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As explained in refs., we can define minimum secondary aberration mixed-level fractional factorial split-plot designs of other types according to different orderings of importance of different types of pencils

Minimum secondary aberration fractional factorial split-plot designs in terms of consulting designs

Science China-mathematics, no. 4 (2006): 494-512

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摘要

It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial...更多

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简介
  • Ractional factorial (FF) designs are widely used in industrial and agricultural experiments.
  • For i = 1, · · · , n, let Ai(d) denote the number of distinct i-factor interaction pencils appearing in the defining contrast subgroup of a regular FFSP sn1−p1 · sn2 design d(C1, C2).
  • By Lemmas 1–3 and identity (3), the authors get the following Theorem 1, which establish the connection between the secondary wordlength patterns of a regular FFSP design and its consulting design.
重点内容
  • Ractional factorial (FF) designs are widely used in industrial and agricultural experiments
  • As discussed later in more details, the design matrix of an fractional factorial split-plot (FFSP) design looks exactly like a classical fractional factorial (FF)’s, and the difference lies in the following two special features: (a) not all factors have the same status, (b) the inference is possible at two distinct levels of accuracy
  • We introduce some concepts in coding theory which will later be used to establish the connection between FFSP design theory and coding theory
  • By Lemmas 1–3 and identity (3), we get the following Theorem 1, which establish the connection between the secondary wordlength patterns of a regular FFSP design and its consulting design
  • As explained in refs. [5,9], we can define minimum secondary aberration (MSA) mixed-level FFSP designs of other types according to different orderings of importance of different types of pencils
  • It is not hard to extend these approaches to the general mixed-level FFSP designs containing more than two more-level SP factors and get more general results
结果
  • The secondary wordlength patterns {Bi(d)} and {Bi(dR)} of a regular FFSP sn1−p1 · sn2 design d and its consulting design dR satisfy the following equations: i i−2
  • The secondary wordlength patterns {Bij(d)} and {Bij(dR)} of a regular FFSP sn1−p1 ·sn2 design d = d(C1, C0, C2) and its consulting design dR
  • Let {Bij(d)} and {Bij(dR)} be the secondary wordlength patterns of a regular FFSP sn1−p1 ·sn2 design d = d(C1, C0, C2) and its consulting design dR, respectively.
  • MSA regular FFSP sn1−p1 ·sn2 designs through the wordlength patterns of their consulting designs.
  • Let {Ai,j(d)}, {Bi,j(d)} and {Ai,j(dR)}, {Bi,j(dR)} be the word length patterns and secondary word lengthpatterns of a regular FFSP sn1−p1 ·sn2 design d = d(C1, C0, C2) and its consulting design dR, respectively.
  • The secondary wordlength patterns {Bi,j(d)} and {Bi,j(dR)} of a regular FFSP sn1−p1 ·(sr2 )sn2 design d = d(C1, C01, C02, C2) and its consulting design dR satisfy the following equations: Bi,0(d) + η1Bi+1,1(d) + η2Bi+1,2(d) + η3Bi+2,3(d)
  • Let {Bi,j(d)} and {Bi,j(dR)} be the secondary wordlength patterns of a regular FFSP sn1−p1 ·(sr1 )(sr2 )sn2 design d = d(C1, C01, C02, C2) and its consulting design dR, respectively.
  • [9], the authors get the following rule for identifying MSA regular FFSP designs with respect to the definite triplet of subflats of P through the wordlength patterns of their consulting designs.
  • Let {Ai,j(d)}, {Bi,j(d)} and {Ai,j(dR)}, {Bi,j(dR)} be the wordlength patterns and secondary wordlength patterns of a regular FFSP sn1−p1 ·(sr2 )sn2 design d = d(C1, C01, C02, C2) and its consulting design dR, respectively.
结论
  • Let {Ai,j(d)}, {Bi,j(d)} and {Ai,j(dR)}, {Bi,j(dR)} be the wordlength patterns and secondary wordlength patterns of a regular FFSP sn1−p1 ·2sn2 design d = d(C1, C01, C02, C2) and its consulting design dR, respectively.
  • This paper obtains some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular FFSP design to that of its consulting design.
  • It is not hard to extend these approaches to the general mixed-level FFSP designs containing more than two more-level SP factors and get more general results
基金
  • This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10231030 & 10571093) and Specialized Research Fund for the Doctoral Program of Higher Education (Grant No 20050055038)
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