On the optimality of orthogonal and balanced arrays with $N\equiv 0$ $(\text {mod}$ 9) runs

Abstract

We investigate the role of orthogonal arrays (OAs) and balanced arrays (BAs) in both full and fractional factorial designs with N runs and m three-level quantitative factors. Firstly, due to the non-existence of the OA(18, 8, 3, 2), we find and construct a BA(18, 8, 3, 2) that represents the E-, A-, D-optimal design with $N=18$ runs and $m=8$ three-level factors under the main-effect model. Also, we are interested in comparing the OA(N, m, 3, 2)s with the BA(N, m, 3, 2)s, when they represent designs with $N\equiv 0$ $(\text {mod}$ 9) runs and m three-level factors with respect to the E-, A-, D-criteria under the second-order model. We provide a generalized definition of balanced arrays. Moreover, we find and construct the OA(N, m, 3, 2)s and the BA(N, m, 3, 2)s that represent the E-, A-, D-optimal designs with $N=9$, 18, 27, 36 runs and $m=2$ three-level factors under the second-order model. Furthermore, it is shown that the BA(18, m, 3, 2)s, $m=3$, 4 and a BA(27, 3, 3, 2) perform better than the OA(18, m, 3, 2)s, $m=3$, 4 and the OA(27, 3, 3, 3), respectively, when they represent the corresponding designs with respect to the E-, A-, D-criteria under the second-order model.

Appendices

Appendix 1

Algorithm 1

Step 1: The first factor of the first block is (0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2) and the first factor of the second block is (0 0 1 1 2 2 0 0 1 1 2 2 0 0 1 1 2 2).

Step 2: Write all factors having in the first 6 places 0 0 1 1 2 2, in the next 6 elements 0 0 1 1 2 2 and in the last 6 places 0 0 1 1 2 2. This is set S and has $\left( \left( {\begin{array}{c}6\\ 2\end{array}}\right) \left( {\begin{array}{c}6-2\\ 2\end{array}}\right) \left( {\begin{array}{c}6-2-2\\ 2\end{array}}\right) \right) ^{3}=90^{3}=729000$ factors.

Step 3: Find from the set S the factors that satisfy, with the first factor of the second block, the relations $n _{12}^{pq} =2$, p, $q=0$, 1, 2. This is set G.

Step 4: From the set G form groups of factors that satisfy between them the relations $n_{11}^{pq}=2$, p, $q=0$, 1, 2. If a group has 5 factors then we have the first block.

Step 5: Find from the set S the factors that satisfy, with the last 5 factors of the first block, the relations $n_{12}^{pq}=2$, p, $q=0$, 1, 2 and, with the first factor from the second block, the relations $n_{22}^{11}=\textit{n}_{22}^{02}=\textit{n}_{22}^{20}=0$, $n_{22}^{00}=n_{22}^{22} =n_{22}^{01}=n_{22}^{10}=n_{22}^{12}=n_{22}^{21}=3$. If there is one factor, then we have the BA(18, 8, 3, 2), $s_1=6$, $s_2=2$.

Algorithm 2

Step 1: Choose a value of N and find all the $1\times 9$ vectors $\mathbf{x} =(x_1, \ldots , x_9)$ of integers such that $\sum _{i=1}^{9}x_i=N$.

Step 2: For each vector $\mathbf{x}$ construct the vectors $\mathbf{v} =(v_1, v_2, v_3)$ and $\mathbf{r} =(r_1, r_2, r_3)$, such that $v_1=\sum _{i=1}^{3}x_i$, $v_2=\sum _{i=4}^{6}x_i$, $v_3=\sum _{i=7}^{9}x_i$ and $r_1=\sum _{i=1,4,7}x_i$, $r_2=\sum _{i=2,5,8}x_i$, $r_3=\sum _{i=3,6,9}x_i$.

Step 3: Keep the vectors x for which $v_i\ge 1$ and $r_i\ge 1$ for any $i=1$, 2, 3, and at least two elements of both v and r are equal. For these vectors x it holds that $\mathbf{x} =(n^{00}, n^{01}, n^{02}, n^{10}, n^{11}, n^{12}, n^{20}, n^{21}, n^{22})$ or $\mathbf{x} =(n_{12}^{00}, n_{12}^{01}, n_{12}^{02}, n_{12}^{10}, n_{12}^{11}, n_{12}^{12}, n_{12}^{20}, n_{12}^{21}, n_{12}^{22})$.

