Some Results on Minimum Aberration Mixed-level (2~r)2~n Factorial Designs

Mukerjee and Wu（2001） employed projective geometry theory to find the wordlength pattern of a regular mixed factorial design in terms of its complementary set, but only for the numbers of words of length 3 or 4. In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain some combinatorial identities that relate the wordlength pattern of a regular mixed-level （2^r）2^n factorial design to that of its consulting design. Consequently, a general rule for identifying minimum aberration （2^r）2^n factorial designs through their consulting designs is established. It is an improvement and generalization of the related result in Mukerjee and Wu（2001）.

ON WEAK MINIMUM ABERRATION AND NUMBER OF CLEAR TWO-FACTOR INTERACTION COMPONENTS IN s_Ⅳ～(m-p) DESIGNS

This article obtains some theoretical results on the number of clear two-factor interaction components and weak minimum aberration in an sm-pIVdesign, by considering the number of not clear two-factor interaction components of the design.

Weak minimum aberration and maximum number of clear two-factor interactions in 2_(IV)^(m-p)designs

Both the clear effects and minimum aberration criteria are the important rules for the design selection. In this paper, it is proved that some 2IVm-p designs have weak minimum aberration, by considering the number of clear two-factor interactions in the designs. And some conditions are provided, under which a 2IVm-p design can have the maximum number of clear two-factor interactions and weak minimum aberration at the same time. Some weak minimum aberration 2IVm-p designs are provided for illustrations and two non-isomorphic weak minimum aberration 2IV13-6 designs are constructed at the end of this paper.

Bayesian-inspired minimum contamination designs under a double-pair conditional effect model

In two-level fractional factorial designs,conditional main effects can provide insights by which to analyze factorial effects and facilitate the de-aliasing of fully aliased two-factor interactions.Con-ditional main effects are of particular interest in situations where some factors are nested within others.Most of the relevant literature has focused on the development of data analysis tools that use conditional main effects,while the issue of optimal factorial design for a given linear model involving conditional main effects has been largely overlooked.Mukerjee,Wu and Chang[Statist.Sinica 27(2017)997-1016]established a framework by which to optimize designs under a con-ditional effect model.Although theoretically sound,their results were limited to a single pair of conditional and conditioning factors.In this paper,we extend the applicability of their frame-work to double pairs of conditional and conditioning factors by providing the corresponding parameterization and effect hierarchy.We propose a minimum contamination-based criterion by which to evaluate designs and develop a complementary set theory to facilitate the search of minimum contamination designs.The catalogues of 16-and 32-run minimum contamination designs are provided.For five to twelve factors,we show that all 16-run minimum contamination designs under the conditional effect model are also minimum aberration according to Fries and Hunter[Technometrics 22(1980)601-608].

DESIGN EFFICIENCY FOR MINIMUM PROJECTION UNIFORMITY DESIGNS WITH TWO LEVELS

The objective of this paper is to study the issue of design efficiency for minimum projection uniformity designs. The results show that for orthogonal arrays with strength two, the minimum projection uniformity criterion is a good surrogate for the design efficiency criterion proposed by Cheng, Deng and Tang （2002）.

Matrix Image Method for Ranking Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. In this paper,we study matrix image theory and present a new method for distinguishing and assessing nonregular designs with complex alias structure, which works for all symmetrical and asymmetrical, regular and nonregular orthogonal arrays. Based on the matrix image theory, our proposed method captures orthogonality and projection properties. Empirical studies show that the proposed method has a more precise differentiation capacity when comparing with some other criteria.

Construction of Multi-level Space-filling Designs via Code Mappings

Space-filling designs are widely used in various fields because of their nice space-filling properties.Uniform designs are one of space-filling designs,which desires the experimental points to scatter uniformly over the experimental area.For practical need,the construction and their properties of nine-level uniform designs are discussed via two code mappings in this paper.Firstly,the algorithm of constructing nine-level uniform designs is presented from an initial three-level design by the Type-I code mapping and tripling technique.Secondly,the algorithm of constructing nine-level uniform designs is presented from a three-level base design by the Type-II code mapping and generalized orthogonal arrays.Moreover,relative properties are discussed based on the two code mappings.Finally,some numerical examples are given out for supporting our theoretical results.

Connection Among Some Optimal Criteria for Symmetrical Fractional Factorial Designs

A fundamental and practical question for fractional factorial designs is the issue of optimal factor assignment. Recently, some new criteria, such as generalized minimum aberration, WV-criterion, NB-criterion and uniformity criterion are proposed for comparing and selecting fractions. In this paper, we indicate that these criteria agree quite well for symmetrical fraction factorial designs.

Construction of 2_Ⅲ^(m-(m-k)) Designs with the Maximum Number of Clear Two-factor Interactions

It is useful to know the maximum number of clear two-factor interactions in a 2Ⅲ^[m-（m-k）] design. This paper provides a method to construct a 2Ⅲ^[m-（m-k）] design with the maximum number of clear two-factor interactions. And it is proved that the resulting designs have more dear two-factor interactions than those constructed by Tang et al. Moreover, the designs constructed are shown to have concise grid representations.

Lee discrepancy on asymmetrical factorials with two-and three-levels

Lee discrepancy has been employed to measure the uniformity of fractional factorials.In this paper,we further study the statistical justification of Lee discrepancy on asymmetrical factorials.We will give an expression of the Lee discrepancy of asymmetrical factorials with two-and three-levels in terms of quadric form,present a connection between Lee discrepancy,orthogonality and minimum moment aberration,and obtain a lower bound of Lee discrepancy of asymmetrical factorials with two-and three-levels.

Uniformity pattern and related criteria for two-level factorials

In this paper,the study of projection properties of two-level factorials in view of geometry is reported.The concept of uniformity pattern is defined.Based on this new concept,criteria of uniformity resolution and minimum projection uniformity are proposed for comparing two-level factorials.Relationship between minimum projection uniformity and other criteria such as minimum aberration,generalized minimum aberration and orthogonality is made explict.This close relationship raises the hope of improving the connection between uniform design theory and factorial design theory.Our results provide a justification of orthogonality,minimum aberration,and generalized minimum aberration from a natural geometrical interpretation.

Theory of optimal blocking for fractional factorial split-plot designs

The issue of optimal blocking for fractional factorial split-plot (FFSP) designs is considered under the two criteria of minimum aberration and maximum estimation capacity. The criteria of minimum secondary aberration (MSA) and maximum secondary estimation capacity (MSEC) are developed for discriminating among rival nonisomorphic blcoked FFSP designs. A general rule for identifying MSA or MSEC blocked FFSP designs through their blocked consulting designs is established.

A Note on Supersaturated Design

Supersaturated designs are common choice for screening experiments.This paper studies the properties of supersaturated designs.We give new lower bounds of E(s^(2))-criterion and E(f_(NOD))-criterion.Some linkages between the combined/double design and its original design are firstly provided,and the lower bounds of E(s^(2)) and E(f_(NOD)) for the combined/double design are also given.Furthermore,the close relationship between the minimum Lee-moment aberration criterion and the criteria for optimal supersaturated designs is revealed.These theoretical results can be used to construct or search for optimal supersaturated designs in practice.Numerical results are also provided,which lend further support to our theoretical findings.