eBook

- Presents recent developments and results not found in similar books
- Builds a general and unifying theory after covering basic experimental design
- Uses many examples to illustrate the theory
- Requires minimal prior knowledge of experimental design

Bringing together both new and old results, **Theory of Factorial Design: Single- and Multi-Stratum Experiments** provides a rigorous, systematic, and up-to-date treatment of the theoretical aspects of factorial design. To prepare readers for a general theory, the author first presents a unified treatment of several simple designs, including completely randomized designs, block designs, and row-column designs. As such, the book is accessible to readers with minimal exposure to experimental design. With exercises and numerous examples, it is suitable as a reference for researchers and as a textbook for advanced graduate students.

In addition to traditional topics and a thorough discussion of the popular minimum aberration criterion, the book covers many topics and new results not found in existing books. These include results on the structures of two-level resolution IV designs, methods for constructing such designs beyond the familiar foldover method, the extension of minimum aberration to nonregular designs, the equivalence of generalized minimum aberration and minimum moment aberration, a Bayesian approach, and some results on nonregular designs. The book also presents a theory that provides a unifying framework for the design and analysis of factorial experiments with multiple strata (error terms) arising from complicated structures of the experimental units. This theory can be systematically applied to various structures of experimental units instead of treating each on a case-by-case basis.

**Introduction **

**Linear Model Basics **Least squares

Estimation of σ

F-test

One-way layout

Estimation of a subset of parameters

Hypothesis testing for a subset of parameters

Adjusted orthogonality

Additive two-way layout

The case of proportional frequencies

**Randomization and Blocking **Randomization

Assumption of additivity and models for completely randomized designs

Randomized block designs

Randomized row-column designs

Nested row-column designs and blocked split-plot designs

Randomization model

**Factors**Factors as partitions

Block structures and Hasse diagrams

Some matrices and spaces associated with factors

Orthogonal projections, averages, and sums of squares

Condition of proportional frequencies

Supremums and infimums of factors

Orthogonality of factors

**Analysis of Some Simple Orthogonal Designs **A general result

Completely randomized designs

Null ANOVA for block designs

Randomized complete block designs

Randomized Latin square designs

Decomposition of the treatment sum of squares

Orthogonal polynomials

Orthogonal and nonorthogonal designs

Models with fixed block effects

**Factorial Treatment Structure and Complete Factorial Designs **Factorial effects for two and three two-level factors

Factorial effects for more than three two-level factors

The general case

Analysis of complete factorial designs

Analysis of unreplicated experiments

Defining factorial effects via finite geometries

Defining factorial effects via Abelian groups

More on factorial treatment structure

**Blocked, Split-Plot, and Strip-Plot Complete Factorial Designs **An example

Construction of blocked complete factorial designs

Analysis

Pseudo factors

Partial confounding

Design keys

A template for design keys

Construction of blocking schemes via Abelian groups

Complete factorial experiments in row-column designs

Split-plot designs

Strip-plot designs

**Fractional Factorial Designs and Orthogonal Arrays **Treatment models for fractional factorial designs

Orthogonal arrays

Examples of orthogonal arrays

Regular fractional factorial designs

Designs derived from Hadamard matrices

Mutually orthogonal Latin squares and orthogonal arrays

Foldover designs

Difference matrices

Enumeration of orthogonal arrays

Some variants of orthogonal arrays

**Regular Fractional Factorial Designs **Construction and defining relation

Aliasing and estimability

Analysis

Resolution

Regular fractional factorial designs are orthogonal arrays

Foldovers of regular fractional factorial designs

Construction of designs for estimating required effects

Grouping and replacement

Connection with linear codes

Factor representation and labeling

Connection with finite projective geometry

Foldover and even designs revisited

**Minimum Aberration and Related Criteria **Minimum aberration

Clear two-factor interactions

Interpreting minimum aberration

Estimation capacity

Other justifications of minimum aberration

Construction and complementary design theory

Maximum estimation capacity: a projective geometric approach

Clear two-factor interactions revisited

Minimum aberration blocking of complete factorial designs

Minimum moment aberration

A Bayesian approach

**Structures and Construction of Two-Level Resolution IV Designs **Maximal designs

Second-order saturated designs

Doubling

Maximal designs with

Maximal designs with

Partial foldover

More on clear two-factor interactions

Applications to minimum aberration designs

Minimum aberration even designs

Complementary design theory for doubling

Proofs of Theorems 11.27 and 11.28

Coding and projective geometric connections

**Orthogonal Block Structures and Strata **Nesting and crossing operators

Simple block structures

Statistical models

Poset block structures

Orthogonal block structures

Models with random effects

Strata

Null ANOVA

Nelder’s rules

Determining strata from Hasse diagrams

Proofs of Theorems 12.6 and 12.7

Models with random effects revisited

Experiments with multiple processing stages

Randomization justification of the models for simple block structures

Justification of Nelder’s rules

**Complete Factorial Designs with Orthogonal Block Structures **Orthogonal designs

Blocked complete factorial split-plot designs

Blocked complete factorial strip-plot designs

Contrasts in the strata of simple block structures

Construction of designs with simple block structures

Design keys

Design key templates for blocked split-plot and strip-plot designs

Proof of Theorem 13.2

Treatment structures

Checking design orthogonality

Experiments with multiple processing stages: the nonoverlapping case

Experiments with multiple processing stages: the overlapping case

**Multi-Stratum Fractional Factorial Designs**

A general procedure

Construction of blocked regular fractional factorial designs

Fractional factorial split-plot designs

Blocked fractional factorial split-plot designs

Fractional factorial strip-plot designs

Design key construction of blocked strip-plot designs

Post-fractionated strip-plot designs

Criteria for selecting blocked fractional factorial designs based on modified wordlength patterns

Fixed block effects: surrogate for maximum estimation capacity

Information capacity and its surrogate

Selection of fractional factorial split-plot designs

A general result on multi-stratum fractional factorial designs

Selection of blocked fractional factorial split-plot designs

Selection of blocked fractional factorial strip-plot designs

Geometric formulation

**Nonregular Designs **Indicator functions and J-characteristics

Partial aliasing

Projectivity

Hidden projection properties of orthogonal arrays

Generalized minimum aberration for two-level designs

Generalized minimum aberration for multiple and mixed levels

Connection with coding theory

Complementary designs

Minimum moment aberration

Proof of Theorem 15.18

Even designs and foldover designs

Parallel flats designs

Saturated designs for hierarchical models: an application of algebraic geometry

Search designs

Supersaturated designs

Appendix

References

Index

**Ching-Shui Cheng** is currently a Distinguished Research Fellow and Director of the Institute of Statistical Science, Academia Sinica, in Taiwan, and a retired professor from the University of California, Berkeley. He received his B.S. in mathematics from National Tsing Hua University and both his MS in mathematics and Ph.D. in mathematics from Cornell University. After receiving his Ph.D., he became an assistant professor in the Department of Statistics at the University of California, Berkeley. He was later promoted to associate professor and then professor. He retired on July 1, 2013.

Dr. Cheng’s research interest is mainly in experimental design and related combinatorial problems. He is a fellow of the Institute of Mathematical Statistics and the American Statistical Association and an elected member of the International Statistical Institute. He was an associate editor of the *Journal of Statistical Planning and Inference, Annals of Statistics, Statistica Sinica, Biometrika, *and *Technometrics*. He also served as the chair-editor of *Statistica Sinica* from 1996 to 1999.