Fractional factorial designs are widely used in practice and typically chosen according to the minimum aberration criterion. A sequential algorithm is developed for constructing efficient fractional factorial designs. A construction procedure is proposed that only allows a design to be constructed from its minimum aberration pro jection in the sequential build-up process. To efficiently identify nonisomorphic designs, designs are divided into different categories according to their moment pro jection patterns. A fast isomorphism check procedure is developed by matching the factors using their delete-one-factor pro jections. A method is proposed for constructing minimum aberration designs using only a partial catalog of some good designs. Minimum aberration designs are constructed for 128 runs up to 64 factors, 256 runs up to 28 factors, and 512, 1024, 2048, and 4096 runs up to 23 or 24 factors. Furthermore, this algorithm is used to completely enumerate all 128-run designs of resolution 4 up to 30 factors, all 256-run designs of resolution 4 up to 17 factors, all 512-run designs of resolution 5, all 1024-run designs of resolution 6, and all 2048- and 4096-run designs of resolution 7.

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.

When there is interest to study n chemicals using x dose levels each, factorial designs that require xn treatment groups have been put forward as one of the valuable statistical approaches for hazard assessment of chemical mixtures. Exemplary applications and cost-eciency comparisons of full factorial designs and regular fractional factorial designs in toxicity studies can be found in Nesnow et al. (1995), Narotsky et al. (1995), and Groten et al. (1996,1997). We introduce nonregular fractional factorial designs and show their bene�ts 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 e�ects, 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).

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.

When there is interest to study n chemicals using x dose levels each, factorial designs that require xn 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. (1995), Narotsky et al. (1995), and Groten et al. (1996,1997). 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).

Fractional factorial designs are widely used experimental plans for identifying important factors in screening studies where many factors are involved. Traditionally, Plackett-Burman (PB) and related designs are employed for in such studies because of their cost efficiencies. The caveat with the use of PB designs is that interactions among factors are implicitly assumed to be non-existent. However, there are many practical situations where some interactions are significant and ignoring them can result in wrong statistical inferences, including biased estimates, missing out on important factors and detection of spurious factors. We reanalyze data for three chemical experiments using the Hamada and Wu’s method and show that we are able to identify significant interactions in each of these chemical experiments and improve the overall fit of the model. In addition, we analyze the data using a Bayesian approach that confirms our findings. In both approaches, graphical tools are employed along with easily available software for analysis.