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.
Analysis of Factorial Designs with the Consideration of Interactions via the Stepwise Response Refinement Screener (SRRS)
Factorial designs are widely used experimental plans for identifying important factors in screening studies where many factors are involved. In many practical situations where some interactions are significant, the design is supersaturated and the experimental analysis becomes infeasible due to the lack of degree of freedoms [9]. Recently, a new analysis procedure called the Stepwise Response Refinement Screener (SRRS) method is proposed to screen important effects for supersaturated designs [6]. This paper extends this method to the twolevel factorial designs. The applications to several real-life examples suggest that the SRRS method is able to retrieve similar results as the existing methods do. Simulation studies show that compared to existing methods in the literature, the SRRS method performs well in terms of the true model identification rate and the average model size.
Some Projection Properties of Orthogonal Arrays
The definition of an orthogonal array imposes an important geometric property: the projection of an OA(λ2 t ,2 k ,t), a λ2 t -run orthogonal array with k two-level factors and strength t, onto any t factors consists of λ copies of the complete 2 t factorial. In this article, projections of an OA(N,2 k ,t) onto t + 1 and t + 2 factors are considered. The projection onto any t + 1 factors must be one of three types: one or more copies of the complete 2 t+1 factorial, one or more copies of a half-replicate of 2 t+1 or a combination of both. It is also shown that for k ≥ t+2, only when N is a multiple of 2 t+1 can the projection onto some t+1 factors be copies of a half-replicate of 2 t+1 . Therefore, if N is not a multiple of 2 t+1 , then the projection of an OA(N, 2 k , t) with k ≥ t + 2 onto any t + 1 factors must contain at least one complete 2 t+1 factorial. Some properties of projections onto t + 2 factors are established and are applied to show that if N is not a multiple of 8, then for any OA(N, 2 k , 2) with k ≥ 4, the projection onto any four factors has the property that all the main effects and two-factor interactions of these four factors are estimable when the higher-order interactions are negligible.