A note on minimum aberration and clear criteria

Minimum aberration and clear criteria are two important rules for selecting optimal fractional factorial designs, in both unblocked and blocked cases. In this paper, we first show that under some given conditions, a blocked design DB=(D,B) having blocked minimum aberration is equivalent to D having minimum aberration. Let m=n/4+1 and n=2q. From the results of Tang et al. [Bounds on the maximum number of clear two-factor interactions for 2m-p designs of resolution III and IV. Canad. J. Statist. 30 (2002) 127-136] and Wu and Wu [Clear two-factor interactions and minimum aberration. Ann. Statist. 30 (2002) 1496-1511], we know that the maximum number of clear two-factor interactions (2FIs) in 2IVm-(m-q) designs is n/2-1. Here it is proved that the maximum number of clear 2FIs in 2m-(m-q) designs in 2l blocks, denoted by 2IV-m-(m-q):2l, is also n/2-1 when q-l⩾2. Furthermore, it is shown that any 2IVm-(m-q) design that contains the maximum number of clear 2FIs is not a minimum aberration design, and this conclusion also holds when the design is a 2IV-m-(m-q):2l design with q-l⩾2.