Introduction

Abstract

Experimental design is an important branch of statistics. This chapter concerns with experiments in various fields and indicates their importance, purpose, type of experiments, statistical models, and related designs. Section 1.1 demonstrates several experiments for different purposes and characteristics. This section also presents discussion on two popular types of experiments: (1) physical experiments and (2) computer experiments. Basic terminologies used in experimental design are introduced in Sect. 1.2. Various kinds of experimental designs based on different kinds of statistical models are introduced in Sect. 1.3. They involve the factorial design under ANOVA model, the optimum design under linear regression model, and the uniform design under model uncertainty (or nonparametric regression model). There are many criteria for assessing fractional factorial designs, among which the minimum aberration criterion has been widely used. Section 1.4 gives a brief introduction to this concept and its extensions. Section 1.5 shows the implementation of the uniform design for a multifactor experiment. Readers are recommended to read this chapter carefully so that they can understand the methodology of uniform design and will easily follow the remaining contents of the book.

Exercises

1.1

Compare the physical experiment and the computer experiment and list their difference.

1.2

What are metamodels in computer experiments? Give some requirements for metamodels.

1.3

In an experiment, there are many variables in general. Give the difference between variable and factor in experimental design. Give some examples for quantitative factors and quantitative factors. What is the difference between environmental variables and nuisance variables?

1.4

Consider model for the one-factor experiment and its statistical model (1.3.1)

$$\begin{aligned} y_{ij}=\mu _j+\varepsilon _{ij},~~ j=1, \ldots , q, i=1, \ldots , n_j, \end{aligned}$$

where $\varepsilon _{ij}$ are i.i.d. distributed as $N(0,\sigma ^2)$. Let $n=n_1+\cdots +n_q$ be the number of runs. This model can be expressed as a linear model ${\varvec{y}}={\varvec{X}}{\varvec{\beta }}+{\varvec{\epsilon }}$.

(1) Write down ${\varvec{y}},{\varvec{X}},{\varvec{\beta }}$ in details.

(2) Find ${\varvec{X}}'{\varvec{X}}$ and ${\varvec{X}}'{\varvec{y}}$.

(3) Find the distributions of $y_{ij}$ and ${\varvec{y}}$.

1.5

Model in the previous exercise can be expressed as (1.3.2), i.e.,

$$\begin{aligned} y_{ij}=\mu + \alpha _j+\varepsilon _{ij},~~ j=1, \ldots , q, i=1, \ldots , n_j. \end{aligned}$$

(1) Express this model as a linear model ${\varvec{y}}={\varvec{X}}{\varvec{\beta }}+{\varvec{\epsilon }}$, and give ${\varvec{y}},{\varvec{X}},{\varvec{\beta }}$ in details.

(2) Let $SS_E=\sum _{i=1}^q\sum _{j=1}^{r}(y_{ij}-\bar{y}_i)^2$ be the error sum of squares, where $n_1=\cdots =n_q=r$. Prove $E[SS_E]=q(r-1)\sigma ^2$.

1.6

For a two-factor experiment, its ANOVA model is given by (1.3.3), i.e.,

$$\begin{aligned} y_{ijk}= & {} \mu +\alpha _i+\beta _j+ (\alpha \beta )_{ij}+\varepsilon _{ijk},\\&i=1,\ldots ,q_1;\ j=1,\ldots ,q_2;k=1,\ldots ,r, \end{aligned}$$

with constraints

$$\begin{aligned}&\alpha _1+\cdots + \alpha _{q_1}=0; \beta _1+\cdots +\beta _{q_2}=0;\\&\sum _{i=1}^{q_1}(\alpha \beta )_{ij}=0,\ j=1,\ldots ,q_2; \sum _{j=1}^{q_2}(\alpha \beta )_{ij}=0, i=1,\ldots ,q_1. \end{aligned}$$

Prove

(1) When $q_1=q_2=2$, there is one linearly independent interaction among $\{(\alpha \beta )_{11}, (\alpha \beta )_{12},(\alpha \beta )_{21},(\alpha \beta )_{22}\}$.

(2) There are $(q_1-1)(q_2-2)$ linearly independent interactions among $\{(\alpha \beta )_{ij}, i=1,2; j=1,\ldots ,q_i\}$.

