Three-level Blocked Regular Designs With General Minimum Lower Order Confounding

Factorial design is an important type of experimental design.In practice,because of the experimental cost and time,the experimenters usually only carry out part of the runs.This kind of design is called fractional factorial design.There are some common fractional factorial designs,such as blocked design,regular design,nonregular design,split-plot design,compromise design,etcIt is a concern problem that choosing the most efficient and advantageous design from many designs for the experimenters.Effect hierarchy principle(EHP)shows that lower order effects are more important than higher order effects and same order ef-fects are equally important.Basing on EHP,statisticians has established many optimal criteria for measuring fractional factorial designs from different perspectives,such as maximum resolution(MR)criteria,minimum aberration(MA)criteria,maximum es-timation capacity(MEC)criteria,and clear effects(CE)criteria and so on.There are a lot of research achievements on the merits of these criteria and the construction of corresponding optimal designs.The optimal designs based on different optimal criteria are often different and have their own characteristics.It is hoped that more general cri-teria can be obtained so that lower order effects of the corresponding optimal designs can be confounded least.Thus a new optimal criterion emerges as the times requireIn 2008,for the simplest two-level regular design,Zhang et al.[71]first proposed a new pattern called aliased effect number pattern(AENP),which described the con-founding information between different order effects in detail.Meanwhile AENP was arranged the elements in the order of the effect from low to high and severity from light to heavy,so that it reflected effect hierarchy principle(EHP)more comprehen-sively.Based on AENP,general minimum lower order confounding(GMC)criteria was given.The optimal design obtained from this criteria was called GMC design.It was also proved that the previous optimal criteria can be expressed by AENP,which indicated AENP possessing strong inclusiveness and wide applicability.A series of theories derived from GMC criterion were called GMC theory.In the past ten years,GMC theory has developed rapidly and produced a lot of research fruits.For example,the construction of two-level GMC designs,the definition and construction of two-level blocked GMC designs and three-level GMC designs,the property of s-level and mixed level GMC designs and so on.Moreover,the idea and method of GMC has been respectively applied to the study of split-plot designs,nonregular designs,robust parameter designs and compromise designs.On the basis of the above conclusions of literature,in this paper,we extend GMC criterion to three-level blocked regular designs,and propose the corresponding optimal criterion and designs.This paper is divided into five parts:In the first part,we main-ly introduce the development process of experimental design,some related concepts of factorial design,especially the meaning and application of regular designs and the main research results of existing optimal criteria.In the second part,we review the background and significance of GMC criterion for two-level regular design and the relationship between AENP and other existing criteria,and summarize a large number of research results and development prospects of GMC theory.In the third part,we present the common analytical tools for three-level regular design:orthogonal compo-nent system and linear-quadratic system.Based on the orthogonal component system,we introduce the basic concepts of three-level blocked regular design.The fourth part is the focus of this paper.As to three-level blocked regular designs,we propose a new type of blocked aliased component number pattern,abbreviated as B-ACNP,which is used to reveal the confounding information of factor interaction components.In view of B-ACNP,we establish Bl-GMC criterion in three level case,and the correspond-ing optimal design is called a B1-GMC design.Using B-ACNP as a tool,we analyze the relationship between B1-GMC criterion and other main existing criteria,includ-ing MA-Type,CE and B-GMC criterion and deeply explore the powerful function and widely application of B-ACNP.In the fifth part,we focus on the structural features of three-level blocked regular designs,and propose an effective method to obtain B1-GMC designs quickly.Using this method,we list all the B1-GMC 3n-m:3P designs and B-ACNP s with N = 27,81.,243,n = 4.,5,...,10,p= 1,2.,3.Furthermore,we establish the concept of complementary design for three-level blocked regular designs,and study the relationship between original designs and complementary designs.Last-ly,we obtain some B1-GMC 3n-m:3p designs and B-ACNP's in special case.