In factor screening, often only a few factors among a large pool of potential factors are active. Under such assumption of effect sparsity, in choosing a design for factor screening, it is important to consider projections of the design onto small subsets of factors. Cheng showed that as long as the run size of a two-level orthogonal array of strength two is not a multiple of 8, its projection onto any four factors allows the estimation of all the main effects and two-factor interactions when the higher-order interactions are negligible. This result applies, for example, to all Plackett-Burman designs whose run sizes are not multiples of 8. It is shown here that the same hidden projection property also holds for Paley designs of sizes greater than 8, even when their run sizes are multiples of 8. A key result is that such designs do not have defining words of length three or four. Applications of this result to the construction of $E(s^2)$-optimal supersaturated designs are also discussed. In particular, certain designs constructed by using Wu's method are shown to be $E(s^2)$-optimal. The article concludes with some three-level designs with good projection properties.
"Hidden projection properties of some nonregular fractional factorial designs and their applications." Ann. Statist. 31 (3) 1012 - 1026, June 2003. https://doi.org/10.1214/aos/1056562472