现代因子设计理论与统计分析

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"Modern Theory of Factorial Design (Springer Series in Statistics)" 是由 Rahul Mukerjee 和 C.F.J. Wu 合著的一本书，属于 Springer 出版社的统计学系列。该书的ISBN号是0387319913。这本书深入探讨了现代因子设计的理论，因子设计是实验设计中的一个重要分支，它涉及到在科学研究和工程领域如何有效地安排实验，以优化对多个因子（或变量）影响的探究。因子设计通常用于确定哪些因子对结果有显著影响，以及这些因子之间的交互作用。这种方法在质量控制、化学实验、生物科学、工业工程等众多领域有着广泛的应用。 Rahul Mukerjee 和 C.F.J. Wu 作为作者，可能在这本书中详细阐述了因子设计的基本概念，包括全因子设计、部分因子设计、混合设计、响应曲面设计等，并可能涵盖了正交设计、平衡设计和最优设计等高级主题。他们可能还讨论了如何通过统计分析来解释和解读实验数据，以及如何利用统计模型进行预测和优化。 Springer Series in Statistics 系列的顾问团队包括 P.Bickel、P.Diggle、S.Fienberg、U.Gather、I.Olkin、S.Zeger 等知名统计学家，这表明该书的内容具有高度的专业性和权威性。系列中的其他书籍，如 "Statistical Decision Theory and Bayesian Analysis" 和 "Modern Multidimensional Scaling: Theory and Applications"，进一步反映了这个系列在统计学领域的广泛覆盖和深度。书中可能涵盖了以下关键知识点： 1. 因子设计的基本原理：如何构建和分析不同类型的因子设计。 2. 正交性和平衡性的概念：确保实验条件间的独立性和减少误差。 3. 因子交互作用的识别：如何检测和评估因子间的相互影响。 4. 统计模型的构建：基于因子设计的数据，建立合适的统计模型。 5. 优化方法：如何通过设计找到最佳操作条件。 6. 蒙特卡洛方法：可能用于模拟和验证因子设计的效果。 7. 模型验证和诊断：确保模型的适用性和可靠性。 8. 实际应用案例：可能包括来自不同领域的实际实验设计实例。 "Modern Theory of Factorial Design" 对于那些希望深入了解实验设计及其统计分析的科研人员、工程师和统计学学生来说，是一本宝贵的参考书。它不仅提供了理论基础，还可能包含了实用的工具和技巧，帮助读者在实际工作中解决复杂的问题。

1.3 What Is in the Book? 5

tables can be viewed as the “more applied” part of the book. They also form

a data base for possible use by design researchers.

Chapter 4 extends the above work to s-level factorials, s being a prime

or prime power. It ﬁrst discusses the case of three-level designs and points

out the extra complication in the mathematical treatment and interpretation

of factorial eﬀects while passing from two to three or higher levels. Barring

this complication, the contents and ﬂow of this chapter are similar to those of

Chapter 3. By employing coding-theoretic tools, a major theorem is developed

relating MA designs to their complementary designs. This theorem is one of

the deepest and most signiﬁcant among the recent work in design theory.

Three-level design tables with 27 and 81 runs are given and their practical

use is discussed. These tables, like those in Chapter 3, include MA designs

and other eﬃcient ones.

As the MA criterion became popular, an attempt was made to provide fur-

ther statistical justiﬁcation for it via an alternative criterion that relates more

directly to the estimability property under various model assumptions. This

led to the idea of maximum estimation capacity, which is the focus of Chapter

5. Both two- and general s-level factorials are treated. The main technical tool

is again the use of complementary designs, now called complementary sets.

The designs considered in Chapters 3 to 5 concern symmetrical factorials,

i.e., all the factors have the same number of levels. There are, however, prac-

tical situations in which the intrinsic nature of the factors does not allow such

symmetry. For example, all factors may have two levels except for a quali-

tative factor like part supply with four suppliers. The focus of Chapter 6 is

on asymmetrical (or mixed-level) designs, i.e., those with factors at diﬀerent

numbers of levels. The simplest and most common example is given by the

mixed two- and four-level designs, which are constructed by the method of

replacement. MA designs with one or two four-level factors and a general num-

ber of two-level factors are presented. Designs with s-level and s

-level factors

are then considered. Projective geometry is used for describing such designs

and their MA property is investigated using the technique of complementary

sets. Tables of MA designs are given.

The next three chapters can be viewed as forming the last unit of the book.

They deal with designs that involve two distinct types of factors. In contrast

to Chapter 6, where the distinction among factors is in terms of the number

of factor levels, the distinction here is of a more subtle nature. Additional

care is needed and tools have to be developed to address the new features of

asymmetry. Chapter 7 covers full and fractional factorial designs arranged in

blocks. These designs incorporate a block structure in addition to the factorial

structure for the treatment factors. An extension of the MA criterion is given

for full factorials arranged in blocks. The problem of blocking in fractional

factorial designs is, however, complicated by the existence of two structures:

one deﬁning the fraction for the treatment factors and the other deﬁning

the blocking scheme. Projective geometry is used to describe the necessary

mathematical formulation, and various optimality criteria, motivated by MA,

6 1 Introduction and Overview

are discussed. Tables of block designs with desirable properties are given for

16, 32, 64, and 128 runs in the two-level case, and 27 and 81 runs in the

three-level case.

