Some results on two-level regular designs with multi block variables containing clear effects

Abstract

In this paper, the idea of clear effects is extended to the case of blocked regular $2^{n{-}m}$ designs with multi block variables. Some necessary and sufficient conditions on the existence of clear treatment main effects and clear treatment two-factor interactions (2fi’s) in $2^{n{-}m}:2^l$ designs with resolution III, IV$^-$ and IV are obtained, where the notation $2^{n-m}:2^l$ means that there are n treatment factors, m treatment defining words and l block variables. We discuss how to find $2^{n-m}:2^l$ designs with the maximum number of clear 2fi’s with resolution III, IV$^-$ or IV, and present some 16-, 32-, 64-run designs containing the maximum number of clear 2fi’s in tables.

Appendix

In the following tables, $\mathbf {1,\ldots ,k}$ are the independent columns defined in (1) and the other columns are component-wise products of the k independent columns. Each design contains the independent columns $\mathbf {1,\ldots ,k}$ and m additional columns of $H_k$, where $k=n-m$, and only the additional m columns of the $2^{n-m}$ design $D_0$ are shown in the tables under the column heading Additional columns. BC represents the number of clear 2fi’s in the $2^{n-m}:2^l$ designs, and for all the different n, k and l, $*=2^k-1-n-l$.

Tables 1 and 2 present the 16- and 32-run $2_\mathrm{III}^{n-m}:2^l$ designs, containing the maximum number of clear 2fi’s, respectively. In Table 1, take the $2^{6-2}:2^l$ design $D=(D_0,B)$ for example, where $D_0=(\mathbf {1},\mathbf {2},\mathbf {3},\mathbf {4},\mathbf {12},\mathbf {34})$. For any $1\le l\le 8$, under the column heading Block variables, $b_1,\ldots ,b_l\in H_4\backslash D_0$ means that we can select any l columns from $H_4\backslash D_0$ as block variables B of the $2^{6-2}:2^l$ design. In Table 2, take the $2^{7-2}:2^l$ design $D=(D_0,B)$ for example, where $D_0=(\mathbf {1},\mathbf {2},\mathbf {3},\mathbf {4},\mathbf {5},\mathbf {12},\mathbf {1345})$. For $1\le l\le 6$, any l columns of $\{\mathbf {234},\mathbf {235},\mathbf {245},\mathbf {1234},\mathbf {1235},\mathbf {1245}\}$ under the column heading Block variables make the block variables B; for $7\le l\le 23$, under the column heading Block variables, other$b_i\in H_5\backslash D_0, 7\le i\le l$ means that we take all the 6 columns of $\{\mathbf {234},\mathbf {235},\mathbf {245},\mathbf {1234},\mathbf {1235},\mathbf {1245}\}$ and any other $l-6$ columns from $H_5\backslash D_0$ as block variables B. In the two tables, the $2^{8-4}:2^l$ design and the $2^{10-5}:2^l$ design are similar to the $2^{6-2}:2^l$ design. The other blocked designs are similar to the $2^{7-2}:2^l$ design.

In Tables 3 and 4, under the column heading $n-m$, $n-m.j$ denotes the jth $2^{n-m}$ design in the catalogue. For each given $\{n, m, l\}$, the first $2^{n-m}:2^l$ design $D=(D_0,B)$ is the blocked design containing the maximum number of clear 2fi’s with resolution at least IV$^-$, where $D_0$ is determined by $n-m.1$ under the column heading $n-m$ and B is determined by the corresponding l block variables under the column heading Block variables. These blocked designs are constructed by the method in Section 3. By the algorithm of Section 4, we obtain the $2_\mathrm{IV}^{n-m}:2^l$ design containing the maximum number of clear 2fi’s. For each $2_\mathrm{IV}^{n-m}:2^l$ design, all the columns of the $\varphi$-class are listed under the column heading Block variables. For any given l, we can choose any l columns from $\varphi$-class to form the block variables. Take the $2^{7-2}:2^l$ design $D=(D_0,B)$ for example, where $D_0=(\mathbf {1},\mathbf {2},\mathbf {3},\mathbf {4},\mathbf {5},\mathbf {123},\mathbf {1245})$ and the $\varphi$-class of $D_0$ is $\{\mathbf {134},\mathbf {234},\mathbf {135},\mathbf {235},\mathbf {1345},\mathbf {2345}\}$. When $1\le l\le 6$, we can choose any l columns from the $\varphi$-class as block variables; thus we obtain the $2^{7-2}:2^l$ design $D=(D_0,B)$, which contains 15 clear 2fi’s, the maximum number of clear 2fi’s among all the $2_\mathrm{IV}^{7-2}:2^l$ designs. When $7\le l\le 9$, $b_7,\ldots ,b_l\in UC$-class means that we choose all the 6 columns of the $\varphi$-class and any $l-6$ columns from the UC-classs as block variables; such a $2^{7-2}:2^l$ design has resolution IV$^-$ and 15 clear 2fi’s. When $10\le l\le 23$, other $b_i\in C$-class, $10\le i\le l$ means that we choose all the 9 columns of the $\varphi$-class and the UC-class and any $l-9$ columns from the C-class as block variables; such a $2^{7-2}:2^l$ design has resolution IV$^-$ and $2^k-1-n-l=24-l$ clear 2fi’s.

Table 1 MaxC2fi’s for 16-run $2_\mathrm{III}^{n-m}:2^l$ designs

Table 2 MaxC2fi’s for 32-run $2_\mathrm{III}^{n-m}:2^l$ designs

Table 3 MaxC2fi’s for 32-run $2^{n-m}:2^l$ designs with resolution IV$^-$ or IV

Table 4 MaxC2fi’s for 64-run $2^{n-m}:2^l$ designs with resolution IV$^-$ or IV