For a fixed number of runs, when can designs have clear main effects or clear two-factor interactions (in brief, 2fi' s)? This paper gives the maximum value of n in sn m designs containing clear main effects or clear 2fi' s, where s is any prime or prime power. It is a generalization of the related results in Chen and Hedayat (J. Statist. Plann. Inference 75 (1998) 147-158) for two-level designs. It is further concluded that the weak minimum aberration designs have a maximum number of clear main effects for two-level designs. A collection of designs containing most clear main effects or clear 2fi' s for 16, 32, 27, and 81 runs is given.

Supersaturated design is a form of fractional factorial design and has recently received much interest because of its potential in factor screening experiments. This paper mainly concerns the existing criteria for mixed-level designs, i.e. the w2ðDÞ criterion (for design D) proposed by Yamada and Matsui (J. Statist. Plann. Inference 104 (2002) 459), the EðfNODÞ criterion proposed by Fang et al. (Metrika 58 (2003b) 279) and the minimum moment aberration (MMA) proposed by Xu (Statist. Sinica 13 (2003) 691). A lower bound of χ2 (D) is obtained along with the sufficient and necessary condition for achieving it. The connections between χ2 (D) and other criteria are discussed. Especially, it is shown that χ2 (D) and the second power moment χ2 (D) are in fact equivalent to each other and hence the two criteria, χ2 (D) and MMA, share the same optimal supersaturated designs, which can be constructed from saturated orthogonal arrays and other supersaturated designs.

Based on the work of Wang and Fang (1990a, b), this paper extends the def-inition of discrepancy of a set of points with respect to a distribution. To construct multivariate uniform designs, two transformations, discrepancy-preserving transfor-mation and density-preserving transformation, and a method of separating variables are introduced. Also, for some special domains useful in statistics, some related transformations are given. In particular, as an application of the method of sepa-rating variables, we give an approach of constructing a uniform design in the Stiefel manifold and outline its applications in projection pursuit methods.

This paper extends the 4m2n minimum aberration designs (MA designs) of Wu and Zhang (1993) to the case of (S2)Sn−k, where S is any prime or prime power. Some basic properties of (S2)Sn−k MA designs, including the relations with Sn−k MA designs, are discussed. The (9)3n−k MA designs with 27 runs and (16)4n−k MA designs with 64 runs are tabulated, and some (4)2n−k MA designs are constructed using the above relations.