Chapter 8 is on fractional factorial split-plot designs, where the whole

plot and subplot factors cannot be treated symmetrically because of diﬀerent

error structures. This necessitates supplementing the original MA criterion by

another follow-up criterion. As in the previous chapters, projective geometry

and complementary sets are used as the main technical tools. This chapter

concludes with tables of optimal designs.

Robust parameter design is a statistical/engineering methodology for vari-

ation reduction that works by choosing appropriate settings of the control fac-

tors so as to make the system insensitive to hard-to-control noise variations.

Parameter design experiments are commonly used in quality improvement,

and Chapter 9 gives an account of their planning aspects. Two experimental

formats, cross arrays and single arrays, are considered. A fundamental result

on the estimability property of cross arrays is given. Here the lack of sym-

metry is due to the diﬀerent roles played by the control and noise factors in

the choice of designs and modeling strategies. A new eﬀect ordering principle

is developed to address such asymmetry, taking due cognizance of the exper-

imental priorities. This, in turn, leads to a substantial modiﬁcation of the

MA criterion. The use of the modiﬁed criterion in selecting optimal designs is

discussed.

Since the focus of the book is on design of experiments, the modeling issues

are seldom addressed. For information on modeling, analysis, and applications,

we refer to applied design texts like Montgomery (2000), Wu and Hamada

(2000), and Box, Hunter, and Hunter (2005).

1.4 Beyond the Book

In this section, we brieﬂy indicate some promising recent topics in factorial

designs that are not covered in the book. They are still undergoing rapid

development and hence have yet to crystalize.

All the fractional factorial designs considered in the book are called regu-

lar. A precise deﬁnition is given in Section 2.4; see also Section 2.7. To give

an idea of what a regular fraction entails, let us refer to the design matrix in

Table 1.1. It is easy to see that the sum modulo 2 of any two diﬀerent columns

of this matrix can be found among the remaining columns. In this sense, any

fractional factorial design given by a selection of columns from this matrix

is regular. On the other hand, designs arising from the 12-run matrix given

in Table 1.2 (Plackett and Burman, 1946) are nonregular; observe that the

sum modulo 2 of any two of its columns cannot be found among the remain-

ing columns. Traditionally, regular fractions have been the primary focus of

research in factorial designs. They have a neat mathematical structure that

1.4 Beyond the Book 7

Table 1.2 A 12-run Plackett–Burman design matrix

Column

Run 1234567891011

1 001000111 0 1

2 100100011 1 0

3 010010001 1 1

4 101001000 1 1

5 110100100 0 1

6 111010010 0 0

7 011101001 0 0

8 001110100 1 0

9 000111010 0 1

10 100011101 0 0

11 010001110 1 0

12 111111111 1 1

simpliﬁes the derivation and facilitates the understanding of eﬀect aliasing.

Moreover, they are the most commonly used designs in practice.

Over the years, nonregular designs had received some attention of re-

searchers mainly from the mathematical point of view. In the recent past,

however, there was a realization that they too could be utilized in conducting

eﬃcient experiments with ﬂexibility, run size economy, and ability to exploit

of interest in nonregular designs. Th

question of their

optimal selection was a

natural research topic to follow the parallel work in the

regular case. Various

extensions of the minimum aberration criterion to

nonregular designs were

proposed. Some major ones are minimum G

aberration (Deng and Tang,

1999; Tang and Deng, 1999), generalized minimum

aberration (Xu and Wu,

2001), and minimum moment aberration (Xu, 2003).

Although these criteria

are mathematically equivalent to the MA criterion in

the regular case, some of

them are potentially advantageous even for tackling

regular design problems

with relatively complex structure such as blocking

(Xu, 2006). Through the

use of indicator functions, Ye (2003) and Cheng

and Ye (2004) investigated

nonregular designs from a diﬀerent viewpoint.

Another promising area of research concerns supersaturated designs,where

the run size is not suﬃcient even for estimating all the main eﬀects of the

factors. This can be a realistic scenario when the runs are expensive and the

problem is complex enough to suggest many factors for investigation. None

of the designs considered in the book is supersaturated. For example, the

design matrix in Table 1.1 can be used to study up to 15 two-level factors

with 16 runs. Suppose economic considerations limited the run size to 16 and

the investigators were insistent on including, say, as many as 19 factors in the

experiment. A supersaturated design in the form of a 16×19 array would then

be required. Various construction methods for supersaturated designs have

been proposed; see, e.g., Lin (1993, 1995), Wu (1993), Nguyen (1996), Yamada

This led to a growth

interactions (Hamada and Wu, 1992; Wu andHamada,2000,Chapters 7 and 8).

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