Cheng, C: Theory of Factorial 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 sigma 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 N/4+1 <= n <= N/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 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