On optimal two-level nonregular factorial split-plot designs

This article studies two-level nonregular factorial split-plot designs. The concepts of indicator function and aliasing are introduced to study such designs. The minimum G$G$-aberration criterion proposed by Deng and Tang (1999) [4] for two-level nonregular factorial designs is extended to the split-plot case. A method to construct the whole-plot and sub-plot parts is proposed for nonregular designs. Furthermore, the optimal split-plot schemes for 12$12$-, 16$16$-, 20$20$- and 24$24$-run two-level nonregular factorial designs are searched, and many such schemes are tabulated for practical use.     

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.     

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

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 split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.     

Forms of four-word indicator functions with implications to two-level factorial designs

Indicator functions are new tools to study fractional factorial designs. In this paper, we study indicator functions with four words and provide possible forms of the indicator functions and explain their implications to two-level factorial designs.     

Mixed two- and eight-level fractional factorial split-plot designs containing clear effects

Fractional factorial split-plot (FFSP) designs are useful in practical experiments. When the numbers of levels of the factors are not all equal in an experiment, mixed-level design is selected. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and eight-level factors to have various clear effects, including two types of main effects and three types of two-factor interaction components.     

Optimal balanced 27 fractional factorial designs of resolutionV,withN≦42

Summary In this paper, we present a class of fractional factorial designs of the 2⁷ series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 2⁷ factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).     

Detailed wordlength pattern of regular fractional factorial split-plot designs in terms of complementary sets

With reference to regular fractional factorial split-plot designs, we consider a detailed wordlength pattern taking due cognizance of the distinction between the whole-plot and sub-plot factors. A generalized version of the MacWilliams’ identity is employed to express the detailed wordlength pattern in terms of complementary sets. Several special features make this result intrinsically different from the corresponding one in classical fractional factorial designs where all factors have the same status. An application to robust parameter designs is indicated and examples given.     

Note on balanced fractional 2 m factorial designs of resolution 2l+1

It is shown that the characteristic roots of the information matrix of a balanced fractional 2 ^m factorial designT of resolution 2l+1 are the same as those of its complementary design . Necessary conditions for the existence of such a designT are also given.     

Optimal partially balanced fractional 2 m 1+m 2 factorial designs of resolution IV

Summary This paper investigates some partially balanced fractional 2 ^m ₁+m ₂ factorial designs of resolution IV derived from partially balanced arrays, which permit estimation of the general mean, all main effects, all two-factor interactions within each set of them _k factors (k=1, 2) and some linear combinations of the two-factor interactions between the sets of them _k ones. In addition, optimal designs with respect to the generalized trace criterion defined by Shirakura (1976,Ann. Statist.,4, 723–735) are presented for each pair (m ₁,m ₂) with 2≦m ₁≦m ₂ andm ₁+m ₂≦6, and for values ofN (the number of observations) in a reasonable range. Partially supported in part by Grants 56530009 (C) and 57530010 (C).     

Geometric isomorphism check for symmetric factorial designs

Two designs are geometrically isomorphic if one design can be obtained from the other by reordering the runs, relabeling the factors and/or reversing the level order of one or more factors. In this paper, some new necessary and sufficient conditions for identifying geometric isomorphism of symmetric designs with prime levels are provided. A new algorithm for checking geometric isomorphism is proposed and a searching result for geometrically non-isomorphic 3-level orthogonal arrays of 18 runs is presented.     

Kronecker factorial designs for multiway elimination of heterogeneity

This paper considers the application of Kronecker product for the construction of factorial designs, with orthogonal factorial structure, in a set-up for multiway elimination of heterogeneity. A technique involving the use of projection operators has been employed to show how a control can be achieved over the interaction efficiencies. A modification of the ordinary Kronecker product yielding smaller designs has also been considered. The results appear to have a fairly wide coverage.     

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

As a generalization of alias balanced designs due to Hedayat, Raktoe and Federer [5], we introduce the concept of alias partially balanced designs for fractional 2 ^m factorial designs of resolution 2l+1. All orthogonal arrays of strength 2l yield alias balanced designs. Some balanced arrays of strength 2l yield alias balanced and alias partially balanced designs. In particular, simple arrays which are a special case of balanced arrays yield alias partially balanced designs. At most 2 ^m −1 alias balanced (or alias partially balanced) designs are generated from an alias balanced (or alias partially balanced) design by level permutations. This implies that alias balanced or alias partially balanced designs need not be orthogonal arrays or balanced arrays of strength 2l.     

Two-level fractional factorial designs in blocks of size two for the orthogonal estimation of all main effects and two-factor interactions

In this paper we consider the problem of constructing two-level fractional factorial designs in blocks of size two that allow for the orthogonal estimation of all main effects and two-factor interactions (after adjusting for blocks). This problem has been considered in the literature, e.g., see Yang and Draper (2003), Wang (2004) and Kerr (2006). In this paper we give two systematic methods for the construction of such designs. The first construction method gives in many situations designs requiring fewer runs than those designs previously given whereas the second method gives a systematic method for constructing designs analogous to those illustrated in Yang and Draper (2003) by example.     

Algebraic generation of minimum size orthogonal fractional factorial designs: an approach based on integer linear programming

Generation of orthogonal fractional factorial designs (OFFDs) is an important and extensively studied subject in applied statistics. In this paper we show how searching for an OFFD that satisfies a set of constraints, expressed in terms of orthogonality between simple and interaction effects, is, in many applications, equivalent to solving an integer linear programming problem. We use a recent methodology, based on polynomial counting functions and strata, that represents OFFDs as the positive integer solutions of a system of linear equations. We use this system to set up an optimization problem where the cost function to be minimized is the size of the OFFD and the constraints are represented by the system itself. Finally we search for a solution using standard integer programming techniques. Some applications are also presented in the computational results section. It is worth noting that the methodology does not put any restriction either on the number of levels of each factor or on the orthogonality constraints and so it can be applied to a very wide range of designs, including mixed orthogonal arrays.     

Uniformity pattern and related criteria for two-level factorials

In this paper,the study of projection properties of two-level factorials in view ofgeometry is reported.The concept of uniformity pattern is defined.Based on this new con-cept,criteria of uniformity resolution and minimum projection uniformity are proposed forcomparing two-level factorials.Relationship between minimum projection uniformity andother criteria such as minimum aberration,generalized minimum aberration and orthogo-nality is made explict.This close relationship raises the hope of improving the connectionbetween uniform design theory and factorial design theory.Our results provide a justifi-cation of orthogonality,minimum aberration,and generalized minimum aberration from anatural geometrical interpretation.     

Adding more observations to saturated D‐optimal resolution III two‐level factorial designs

We consider the class of saturated main effect plans for the 2^k factorial. With these saturated designs, the overall mean and all main effects can be unbiasedly estimated provided that there are no interactions. However, there is no way to estimate the error variance with such designs. Because of this and other reasons, we like to add some additional runs to the set of (k+1) runs in the D‐optimal design in this class. Our goals here are: (1) to search for s additional runs so that the resulting design based on (k+s+1) runs yields a D‐optimal design in the class of augmented designs; (2) to classify all the runs into equivalent classes so that the runs in the same equivalent class give us the same value of the determinant of the information matrix. This allows us to trade runs for runs if this becomes necessary; (3) to obtain upper bounds for determinant of the information matrices of augmented designs. In this article we shall address these approaches and present some new results. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 51–77, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10026