Theory of Factorial Design: Single- and Multi-Stratum Experiments

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ISBN 9781466505575
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  • 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.

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




Author Bio(s)

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.

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