1.7

Consider the following model of a three-factor experiment

$$ y_{ijk} = \mu + \alpha _i + \beta _j + \gamma _k + (\beta \gamma )_{jk} + \varepsilon _{ijk},\ i=1, 2, 3, \ j=1, 2, 3,\ k=1, 2, $$

where A, B, C are the factors,

$y_{ijk}$ is the response at $A=A_i, B=B_j$ and $C=C_k$,

$\mu$ is the overall mean,

$\alpha _i$ is the main effect of the factor A at the level $A_i$,

$\beta _j$ is the main effect of the factor B at the level $B_j$,

$\gamma _k$ is the main effect of the factor C at the level $C_k$,

$(\beta \gamma )_{jk}$ is the interaction of B and C at $B_j$ and $C_k$,

$\varepsilon _{ijk}$ is random error at the experiment with $A=A_i$, $B=B_j$ and $C=C_k$, and $\varepsilon _{ijk}\sim N(0,\sigma ^2)$.

Answer the following questions:

(a) Give constrains on $\alpha _i, \beta _j, \gamma _k,$ and $(\beta \gamma )_{jk}$. How many independent parameters among $\alpha _i, \beta _j, \gamma _k,$ and $(\beta \gamma )_{jk}$?

(b) Express this model as of the form ${\varvec{y}}={\varvec{X}}{\varvec{\beta }}+{\varvec{\epsilon }}$ where ${\varvec{\beta }}$ is formed by independent parameters discussed in question a) and indicate ${\varvec{y}}, {\varvec{X}}, {\varvec{\beta }},$ and ${\varvec{\epsilon }}$.

(c) Give the degrees of freedom for the sum of squares: $SS_A, SS_B, SS_C, SS_{B\times C}, SS_E,$ and $SS_T$.

(d) Give formulas for $SS_A, SS_B, SS_C, SS_{B\times C}, SS_E,$ and $SS_T$.

1.8

The concept “orthogonality” has been appeared in different fields. Answer the following questions:

(1) Give definition for two line segments in $R^d$ be orthogonal.

(2) Give definition for two planes be orthogonal.

(3) Give definition for two linear spaces be orthogonal.

(4) Let ${\varvec{X}}$ be the matrix of the orthogonal design table $L_9(3^4)$. Denote by $\mathcal {L}_i, i=1,2,3,4$ the linear subspace generated by the ith column of ${\varvec{X}}$; denote by $\mathcal {L}_{ij}, 1\leqslant i<j\leqslant 4$ the linear subspace generated by the ith and jth columns of ${\varvec{X}}$. Prove that $\mathcal {L}_i$ and $\mathcal {L}_j$ are orthogonal if $i\ne j$ and $\mathcal {L}_{12}$ and $\mathcal {L}_{34}$ are orthogonal.

1.9

Give the word-length pattern and resolution for the following designs:

(a) A design $2^{6-2}$ with defining relations $I=ABCE=BCDF$;

(b) A design $2^{7-2}$ with defining relations $I=ABCDF=ABDEG$;

(c) A design $2^{7-3}$ with defining relations $I=ABCE=BCDF=ACDG$.

1.10

Answer the following questions:

• 1. There is a command “hadamard” to generate a Hadamard matrix of order n. Use this command to find a Hadamard matrix of order 8 by which we can obtain $L_8(2^7)$.

• 2. If ${\varvec{H}}$ is a Hadamard matrix, then $-{\varvec{H}}$ is a Hadamard matrix.

• 3. If ${\varvec{H}}$ is a Hadamard matrix, let

  $$\begin{aligned}{\varvec{V}}=\left[ \begin{array}{rr} {\varvec{H}}&{} -{\varvec{H}}\\ {\varvec{H}}&{} {\varvec{H}}\\ \end{array} \right] . \end{aligned}$$

  Prove that ${\varvec{V}}$ is a Hadamard matrix.

1.11

Calculate the Hamming distances between any two different runs designed by $L_9(3^4)$ below. Give your finding and conjecture.

No	1	2	3	4
1	1	1	1	1
2	1	2	2	2
3	1	3	3	3
4	2	1	2	3
5	2	2	3	1
6	2	3	1	2
7	3	1	3	2
8	3	2	1	3
9	3	3	2	1

1.12

In economics, the so-called Lorenz curve is a graphical representation of the distribution of income. Give a review on the Lorenz curve and relationship between the Lorenz curve and the majorization theory.