Step 4: For each vector of Step 3 construct the corresponding OA(N, 2, 3, 2) or BA(N, 2, 3, 2), according to Definition 3, and obtain the array (orthogonal or balanced) that represents the E-, A-, D-optimal design with N runs and $m=2$ three-level factors under model (2).

Appendix 2

Let $n_{ijk}^{pqr}$ be the number of times factors $F_i$, $F_j$, $F_k$ appear together at levels p, q, r, respectively, and $n_{ijko}^{pqrt}$ be the number of times factors $F_i$, $F_j$, $F_k$, $F_o$ appear together at levels p, q, r, t, respectively, in the N runs, where p, q, r, $t=0$, 1, 2, $i\ne j\ne k\ne o=1,\ldots ,m$.

Let B be a $m\times \left( {\begin{array}{c}m\\ 2\end{array}}\right)$ matrix with elements for $i\ne j<k=1,\ldots ,m$ equal to

$$\begin{aligned} \sum _{s=1}^{N}g_{si}g_{sj}g_{sk}=\frac{(-n_{ijk}^{000}+n_{ijk}^{002}+n_{ijk}^{020}-n_{ijk}^{022}+n_{ijk}^{200}-n_{ijk}^{202}-n_{ijk}^{220}+n_{ijk}^{222})}{2\sqrt{2}} \end{aligned}$$

(4)

and for $i\ne j=1,\ldots ,m$ equal to

$$\begin{aligned}&\sum _{s=1}^{N}g_{si}^2g_{sj}=\frac{(-n_{ij}^{00}+n_{ij}^{02}-n_{ij}^{20}+n_{ij}^{22})}{2\sqrt{2}},\\&\sum _{s=1}^{N}g_{sj}^2g_{si}=\frac{(-n_{ij}^{00}+n_{ij}^{20}-n_{ij}^{02}+n_{ij}^{22})}{2\sqrt{2}}, \end{aligned}$$

$\mathbf{B} ^{(1)}$ be a $m\times \left( {\begin{array}{c}m\\ 2\end{array}}\right)$ matrix with elements for $i=1,\ldots ,m$, $j<k=1,\ldots ,m$ equal to

$$\begin{aligned} \sum _{s=1}^{N}g_{si}\sum _{s=1}^{N}g_{sj}g_{sk}=\frac{(n_{i}^{2}-n_{i}^{0})(n_{jk}^{00}-n_{jk}^{02}-n_{jk}^{20}+n_{jk}^{22})}{2\sqrt{2}}, \end{aligned}$$

C be a $m\times \left( {\begin{array}{c}m\\ 2\end{array}}\right)$ matrix with elements for $i\ne j<k=1,\ldots ,m$ equal to

$$\begin{aligned} 3\sum _{s=1}^{N}h_{si}g_{sj}g_{sk}=\frac{\sqrt{3}(n_{ijk}^{000}-n_{ijk}^{002}-n_{ijk}^{020}+n_{ijk}^{022}+n_{ijk}^{200}-n_{ijk}^{202}-n_{ijk}^{220}+n_{ijk}^{222})}{2\sqrt{2}} \end{aligned}$$

(5)

and for $i\ne j=1,\ldots ,m$ equal to

$$\begin{aligned} 3\sum _{s=1}^{N}h_{si}g_{si}g_{sj}=\frac{\sqrt{3}(n_{ij}^{00}-n_{ij}^{02}-n_{ij}^{20}+n_{ij}^{22})}{2\sqrt{2}}, \end{aligned}$$

$\mathbf{C} ^{(1)}$ be a $m\times \left( {\begin{array}{c}m\\ 2\end{array}}\right)$ matrix with elements for $i=1,\ldots ,m$, $j<k=1,\ldots ,m$ equal to

$$\begin{aligned} 3\sum _{s=1}^{N}h_{si}\sum _{s=1}^{N}g_{sj}g_{sk}=\frac{\sqrt{3}(N-n_{i}^{1})(n_{jk}^{00}-n_{jk}^{02}-n_{jk}^{20}+n_{jk}^{22})}{2\sqrt{2}}, \end{aligned}$$

