http://www3.stat.sinica.edu.tw/statistica/oldpdf/A18n425.pdf

On general minimum lower order confounding criterion for s-level regular designs

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.

Blocked Two-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 and 243-run GMC designs with resolution or higher are tabulated. The Canadian Journal of Statistics 41: 192–210; 2013 © 2012 Statistical Society of Canada



Dans cet article, nous generalisons le critere de l'amalgame general minimal d'ordre inferieur (GMC) aux plans d'experience a trois niveaux. Dans un premier temps, nous passons en revue les relations entre le GMC et les autres criteres. Par la suite, nous presentons une representation numerique du nombre de composantes de repliement (ACNP) et un critere GMC a trois niveaux en considerant l'effet des composantes. Nous presentons aussi quelques resultats sur ce nouveau critere. Nous avons catalogue tous les devis GMC a 27 iterations et a 81 iterations ayant un nombre de facteurs egaux a et les devis GMC a 243 iterations ayant une resolution ou superieure. La revue canadienne de statistique 41: 192–210; 2013 © 2012 Societe statistique du Canada

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

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 \(r=2^m-3,2^m-2\).

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

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.

Jianwei Hu

Papers

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