can be used to study partially GMC designs and blocked GMC designs.

Construction of some 3n-m regular designs with general minimum lower order confounding

For three-level designs, the general minimum lower order confounding (GMC) criterion aims to choose optimal designs by treating aliased component-number pattern (ACNP) as a set. In this article, we develop some theoretical results of a three-level GMC criterion. The characterizations of three-level GMC designs are studied in terms of complementary sets. All GMC 3n–m designs with N = 3n–m runs and the factor number n = (N – 3r)/2 + i are constructed for r< n – m and i = 0,1,2,3. Furthermore, the confounding information of lower order component effects of GMC 3n–m designs is obtained.

A note on constructing clear compromise plans

The regular two-level fractional factorial designs of n factors and N runs, having resolution IV and allowing experimenters to clearly estimate all main effects and a set of required two-factor interactions (2fi’s), are called clear compromise plans. Four classes of clear compromise plans have been discussed in the literature. The general minimum lower order confounding (GMC) is an elaborate criterion, which was proposed to select optimal fractional factorial designs. This paper gives a theory on constructing a set of class three clear compromise plans with 5N∕32+3≤n≤N∕4+1, called partially general minimum lower order confounding (P-GMC) designs. We first prove that each P-GMC design is constructed by a GMC design and two specified columns. Then we study the properties of these designs. For N=32,64 and 128, we illustrate that the P-GMC designs are admissible designs. Furthermore, they all have GMC, except for the P-GMC 213−7 and 223−16 designs.

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

ABSTRACTMixed-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) 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.

A note on the construction of blocked two-level designs with general minimum lower order confounding

Blocked designs are useful in experiments. The general minimum lower order confounding (GMC) is an elaborate criterion which was proposed for selecting optimal fractional factorial designs. Zhang and Mukerjee (2009b) extended the GMC criterion to the B-GMC criterion for selecting a 2n−m:2r design, where 2n−m:2r denotes a two-level regular blocked design with N=2n−m runs, n treatment factors and 2r blocks. This paper gives the first construction method of B-GMC 2n−m:2r designs with 5N/16+1≤n≤N/2. The results indicate that under isomorphism, with suitable choice of the blocking factors, each B-GMC blocked design has a common specific structure. Examples are included to illustrate the developed theory.

Analysis on $$s^{n-m}$$ s n - m designs with general minimum lower-order confounding

An optimal design should minimize the confounding among factor effects, especially the lower-order effects, such as main effects and two-factor interaction effects. Based on the aliased component-number pattern, general minimum lower-order confounding (GMC) criterion can provide the confounding information among factors of designs in a more elaborate and explicit manner. In this paper, we extend GMC theory to s-level regular designs, where s is a prime or prime power. For an \(s^{n-m}\) design D with \(N=s^{n-m}\) runs, the confounding of design D is given by complementary set. Further, according to the factor number n, we discuss two cases: (i) \(N/s<n\le (N-1)/(s-1)\), and (ii) \((N/s+1)/2<n\le N/s\). We not only provide the lower-order confounding information among component effects of D, but also obtain three necessary conditions for design D to have GMC.

Results for Two‐Level Designs with General Minimum Lower‐Order Confounding

The general minimum lower-order confounding (GMC) criterion for two-level design not only reveals the confounding information of factor effects but also provides a good way to select the optimal design, which was proposed by Zhang et al. (2008). The criterion is based on the aliased effect-number pattern (AENP). Therefore, it is very important to study properties of AENP for two-level GMC design. According to the ordering of elements in the AENP, the confounding information between lower-order factor effects is more important than that of higher-order effects. For two-level GMC design, this paper mainly shows the interior principles to calculate the leading elements 1 # C 2 and 2 # C 2 in the AENP. Further, their mathematical formulations are obtained for every GMC 2n−m design with N = 2n−m according to two cases: (i) 5N/16 + 1 ≤ n < N/2 and (ii) N/2 ≤ n ≤ N − 1.

On construction of blocked general minimum lower-order confounding [formula omitted] designs with [formula omitted

As an extension of the general minimum lower-order confounding (GMC) criterion to the case of blocked design, a B 1-GMC criterion was proposed (see Wei et al. (2014)). Zhao et al. (2013) constructed all the B 1-GMC 2n−m:2r designs with 5N/16+1≤n≤N−1, where N=2n−m, n and r are the numbers of runs, treatment factors and block factors respectively. In this paper, we establish a construction theory to obtain all the B 1-GMC 2n−m:2r designs with N/4+1≤n≤5N/16. For application, all the B 1-GMC 2n−m:2r designs with N/4+1≤n≤5N/16 and N=16,32,64 and 128 respectively are tabulated.

Blocked Two-Level Regular Designs With General Minimum Lower Order Confounding

Blocked designs are widely used in experiments with heterogeneous experimental units. In this article, we establish an optimality theory of blocked designs when the experimenter has prior information about the order of importance of the treatment factors. First, as an extension of aliased effect-number pattern (AENP) in the unblocked case, a blocked aliased effect-number pattern (B-AENP) is introduced for classifying blocked designs. Then, based on the B-AENP, a blocked general minimum lower order confounding (denoted as B -GMC) criterion for selecting optimal designs is proposed. Some theoretical results are obtained and comparisons are made with several existing criteria. For application and comparison, optimal blocked designs in 16, 32, and 64 runs are tabulated under the B -GMC and several existing criteria.

