Theory Of S-level Regular Designs With General Minimum Lower Order Confounding

Factorial designs have wide applications in many fields. For enhancing the effi-ciency of an experiment, it is important to employ optimal designs. Up to now, quite afew criteria for selecting optimal factorial designs are proposed: maximum resolution(MR), minimum aberration (MA), clear effects (CE), maximum estimation capacity(MEC) and general minimum lower order confounding (GMC) criteria. Note that MRand MA criteria are based on the word-length pattern (WLP). MEC criterion appearsto have no direct connection with the WLP. However,[31] proves that if attention isrestricted to main effects and two-factor interactions (2fi’s), then an MEC design hasMA. As for CE criterion, it selects designs by the numbers of clear main effects andclear2fi’s.To deeply provide the confounding information among factors of designs and re-veal the relationships of the above criteria,[56] introduces an aliased effect-numberpattern (AENP) for two-level regular design. Based on the AENP, it provides a newoptimality criterion–GMC criterion. The AENP contains the basic information of thedifferent order effects aliased with the other order effects at various severity degrees.Thus, the WLP and the number of clear factors are respectively the functions of theAENP (see [56]). GMC criterion aims at finding optimal designs under effect hierarchyprinciple (EHP) in a more elaborate and explicit manner. GMC is flexible, accommo-dating to the prior information about the relative importance of factors. This kind ofprior information is often available in practice, so GMC designs are widely applicable.Now the AENP and GMC criterion have been developed well and formed GMC the-ory, which is applied in regular designs, block designs, split-plot designs, mixed-leveldesigns and orthogonal designs. The literature in two-level designs has grown copi-ously and included many results. However, the articles for three-or general-level GMCdesigns are few.The doctoral dissertation aims at extending GMC theory to the cases of three-and prime or prime power s-level regular designs. However, we can not directly apply the AENP and GMC criterion of two-level case to three-or s-level designs since thedistinction lies in the correspondence between factorial effects and components. Inthe former, there is a one-to-one correspondence and use of components to representfactorial effects is unnecessary. In the latter, a2fi of three-level case corresponds totwo orthogonal components, and a2fi corresponds to s1components for s-level case.Moreover, we use the alias relation of components to reveal the confounding amongfactors. Since lower order components are more likely to be important than higher orderones and the same-order components are equally likely to be important, we establish thecomponent hierarchy principle (CHP). Based on the CHP, we introduce the concept ofthe alias component-number pattern (ACNP), especially, the AENP of two-level case isalso called the ACNP. Based on the ACNP we provide three-or s-level GMC criterion.The main contributions of this thesis are five parts:The first part is formed by Chapter2. Based on the CHP, we introduce the conceptof ACNP as follows:and three-level GMC criterion. Here#)iC(kjmeans the number of ith-order interactioncomponents that are aliased with jth-order interaction components at degree k. Espe-cially, the cases i=0,1respectively correspond to the grand mean and a main effect.Theorem2.2.1gets the relationship of the ACNP and the WLP:where Aiis the number of words with length i in the defining contrast identity relations.For any three-level designs with resolution R≥III, Theorem2.2.2gives the numberof clear main effects C1=#0)2. Theorem2.3.3and Theorem2.3.4prove some necessary conditions for three-level designs to have GMC. In the last section, we list a catalogue of designs withall27-run GMC designs,81-run GMC designs with n=5,...,20and243-run GMCdesigns with resolution IV or higher, which includes the comparative case of GMCdesigns and MA designs.Second part contains Chapter3. For GMC two-level designs, there are some prin-ciples for calculating the value of#2C2. To reveal the characters, we analyze the structure of the constructed2n mGMC designs with5N/16+1≤n <N/2andn≥N/2, and obtain the calculation formulation of##1C2and2C2, where N=2n m.Main results are Theorem3.2.6, Theorem3.2.7, Theorem3.3.1and Theorem3.3.2.Third part consists of Chapter4. We study prime or prime power s-level GMCdesigns. Theorem4.1.1obtains the calculation formula of the ACNP for any sn mdesign D below:D, lt(=0)∈GF(s1),1≤t≤i} denotes the number of ith-order interaction compo-nents aliased with γ (γ∈Hq). Here Hqis an s-level saturated design.From Theorem4.2.1to Theorem4.2.8, we analyze the relationships betweenGMC criterion and other criteria for s-level case. Theorem4.2.1and Theorem4.2.2an-alyze the connection among the ACNP and resolution R. Theorem4.2.3gives the factthat the WLP is a function of the ACNP. Theorem4.2.4reveals the inherent property ofMA designs, which has the average minimum lower order confounding property. Theconnections between GMC and CE criteria are obtained in Theorem4.2.5and Theorem4.2.6. Moreover, Theorem4.2.7and Theorem4.2.8obtain the relationship of GMC andMEC criteria. By the corresponding complementary set, we get some results for s-levelGMC designs, such as Theorem4.1.3and Theorem4.3.4. To develop the s-level GMCtheory, we study sn mGMC designs for two cases:(1)(N/s+1)/2<n≤N/s and (2)N/s <n≤(N1)/(s1), where N=sn m. Main results are given in Theorem4.4.3and Theorem4.4.6.Forth part is formed by Chapter5. It first provides a useful theorem to constructthree-level GMC designs. By this theorem, we know that any design Sqr=Hq\Hr(r <q) is a GMC design. Furthermore, we construct GMC3n mdesign with these cases:n=(N-3~r)/2+i,i=1,2,3and obtain the corresponding values of#1C2,#2C2. Mainresults are Theorem5.2.1, Theorem5.3.1, Theorem5.3.3, Theorem5.3.4and Theorem5.3.5.Chapter6is fifth part. An experimenter who has prior knowledge may suggestthat main effects and2fi’s are potentially important and should be estimated. To solve this problem,[47] introduces CE criterion. However, CE criterion can not distinguishdesigns with the same numbers of clear main effects and2fi’s or no clear effects. Ac-cording to the confounding degree of components, a new ordering of three-level ACNPsolves this problem:0. It is called the aliased component-number pattern based on degrees (ACNP-D).Based on the ACNP-D we introduce the general minimum lower order confoundingbased on degrees (GMC-D) criterion. Main results are obtained in Theorem6.2.4andTheorem6.2.5. In the last section, we provide a catalogue of all27-run GMC-D de-signs,81-run GMC-D designs with n=5,...,20and243-run GMC-D designs for res-olution IV or higher, which contains the comparison of GMC-D and GMC designs.