Research On Split-plot Designs Under The General Minimum Lower-order Confounding Criterion

Factorial designs,as the main type of designs for analyzing and studying whether the experimental indicators(responses)are significantly affected by various factors,have received increasing attention in theories and applications.As a type of factorial designs,split-plot designs can simplify the experimental process and save costs through reducing the runs difficult to repeat and increasing the runs easy to repeat.It is widely used in complex designs involving more factors.A split-plot design divides the factors into two categories,the factors whose levels are difficult to change are called whole-plot(WP)factors,and those whose levels are relatively easy to change are called subplot(SP)factors.Due to the limitations of time,outlays and other practical conditions,the full factorial designs are often infeasible in practice.The experimenter can only choose a fraction from all level combinations to run an experiment.How to choose a "good" fraction has become an important issue.At present,the most popular optimality criteria include maximum resolution(MR)criterion,minimum aberration(MA)criterion,clear effect(CE)criterion,maximum estimation capacity(MEC)criterion and general minimum lower order confounding(GMC)criterion,etc.The GMC criterion was first proposed by Zhang et al.[93].It started from the alias sets of designs in order to find a design that minimizes the aberration between lower-order effects.When the experimenter has prior information about the importance of factors,the GMC criterion is applicable.At present,there is little literature on split-plot designs under the GMC criterion,and few people have studied the split-plot designs with prior information on the importance of wholeplot factors.This dissertation mainly studies the regular fractional factorial splitplot designs(FFSP)under the GMC criterion when the experimenter has prior information about the importance of whole-plot factors in the split-plot design.It is divided into seven chapters.Chapter 1 introduces the background,the existing optimality criteria and their main research results,especially the development of the GMC theory.Chapter 2 introduces the basic knowledge of regular fractional factorial designs(including the split-plot designs),reviews the aliased effect number pattern(AENP)and GMC criterion proposed for regular two-level fractional factorial designs,and establishes the relationships between the finite projection geometry and the splitplot designs.Chapter 3 discusses the two-level fractional factorial split-plot designs with more important WP factors when the experimenter is interested in the interaction effects between WP factors and SP factors.Firstly,we propose a new effect hierarchy principle of WP type(WP-EHP)of the split-plot designs.Based on the WP-EHP,the GMC criterion of regular two-level fractional factorial design is extended to the split-plot designs,and the corresponding aliased effect-number pattern and GMC criterion are proposed,denoted as WP-AENP and WP-GMC criterion,respectively.The optimal FFSP designs selected under the WP-GMC criterion are called WPGMC FFSP designs.Then the WP-GMC criterion is compared with the minimum aberration of type WP criterion proposed by Wang et al.[69]and an example is given to illustrate the application environment of the two criteria in detail.Finally,some WP-GMC FFSP designs are constructed.For k1=1 and k2=0,k1=0 and k2=1,k1=1 and k2=1,we construct the WP-GMC FFSP designs.Furthermore,using the structures of two-level GMC fractional factorial designs,we construct the WPGMC 2(n1+n2)-(k1+0)designs for 5N1/16+1≤n≤N1-1,where N1=2n1-k1.In addition,we also construct the WP-GMC 2(n1+n2)-(0+k2)designs for the parameters n1=1,2 and 5N/32+1 ≤n2≤N/4-1 or n1=3 and 5N/64+1 ≤n2 N/8-1,respectively,where N=2(n1+n2)-(k1+k2).In Chapter 4,the WP-EHP is modified and denoted as WP1-EHP,and WP1AENP is defined,and then WP1-GMC criterion for FFSP design is proposed.The relationships between the points in finite projection geometry and alias sets of twolevel FFSP designs are established and then the terms in WP1-AENP are represented by finite projection geometry.After that,the connection between the WP1-AENP of the original design and that of its complementary design is established.We take complementary sets as the main technical tool to derive explicit formulae connecting the leading terms for the WP1-AENP.These results are then applied to find optimal FFSP designs under the WP-GMC criterion.Finally,we give some necessary conditions for a design to be WP1-GMC design,and construct the WP1-GMC FFSP designs with f1+f2=4.Here f1 and f2 are the cardinalities of the complementary sets corresponding to whole-plot and subplot factors in the finite projection geometry,respectively.Also,the necessary and sufficient conditions for a design to be the WP1-GMC FFSP design for some special parameters are also given.In Chapter 5,we extend the WP1-GMC criterion of two-level FFSP designs proposed in Chapter 4 to three-level FFSP designs.The orthogonal components system of regular three-level fractional factorial designs is introduced.Based on the orthogonal components system,we extend the WP1-GMC criterion to the threelevel FFSP designs.Aliased component-number pattern(ACNP)for three-level FFSP design is proposed.It is denoted as WP1-ACNP,and an example is given to further illustrate the superiority of this criterion in selecting the designs.We also establish the relationship between the WP1-ACNP’s of the original design and its complementary design,and give explicit formulae of the leading terms for the WP1ACNP.In addition,we give a method to calculate the quantities of some terms in the WP1-ACNP.By this method,we can easily calculate these cumbersome quantitative relations.Finally,we give some necessary conditions for a three-level design to be a WP1-GMC FFSP design,and construct the three-level WP1-GMC FFSP designs with f1+f2 ≤3.Here f1 and f2 are the cardinalities of the complementary sets corresponding to whole-plot and subplot factors in the finite projection geometry,respectively.At the same time,the necessary and sufficient conditions for a threelevel design to be the WP1-GMC FFSP design for some special parameters are also given.In Chapter 6,we extend the WP1-GMC criterion to the s-level FFSP designs and derive explicit formulae of the leading terms of the WP1-ACNP.Define f1 an f2 to be the cardinalities of the complementary sets corresponding to the whole-plot and subplot factors in the finite projection geometry,and t1 and t2 to be the numbers of independent points in the sets corresponding to the whole-plot and subplot factors,respectively.When f1=(sw-1)/(s-1),f2=0,1,2 ≤w≤t1-1 or f1=0,f2=(sw-1)/(s-1),2≤w≤t2 or f1=(sw-1)/(s-1),f2=(sv-sw)/(s-1),1 ≤w<v<t,w<t1,we obtain the necessary and sufficient conditions for a design to be the WP1-GMC FFSP design.In Chapter 7,we first discuss the relationship between the WP-GMC criterion and the WP1-GMC criterion.When experimenters are only interested in the main effects and WP-type two-factor interaction,then the optimal designs under the two criteria are the same.In addition,it is found that all WP1-GMC FFSP designs constructed in Chapter 3 are also WP-GMC FFSP designs.Then,we summarize the work of this dissertation and prospect the future work.When f1+f2 ≤4 or f1=3,4,…,15 and f2=0,1,some WP1-GMC 2(n1+n2)-(k1+k2)designs are tabulated in Appendix.