Connections Between Uniformity and Other Design Criteria

Abstract

Most experimental designs, such as simple random design, random block design, Latin square design, fractional factorial design (FFD, for short), optimal design, and robust design are concerned with randomness, balance between factors and levels of each factor, orthogonality, efficiency, and robustness. From the previous chapters, we see that the uniformity has played an important role in the evaluation and construction of uniform designs. In this chapter, we shall show that uniformity is intimately connected with many other design criteria.

Exercises

6.1

Prove that the following two designs, $D_1$ and $D_2$, are isomorphic

$$ D_1= \left[ \begin{array}{rrr} 1 &{}\quad 1 &{}\quad 1 \\ 1 &{}\quad -1 &{}\quad 1\\ 1 &{}\quad 1 &{}\quad -1\\ -1 &{}\quad -1 &{}\quad -1\\ -1 &{}\quad 1 &{}\quad -1\\ -1 &{}\quad -1 &{}\quad 1 \end{array} \right] \, D_2= \left[ \begin{array}{rrr} -1 &{}\quad -1 &{}\quad -1 \\ 1 &{}\quad -1 &{}\quad -1\\ -1 &{}\quad 1 &{}\quad -1\\ 1 &{}\quad 1 &{}\quad 1\\ -1 &{}\quad 1 &{}\quad 1\\ 1 &{}\quad -1 &{}\quad 1 \end{array} \right] . $$

Indicate that these designs belong to:

(i) U-type design; (ii) orthogonal design; (iii) fractional factorial design.

6.2

Denote by $\mathcal {U}_9(3^4)$ all the possible orthogonal designs $L_9(3^4)$ with levels 1, 2 and 3. Answer the following questions:

(1) Show that all these designs in $\mathcal {U}_9(3^4)$ form only one isomorphic group.

(2) Calculate WD, CD and MD for all designs in $\mathcal {U}_9(3^4)$. Give your conclusion.

(3) Table 6.4 list two designs $L_9(3^4)$, where $\varvec{U}_2$ was obtained by minimizing CD on $\mathcal {U}_9(3^4)$. The $\varvec{U}_2$ is called uniformly orthogonal design under CD. Find uniformly orthogonal design under MD/WD.

(4) Calculate the discrete discrepancy for all designs in $\mathcal {U}_9(3^4)$. Give your conclusion.

(5) Give a discussion on two concepts: the uniformly orthogonal design and uniform minimum aberration design.

(6) Calculate the projection discrepancy pattern defined in Definition 6.5.1.

6.3

Calculate the uniform pattern (refer to Definition 6.5.2) for all subdesigns $L_8(2^5)$ of $L_8(2^7)$ in Table 1.3.2.

6.4

Table 6.5 gives a uniform minimum aberration design $UL_{27}(3^{13})$ under MD. This design involves several uniform minimum aberration subdesigns $UL_{27}(3^{s})$ for $s<13$ under MD. Give two such subdesigns for $s=6$ and $s=10$ with detailed calculation, respectively.

6.5

There are five non-isomorphic $L_{16}(2^{15})$ designs. This chapter lists two of them. Give other three from the literature.

6.6

There are two non-isomorphic $L_{27}(3^{13})$ designs (refer to Fang and Zhang 2004). Apply Algorithm 6.1.2 to detect their non-isomorphism.

6.7

Table 6.2 list four orthogonal designs $L_{18}(3^7)$. Ma and Fang (2001) pointed out that there are at least three non-isomorphic $L_{18}(3^{7})$ designs. Furthermore, Evangelaras et al. (2007) confirmed that there exist exactly three non-isomorphic three-level orthogonal arrays with 18 runs and 7 columns. They obtained several minimum aberration subdesigns $L_{18}(3^s), s\leqslant 7.$ List these minimum aberration subdesigns as many as possible.