Some results on constructing general minimum lower order confounding $$2^{n-m}$$ 2 n - m designs for $$n\le 2^{n-m-2}$$ n ≤ 2 n - m - 2

Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC \(2^{n-m}\) designs with \(N/4+1\le n\le N-1\) were constructed by Li et al. (Stat Sinica 21:1571–1589, 2011), Zhang and Cheng (J Stat Plan Inference 140:1719–1730, 2010) and Cheng and Zhang (J Stat Plan Inference 140:2384–2394, 2010), where \(N=2^{n-m}\) is run number and \(n\) is factor number. In this paper, we first study some further properties of GMC design, then we construct all the GMC \(2^{n-m}\) designs respectively with the three parameter cases of \(n\le N-1\): (i) \(m\le 4\), (ii) \(m\ge 5\) and \(n=(2^m-1)u+r\) for \(u>0\) and \(r=0,1,2\), and (iii) \(m\ge 5\) and \(n=(2^m-1)u+r\) for \(u\ge 0\) and \(r=2^m-3,2^m-2\).

On simplifying the calculations leading to designs with general minimum lower-order confounding

Motivated by the effect hierarchy principle, Zhang et al. (Stat Sinica 18:1689–1705, 2008) introduced an aliased effect number pattern (AENP) for regular fractional factorial designs and based on the new pattern proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal \(2^{n-m}\) designs. Zhang et al. (Stat Sinica 18:1689–1705, 2008) proved that most existing criteria can be obtained by functions of the AENP. In this paper we propose a simple method for the calculation of AENP. The method is much easier than before since the calculation only makes use of the design matrix. All 128-run GMC designs with the number of factors ranging from 8 to 32 are provided for practical use.

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

AbstractIn 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

Some theory for constructing general minimum lower order confounding designs

General minimum lower order confounding (GMC) is a newly proposed design criterion that aims at keeping the lower order effects unaliased with one another to the extent possible. This paper shows that for 5N/16 < n ≤ N/2, 9N/32 < n ≤ 5N/16, and 17N/64 < n ≤ 9N/32, all GMC designs with N runs and n two-level factors are projections of maximal designs with N/2, 5N/16, and 9N/32 factors, respectively. Furthermore, it provides immediate approaches to construct- ing these GMC designs from the respective maximal designs; these approaches can produce many more GMC designs than the existing computer search method.

A theory on constructing 2n-m designs with general minimum lower order confounding

When designing an experiment, it is important to choose a design that is optimal under model uncertainty. The general minimum lower-order confounding (GMC) criterion can be used to control aliasing among lower-order factorial effects. A characterization of GMC via complementary sets was considered in Zhang and Mukerjee (2009a); however, the problem of constructing GMC designs is only par- tially solved. We provide a solution for two-level factorial designs with n factors and N = 2 n−m runs subject to a restriction on (n, N ): 5N/16 + 1 ≤ n ≤ N − 1. The construction is quite simple: every GMC design, up to isomorphism, consists of the last n columns of the saturated 2 (N −1)−(N −1−n+m) design with Yates order. In addition, we prove that GMC designs differ from minimum aberration designs when (n, N ) satisfies either of the following conditions: (i) 5N/16 + 1 ≤ n ≤ N/2 − 4, or (ii) n ≥ N/2, 4 ≤ n + 2 r − N ≤ 2 r−1 − 4 with r ≥ 4.

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

Zhang et al. (2008) introduced an aliased effect-number pattern (AENP) for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. By using a finite projective geometric formulation, Zhang and Mukerjee (2009a) characterized GMC designs via complementary designs for general s-level case, and to find GMC designs, for some special cases they proved a result that a design T can have GMC only if T¯ is contained in a specific flat. In this paper, we first generalize the result to general cases for s=2. Then, we prove that, for any given n and m, a GMC design minimizes A3, the first term of the wordlengh pattern of regular 2n−m designs. Furthermore, we find out the unique optimal confounding structure between main effects and two-factor interactions, and prove that minimizing A3 is a sufficient and necessary condition for a regular design to have the structure.

On construction of general minimum lower order confounding 2n−m designs with [formula omitted

The construction of optimal 2n−m designs with N/4+1≤n≤9N/32, where N=2n−m is the run size, and their comparison under different criteria have received significant attention in recent years. In this paper, we first prove that the MaxC2 design with n=N/4+1 is unique up to isomorphism and has general minimum lower order confounding (GMC). Then, by utilizing the theory of doubling and second order saturated resolution IV designs extended by Zhang and Cheng (2010), we propose a method of constructing GMC design and obtain all the GMC designs with N/4+1≤n≤9N/32 up to isomorphism. Finally, we show that, for all N and n in the above range, the MA and GMC designs are different.