### Some results on 4 m 2 n designs with clear two-factor interaction components

ShengLi Zhao, RunChu Zhang, MinQian Liu
2008 Science in China Series A: Mathematics
Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi's) in 2 n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem
more » ... r the asymmetrical case. It improves the construction method of Tang et al. for 2 n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear twofactor interaction components (2fic's) in 4 m 2 n designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic's in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4 m 2 n designs under the clear effects criterion. ZHAO ShengLi et al. by replacing three 2-level factors {a i1 , a i2 , a i3 } with a 4-level factor A i , where a i1 a i2 a i3 = I, i = 1, . . . , m and I is the column with all entries zero. Such a design is determined by B = {a 11 , a 12 , a 13 , . . . , a m1 , a m2 , a m3 , b 1 , . . . , b n }, where a i1 a i2 a i3 = I, i = 1, . . . , m, in the following sections. We call a i1j1 the main-effect component of A i1 , and a i1j1 a i2j2 (or a i1j1 b l ) the two-factor interaction component (2fic) of A i1 and A i2 (or A i1 and b l ), where i 1 , i 2 = 1, . . . , m, i 1 = i 2 , j 1 , j 2 = 1, 2, 3, l = 1, . . . , n. For convenience, we call both the main effects of 2-level factors and the main-effect components of 4-level factors the main-effect components. For the same reason, the two-factor interactions (2fi's) of two 2-level factors, the 2fic's of two 4-level factors, and 2fic's of a 2-level factor and a 4-level factor are all called 2fic's. When the experimenter's knowledge is diffuse, a reasonable assumption people can make is the effect hierarchical assumption. Under such circumstances, resolution in [4] and minimum aberration in [5] are the most often used criteria for selecting good designs. Extending them to the mixed-level case, [6] gave the definitions of resolution and minimum aberration criteria for selecting good 4 m 2 n designs. For m = 1, suppose that a 1 , a 2 , a 3 , b 1 , . . . , b n are columns chosen from the 2 k − 1 columns of a saturated design with 2 k runs such that a 1 a 2 a 3 = I. A 4 1 2 n design can be obtained by replacing {a 1 , a 2 , a 3 } with a 4-level factor. It is easy to see that there are two types of defining contrasts for this design. The first involves only the b j 's, which is called type 0. The second involves one of the a i 's and some of the b j 's, which is called type 1. For a 4 1 2 n design D, let W i0 (D) and W i1 (D) be the numbers of type 0 and type 1 words of length i in the defining contrasts of D, respectively. The resolution of D is defined to be the smallest i such that W ij (D) is positive for at least one j. For m = 2, the resolution for 4 2 2 n designs is defined similarly as that of 4 1 2 n designs. Furthermore, [7] deliberated a method for constructing this class of asymmetric minimum aberration designs through symmetric minimum aberration ones, [8] obtained two types of minimum aberration designs with mixed levels in terms of complementary sets, and [9] improved the results in [8] .