Ching-Shui Cheng

Minimum aberration and model robustness for two‐level fractional factorial designs

The performance of minimum aberration two‐level fractional factorial designs is studied under two criteria of model robustness. Simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived. It is also shown that a minimum aberration design of resolution III or higher maximizes the number of two‐factor interactions which are not aliases of main effects and, subject to that condition, minimizes the sum of squares of the sizes of alias sets of two‐factor interactions. This roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform. It follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here. Examples are given to show that minimum aberration designs are indeed highly efficient.

The Construction of Trend-Free Run Orders of Two-Level Factorial Designs

Abstract In experimental situations where a factorial design with all factors occurring at two levels is to be run in a time sequence, the usual advice given to the experimenter is that the order of runs should be randomized before the experiment is performed; however, randomization may lead to an undesirable run order. For example, in a factory experiment, there may be a certain learning process that occurs over time as a result of making level changes in the factors being studied, or there may be equipment wear-out. In either case the observations obtained will be affected by uncontrollable variables that are highly correlated with the time or position in which they occur, and randomization may lead to a run order in which the estimates of factor effects are adversely effected by the presence of the trend. Joiner and Campbell (1976) gave some more specific examples as to when trend effects can occur in sequential experiments. Therefore, it is important to consider systematic designs in which the estimat...

Maximizing the total number of spanning trees in a graph: Two related problems in graph theory and optimum design theory

The purpose of this paper is to discuss two related problems in graph theory and optimum design theory: maximizing the number of spanning trees in a graph and finding a D -optimum incomplete block design. A regular complete multipartite graph is shown to have the maximum number of spanning trees among all the simple graphs with the same numbers of vertices and edges.

Doubling and projection : A method of constructing two-level designs of resolution IV

Given a two-level regular fractional factorial design of resolution IV, the method of doubling produces another design of resolution IV which doubles both the run size and the number of factors of the initial design. On the other hand, the projection of a design of resolution IV onto a subset of factors is of resolution IV or higher. Recent work in the literature of projective geometry essentially determines the structures of all regular designs of resolution IV with n ≥ N/4 + 1 in terms of doubling and projection, where N is the run size and n is the number of factors. These results imply that, for instance, all regular designs of resolution IV with 5N/ 16 < n ≤ N /2 must be projections of the regular design of resolution IV with N/2 factors. We show that, for 9N/32 < n ≤ 5 N/16, all minimum aberration designs are projections of the design with 5N/16 factors which is constructed by repeatedly doubling the 2 5-1 design defined by I = ABCDE. To prove this result, we also derive some properties of doubling, including an identity that relates the wordlength pattern of a design to that of its double and a result that does the same for the alias patterns of two-factor interactions.

New E(s2)-Optimal Supersaturated Designs Constructed From Incomplete Block Designs

We present a method for constructing two-level supersaturated designs (SSDs) from incomplete block designs. A lower bound of E(s2) that also covers the case of odd run sizes is given. This bound is attained by SSDs constructed from balanced incomplete block designs. We study SSDs that can be constructed from regular graph designs when balanced incomplete block designs do not exist. A computer search is conducted to find SSDs with 5 ≤ n ≤ 50 and n ≤ m ≤ 2n that can be constructed from regular graph designs, where m is the number of factors and n is the run size. Many SSDs derived from regular graph designs are optimal. The best E(s2)-optimal SSDs with respect to additional optimality criteria are tabulated. Some notes on the construction of saturated designs also are given.

Optimal regression designs in the presence of random block effects

Abstract A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.

A general theory of minimum aberration and its applications

Minimum aberration is an increasingly popular criterion for comparing and assessing fractional factorial designs, and few would question its importance and usefulness nowadays. In the past decade or so, a great deal of work has been done on minimum aberration and its various extensions. This paper develops a general theory of minimum aberration based on a sound statistical principle. Our theory provides a unified framework for minimum aberration and further extends the existing work in the area. More importantly, the theory offers a systematic method that enables experimenters to derive their own aberration criteria. Our general theory also brings together two seemingly separate research areas: one on minimum aberration designs and the other on designs with requirement sets. To facilitate the design construction, we develop a complementary design theory for quite a general class of aberration criteria. As an immediate application, we present some construction results on a weak version of this class of criteria.

A Complementary Design Theory for Doubling

Chen and Cheng (2006a) discussed the method of doubling for constructing two-level fractional factorial designs. They showed that for 9N/32<= n<= 5N/16, all minimum aberration designs with N runs and n factors are projections of the maximal design with 5N/16 factors which is constructed by repeatedly doubling the 2^{5-1} design defined by I=ABCDE. This paper develops a general complementary design theory for doubling. For any design obtained by repeated doubling, general identities are established to link the wordlength patterns of each pair of complementary projection designs. A rule is developed for choosing minimum aberration projection designs from the maximal design with 5N/16 factors. It is further shown that for 17N/64<=n<=5N/16, all minimum aberration designs with N runs and n factors are projections of the maximal design with N runs and 5N/16 factors.

Optimal Step-Type Designs for Comparing Test Treatments with a Control

Abstract The problem of obtaining A-optimal designs for comparing v test treatments with a control in b blocks of size k each is considered. A condition on the parameters (u, b, k) is identified for which optimal step-type designs can be obtained. Families of such designs are given. Methods of searching for highly efficient designs are proposed for situations in which it is difficult to determine an A-optimal design. Under the usual additive homoscedastic model, an A-optimal design minimizes the average variance of the least squares estimators of the control-test treatment comparisons. Majumdar and Notz (1983) gave a method for finding A-optimal designs. Their optimal designs can basically be of two types, using the terminology of Hedayat and Majumdar (1984): rectangular (R), in which every block has the same number of replications of the control, and step (5), in which some blocks contain the control t times and the others t + 1 times. Optimal R-type designs were studied by Hedayat and Majumdar (1985). F...

OPTIMAL WEIGHING DESIGNS

A technique is developed for finding optimum designs for weighing n objects in N weighings