Hongquan Xu

An Algorithm for Constructing Orthogonal and Nearly-Orthogonal Arrays With Mixed Levels and Small Runs

Orthogonal arrays are used widely in manufacturing and high-technology industries for quality and productivity improvement experiments. For reasons of run size economy or flexibility, nearly-orthogonal arrays are also used. The construction of orthogonal or nearly-orthogonal arrays can be quite challenging. Most existing methods are complex and produce limited types of arrays. This article describes a simple and effective algorithm for constructing mixed-level orthogonal and nearly-orthogonal arrays that can construct a variety of small-run designs with good statistical properties efficiently.

Construction of optimal multi-level supersaturated designs

A supersaturated design is a design whose run size is not large enough for estimating all the main effects. The goodness of multi-level supersaturated designs can be judged by the generalized minimum aberration criterion proposed by Xu and Wu (2001). Optimal supersaturated designs are shown to have a periodic property and general methods for constructing optimal multilevel supersaturated designs are proposed. Inspired by the Addelman-Kempthorne construction of orthogonal arrays, optimal multi-level supersaturated designs are given in an explicit form: columns are labeled with linear or quadratic polynomials and rows are points over a finite field. Additive characters are used to study the properties of resulting designs. Some small optimal supersaturated designs of 3, 4 and 5 levels are listed with their properties.

A system of high-dimensional, efficient, long-cycle and portable uniform random number generators

We propose a system of multiple recursive generators of modulus <i>p</i> and order <i>k</i> where all nonzero coefficients of the recurrence are equal. The advantage of this property is that a single multiplication is needed to compute the recurrence, so the generator would run faster than the general case. For <i>p</i> = 2³¹ − 1, the most popular modulus used, we provide tables of specific parameter values yielding maximum period for recurrence of order <i>k</i> = 102 and 120. For <i>p</i> = 2³¹ − 55719 and <i>k</i> = 1511, we have found generators with a period length approximately 10^14100.5.

Optimal projective three-level designs for factor screening and interaction detection

Orthogonal arrays (OAs) are widely used in industrial experiments for factor screening. Suppose that only a few of the factors in the experiments turn out to be important. An OA can be used not only for screening factors, but also for detecting interactions among a subset of active factors. In this article a set of optimality criteria is proposed to assess the performance of designs for factor screening, projection, and interaction detection, and a three-step approach is proposed to search for optimal designs. Combinatorial and algorithmic construction methods are proposed for generating new designs. Permutations of levels are used for improving the eligibility and estimation efficiency of the projected designs. The techniques are then applied to search for best three-level designs with 18 and 27 runs. Many new, efficient, and practically useful nonregular designs are found and their properties are discussed.

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.

Application of fractional factorial designs to study drug combinations

Herpes simplex virus type 1 (HSV-1) is known to cause diseases of various severities. There is increasing interest to find drug combinations to treat HSV-1 by reducing drug resistance and cytotoxicity. Drug combinations offer potentially higher efficacy and lower individual drug dosage. In this paper, we report a new application of fractional factorial designs to investigate a biological system with HSV-1 and six antiviral drugs, namely, interferon alpha, interferon beta, interferon gamma, ribavirin, acyclovir, and tumor necrosis factor alpha. We show how the sequential use of two-level and three-level fractional factorial designs can screen for important drugs and drug interactions, as well as determine potential optimal drug dosages through the use of contour plots. Our initial experiment using a two-level fractional factorial design suggests that there is model inadequacy and that drug dosages should be reduced. A follow-up experiment using a blocked three-level fractional factorial design indicates that tumor necrosis factor alpha has little effect and that HSV-1 infection can be suppressed effectively by using the right combination of the other five antiviral drugs. These observations have practical implications in the understanding of antiviral drug mechanism that can result in better design of antiviral drug therapy.

Moment Aberration Projection for Nonregular Fractional Factorial Designs

Nonregular fractional factorial designs, such as Plackett–Burman designs, are widely used in industrial experiments for run size economy and flexibility. A novel criterion, called moment aberration projection, is proposed to rank and classify nonregular designs. It measures the goodness of a design through moments of the number of coincidences between the rows of its projection designs. The new criterion is used to rank and classify designs of 16, 20, and 27 runs. Examples are given to illustrate that the ranking of designs is supported by other design criteria.

Use of Fractional Factorial Designs in Antiviral Drug Studies

Experimental design and analysis is an effective and commonly used tool in scientific investigations and industrial applications. Many successful applications have been reported in engineering domains, such as chemical engineering, electrical engineering, and mechanical engineering. However, few cases have been reported in biological research, particularly in virology study. Antiviral drug combinations are increasingly used to reduce possible drug-resistant viral mutant and reduce cytotoxicity. Drug combinations have often been reported to have higher efficacy and lower individual drug dosage. However, the combined antiviral drug effect is generally hard to assess. One important reason is due to the complex interactions between biological systems and drug molecules. We report a study using fractional factorial designs to investigate a biological system with Herpes simplex virus type 1 and five antiviral drugs. The experiment uses a novel composite design that consists of a 16-run fractional factorial design and an 18-run orthogonal array. The results indicate that two chemical drugs, Ribavirin and Acyclovir, are more effective than three Interferon drugs. Furthermore, significant interactions exist within the Interferon drug group and within the Ribavirin-Acyclovir chemical drug group, but the interactions between the Interferon group and the chemical group are not significant. These observations have major implications in the understanding of antiviral drug mechanism towards better design of combinatorial antiviral drug therapy. Copyright

Uniform fractional factorial designs

The minimum aberration criterion has been frequently used in the selection of fractional factorial designs with nominal factors. For designs with quantitative factors, however, level permutation of factors could alter their geometrical structures and statistical properties. In this paper uniformity is used to further distinguish fractional factorial designs, besides the minimum aberration criterion. We show that minimum aberration designs have low discrepancies on average. An efficient method for constructing uniform minimum aberration designs is proposed and optimal designs with 27 and 81 runs are obtained for practical use. These designs have good uniformity and are effective for studying quantitative factors.

Blocked regular fractional factorial designs with minimum aberration

This paper considers the construction of minimum aberra- tion (MA) blocked factorial designs. Based on coding theory, the con- cept of minimum moment aberration due to Xu (2003) for unblocked designs is extended to blocked designs. The coding theory approach studies designs in a row-wise fashion and therefore links blocked designs with nonregular and supersaturated designs. A lower bound on blocked wordlength pattern is established. It is shown that a blocked design has MA if it originates from an unblocked MA design and achieves the lower bound. It is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros. Sufficient conditions are given for constructing MA blocked designs from unblocked MA designs. The theory is then applied to construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all 81 runs with respect to four combined wordlength patterns.