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.     

On asymptotic tests in three-factor factorial designs with no replications

We revisit the problems of testing three-factor classification models with a single observation per cell. A common approach in analyzing such nonreplicated data is to omit the highest order interaction and regard it as error. This paper discusses the use of a multiplicative model (See and Smith, 1996 and 1998) which is applied on residuals in order to separate the variability due to three-factor interaction from what is counted as random error. In particular, to test the significance of the interaction term, we derived an approximated distribution of the likelihood ratio test statistic based on the quadrilinear model known as Tucker’s three-mode principal component model. The derivation utilizes the distribution of the eigenvalues of the Wishart matrix.     

Dispersion effects in unreplicated factorial designs

Methods for estimation of dispersion effects in two‐level unreplicated factorial designs are studied. The consequences of non‐constant variance are discussed. A natural assumption concerning the form of the covariance of location effects leads to a particular normal model. Some linear combinations of the response variables are constructed in order to make a simple structure for inference. The precision of point estimators of dispersion effects, where one is based on experiments with replicates, are compared. A numerical example is given as an illustration of a test. Finally, estimations in fractional designs are described and discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Optimum two level fractional factorial plans for model identification and discrimination

Model identification and discrimination are two major statistical challenges. In this paper we consider a set of models M_k for factorial experiments with the parameters representing the general mean, main effects, and only k out of all two-factor interactions. We consider the class D of all fractional factorial plans with the same number of runs having the ability to identify all the models in M_k, i.e., the full estimation capacity.The fractional factorial plans in D with the full estimation capacity for k?2 are able to discriminate between models in M_u for u?k^*, where k^*=(k/2) when k is even, k^*=((k-1)/2) when k is odd. We obtain fractional factorial plans in D satisfying the six optimality criterion functions AD, AT, AMCR, GD, GT, and GMCR for 2^m factorial experiments when m=4 and 5. Both single stage and multi-stage (hierarchical) designs are given. Some results on estimation capacity of a fractional factorial plan for identifying models in M_k are also given. Our designs D4.1 and D10 stand out in their performances relative to the designs given in Li and Nachtsheim [Model-robust factorial designs, Technometrics 42(4) (2000) 345-352.] for m=4 and 5 with respect to the criterion functions AD, AT, AMCR, GD, GT, and GMCR. Our design D4.2 stands out in its performance relative the Li-Nachtsheim design for m=4 with respect to the four criterion functions AT, AMCR, GT, and GMCR. However, the Li-Nachtsheim design for m=4 stands out in its performance relative to our design D4.2 with respect to the criterion functions AD and GD. Our design D14 does have the full estimation capacity for k=5 but the twelve run Li-Nachtsheim design does not have the full estimation capacity for k=5.     

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.     

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.     

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     

A note on construction of symmetrical PBIB designs

In this note initial block technique has been used for construction of some classes of 3-associate PBIB designs, known as rectangular designs.     

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 series of search designs for 2m factorial designs of resolution V which permit search of one or two unknown extra three-factor interactions

In the absence of four-factor and higher order interactions, we present a series of search designs for 2^m factorials (m6) which allow the search of at most k (=1,2) nonnegligible three-factor interactions, and the estimation of them along with the general mean, main effects and two-factor interactions. These designs are derived from balanced arrays of strength 6. In particular, the nonisomorphic weighted graphs with 4 vertices in which two distinct vertices are assigned with integer weight (13), are useful in obtaining search designs for k=2. Furthermore, it is shown that a search design obtained for each m6 is of the minimum number of treatments among balanced arrays of strenth 6. By modifying the results for m6, we also present a search design for m=5 and k=2.     

Some optimal criteria of model-robustness for two-level non-regular fractional factorial designs

We present some optimal criteria to evaluate model-robustness of non-regular two-level fractional factorial designs. Our method is based on minimizing the sum of squares of all the off-diagonal elements in the information matrix, and considering expectation under appropriate distribution functions for unknown contamination of the interaction effects. By considering uniform distributions on the symmetric support, our criteria can be expressed as linear combinations of B _s (d) characteristic, which is used to characterize the generalized minimum aberration. We give some empirical studies for 12-run non-regular designs to evaluate our method.     

An algorithmic approach to finding factorial designs with generalized minimum aberration

Factorial designs are arguably the most widely used designs in scientific investigations. Generalized minimum aberration (GMA) and uniformity are two important criteria for evaluating both regular and non-regular designs. The generation of GMA designs is a non-trivial problem due to the sequential optimization nature of the criterion. Based on an analytical expression between the generalized wordlength pattern and a uniformity measure, this paper converts the generation of GMA designs to a constrained optimization problem, and provides effective algorithms for solving this particular problem. Moreover, many new designs with GMA or near-GMA are reported, which are also (nearly) optimal under the uniformity measure.     

On intermediate factorial languages

We prove that factorial languages defined over non-trivial finite alphabets under some natural conditions have intermediate complexity functions, i.e., the number of words in such a language grows faster than any polynomial but slower than any exponential function.     

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.     

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.