D be a $\left( {\begin{array}{c}m\\ 2\end{array}}\right) \times \left( {\begin{array}{c}m\\ 2\end{array}}\right)$ symmetric matrix with diagonal elements for $i\ne j=1,\ldots ,m$ equal to

$$\begin{aligned} \sum _{s=1}^{N}g_{si}^2g_{sj}^2=\frac{(n_{ij}^{00}+n_{ij}^{02}+n_{ij}^{20}+n_{ij}^{22})}{4}, \end{aligned}$$

off-diagonal elements for $i\ne j\ne k=1,\ldots ,m$ equal to

$$\begin{aligned} \sum _{s=1}^{N}g_{si}^2g_{sj}g_{sk}=\frac{(n_{ijk}^{000}-n_{ijk}^{002}-n_{ijk}^{020}+n_{ijk}^{022}+n_{ijk}^{200}-n_{ijk}^{202}-n_{ijk}^{220}+n_{ijk}^{222})}{4} \end{aligned}$$

(6)

and for $i\ne j\ne k\ne o=1,\ldots ,m$ equal to

$$\begin{aligned} \sum _{s=1}^{N}g_{si}g_{sj}g_{sk}g_{so}=\frac{\begin{aligned} \left( n_{ijko}^{0000}-n_{ijko}^{0002}-n_{ijko}^{0020}+n_{ijko}^{0022}-n_{ijko}^{0200}+n_{ijko}^{0202}+n_{ijko}^{0220}-n_{ijko}^{0222}\right. \\ \left. -n_{ijko}^{2000}+n_{ijko}^{2002}+n_{ijko}^{2020}-n_{ijko}^{2022}+n_{ijko}^{2200}-n_{ijko}^{2202}-n_{ijko}^{2220}+n_{ijko}^{2222}\right) \end{aligned}}{4} \end{aligned}$$

(7)

and $\mathbf{D} ^{(1)}$ be a $\left( {\begin{array}{c}m\\ 2\end{array}}\right) \times \left( {\begin{array}{c}m\\ 2\end{array}}\right)$ symmetric matrix with elements for $i<j=1,\ldots ,m$, $k<o=1,\ldots ,m$ equal to

$$\begin{aligned} \sum _{s=1}^{N}g_{si}g_{sj}\sum _{s=1}^{N}g_{sk}g_{so}=\frac{(n_{ij}^{00}-n_{ij}^{02}-n_{ij}^{20}+n_{ij}^{22})(n_{ko}^{00}-n_{ko}^{02}-n_{ko}^{20}+n_{ko}^{22})}{4}. \end{aligned}$$

For the information matrix M in (3) holds

$$\begin{aligned} \mathbf{X} ^{T}{} \mathbf{X} =\left( \begin{array}{ccc} \mathbf{L} &{}\quad \mathbf{A} &{}\quad \mathbf{B} \\ \mathbf{A} ^T &{}\quad \mathbf{Q} &{}\quad \mathbf{C} \\ \mathbf{B} ^{T} &{}\quad \mathbf{C} ^{T} &{}\quad \mathbf{D} \end{array}\right) \end{aligned}$$

and

$$\begin{aligned} \mathbf{X} ^{T}{} \mathbf{J} _{N}{} \mathbf{X} =\left( \begin{array}{ccc} \mathbf{L} ^{(1)} &{}\quad \mathbf{A} ^{(1)} &{}\quad \mathbf{B} ^{(1)} \\ \left( \mathbf{A} ^{(1)}\right) ^T &{}\quad \mathbf{Q} ^{(1)} &{}\quad \mathbf{C} ^{(1)} \\ \left( \mathbf{B} ^{(1)}\right) ^{T} &{}\quad \left( \mathbf{C} ^{(1)}\right) ^{T} &{}\quad \mathbf{D} ^{(1)} \end{array}\right) \end{aligned}$$