Frederick Kin Hing Phoa

Recent developments in 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. The traditional analysis focuses on main e�ffects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.

The use of nonregular fractional factorial designs in combination toxicity studies

When there is interest to study n chemicals using x dose levels each, factorial designs that require x(n) treatment groups have been put forward as one of the valuable statistical approaches for hazard assessment of chemical mixtures. Exemplary applications and cost-efficiency comparisons of full factorial designs and regular fractional factorial designs in toxicity studies can be found in Nesnow et al. [Nesnow, S., Ross, J.A., Stoner, G.D., Mass, M.J., 1995. Mechanistic linkage between DNA adducts, mutations in oncogenes and tumorigenesis of carcinogenic environmental polycyclic aromatic hydrocarbons in strain A/J mice. Toxicology, 105, 403-413] , Narotsky et al. [Narotsky, M.G., Weller, E.A., Chinchilli, V.M., Kevlock, R.J., 1995. Non-additive developmental toxicity in mixtures of trichloroethylene, di(2-ethylhexyl)phthalate and heptachlor in a 5x5x5 design. Fundamental and Applied Toxicology, 27, 203-216], and Groten et al. [Groten, J.P., Schoen, E.D., Feron, V.J., 1996. Use of factorial designs in combination toxicity studies. Food and Chemical Toxicology, 34, 1083-1089], Groten et al. [Groten, J.P., Schoen, E.D., Kuper, C.F., van Bladeren, P.J., Van Zorge, J.A., Feron, V.J., 1997. Subacute toxicity of a mixture of nine chemicals in rats: detecting interactive effects with a fractionated two-level factorial design. Fundamental and Applied Toxicology, 36, 13-29]. We introduce nonregular fractional factorial designs and show their benefits using two studies reported in Groten et al. (1996). Study 1 shows nonregular designs can provide the same amount of information using 75% of the experimental costs required in a regular design. Study 2 demonstrates nonregular designs can additionally estimate some partially aliased effects, which cannot be done using regular designs. We also provide a statistical method to evaluate the quality of an assumption made by experts in Study 2 of Groten et al. (1996).

Quarter-Fraction Factorial Designs Constructed via Quaternary Codes

The research of developing a general methodology for the construction of good nonregular designs has been very active in the last decade. Recent research by Xu and Wong (2007) suggested a new class of nonregular designs constructed from quaternary codes. This paper explores the properties and uses of quaternary codes towards the construction of quarter-fraction nonregular designs. Some theoretical results are obtained regarding the aliasing structure of such designs. Optimal designs are constructed under the maximum resolution, minimum aberration and maximum projectivity criteria. These designs often have larger generalized resolution and larger projectivity than regular designs of the same size. It is further shown that some of these designs have generalized minimum aberration and maximum projectivity among all possible designs.

Optimizing Two-Level Supersaturated Designs Using Swarm Intelligence Techniques

Supersaturated designs (SSDs) are often used to reduce the number of experimental runs in screening experiments with a large number of factors. As more factors are used in the study, the search for an optimal SSD becomes increasingly challenging because of the large number of feasible selection of factor level settings. This article tackles this discrete optimization problem via an algorithm based on swarm intelligence. Using the commonly used E(s2) criterion as an illustrative example, we propose an algorithm to find E(s2)-optimal SSDs by showing that they attain the theoretical lower bounds found in previous literature. We show that our algorithm consistently produces SSDs that are at least as efficient as those from the traditional CP exchange method in terms of computational effort, frequency of finding the E(s2)-optimal SSD, and also has good potential for finding D3-, D4-, and D5-optimal SSDs. Supplementary materials for this article are available online.

A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper shows how a trigonometric approach can facilitate a systematic understanding of such QC designs and lead to new theoretical results covering hitherto unexplored situations. We focus attention on one-eighth and one-sixteenth fractions of two-level factorials and show that optimal QC designs often have larger generalized resolution and projectivity than comparable regular designs. Moreover, some of these designs are found to have maximum projectivity among all designs.

A code arithmetic approach for quaternary code designs and its application to (1/64)th-fractions

The study of good nonregular fractional factorial designs has received significant attention over the last two decades. Recent research indicates that designs constructed from quaternary codes (QC) are very promising in this regard. The present paper aims at exploring the fundamental structure and developing a theory to characterize the wordlengths and aliasing indexes for a general

A Swarm Intelligence Based (SIB) method for optimization in designs of experiments

(1/4)^p

Feature selection in regression tasks using conditional mutual information

th-fraction QC design. Then the theory is applied to (1/64)th-fraction QC designs. Examples are given, indicating that there exist some QC designs that have better design properties, and are thus more cost-efficient, than the regular fractional factorial designs of the same size. In addition, a result about the periodic structure of (1/64)th-fraction QC designs regarding resolution is stated.

Power-law distributions of attributes in community detection

Natural heuristic methods, like the particle swarm optimization and many others, enjoy fast convergence towards optimal solution via inter-particle communications. Many applications of such methods are applied to the optimization in engineering, but only a few to the optimization in statistics. It is especially difficult to implement in the optimization problems of experimental designs as the search space is mostly discrete, while most natural heuristic methods are limited to searching continuous domains. This paper introduces a new natural heuristic method called Swarm Intelligence Based method for optimizing problem with a discrete domain. It includes two new operations, MIX and MOVE, for combining two particles and selecting the best particle respectively. This method is ready for the search of both continuous and discrete domains, and its global best particle is guaranteed to monotonically move towards the optimum. Several demonstrations on the optimization of experimental designs are given at the end of this paper.

Using Swarm Intelligence to Search for Circulant Partial Hadamard Matrices

This paper presents a supervised feature selection method applied to regression problems. The selection method uses a Dissimilarity matrix originally developed for classification problems, whose applicability is extended here to regression and built using the conditional mutual information between features with respect to a continuous relevant variable that represents the regression function. Applying an agglomerative hierarchical clustering technique, the algorithm selects a subset of the original set of features. The proposed technique is compared with other three methods. Experiments on four data-sets of different nature are presented to show the importance of the features selected from the point of view of the regression estimation error (using Support Vector Regression) considering the Root Mean Squared Error (RMSE).