Theory of Factorial Design: Single- and Multi-Stratum Experiments (e-Book) book cover

Theory of Factorial Design

Single- and Multi-Stratum Experiments

By Ching-Shui Cheng

© 2013 – Chapman and Hall/CRC

409 pages | 13 B/W Illus.

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pub: 2013-12-21
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About the Book

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.


"The field of experimental design aims to help practitioners collect their data in a more efficient manner, or more specifically, run their experiments more effectively. There are many good textbooks in this area: the classical ones of the early 50’s (e.g.,Cochran and Cox 1957) focused more on agricultural experimentation; the later ones of the late 70’s (e.g., Box, Hunter, and Hunter 1978) focused more on industrial experimentation, and the recent ones (e.g., Santner, Williams, and Notz 2003; Fang, Li, and Sudjianto 2006) focused more on computer experiments. There are also some theoretical approaches, notably on optimal design (e.g., Pukelsheim 1993) and combinatorics (e.g., Street and Street 1987). This book is clearly one of the very first about design of experiment from a multi-stratum approach… Some topics have never appeared in any other book and the author has produced elegant mathematics accompanied with lucid explanations…I believe that this excellent book will soon become a must read for researchers and educators in experimental design. It could serve as a great reference or textbook for a high-level design course."

—Dennis Lin, Penn State University, in Journal of the American Statistical Association, Volume 111, 2016

"… the book is extremely well written. It is a book on design theory authored by a well-known researcher in the field. As is pointed out by the author, the book provides an elegant and general theory, which once understood is simple to use and can be applied to various structures of experimental units in a unified and systematic way. The book is certainly a necessary reference for Technometrics readers who have an interest in the theory of factorial designs for single- and multi-stratum experiments."

Technometrics, May 2015

"This is a great book on factorial designs, both for academic statisticians and for practitioners. … The style of the presentation, based on the discussion of a large number of real-life examples, supports the overall clarity and readability of the text. … many chapters also contain some interesting topics usually not reported in books. In particular, I would like to mention the construction of two-level resolution IV designs in Chapter 11."

Zentralblatt MATH 1306

Table of Contents


Linear Model Basics

Least squares

Estimation of σ


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


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 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


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



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


Maximal designs with N/4+1 ≤ nN/2

Maximal designs with n = N/4+1

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



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


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




About the Author

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.

About the Series

Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General