155,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
A indispensable guide to understanding and designing modern experiments
The tools and techniques of Design of Experiments (DOE) allow researchers to successfully collect, analyze, and interpret data across a wide array of disciplines. Statistical Analysis of Designed Experiments provides a modern and balanced treatment of DOE methodology with thorough coverage of the underlying theory and standard designs of experiments, guiding the reader through applications to research in various fields such as engineering, medicine, business, and the social sciences.
The book supplies a foundation for the subject, beginning with basic concepts of DOE and a review of elementary normal theory statistical methods. Subsequent chapters present a uniform, model-based approach to DOE. Each design is presented in a comprehensive format and is accompanied by a motivating example, discussion of the applicability of the design, and a model for its analysis using statistical methods such as graphical plots, analysis of variance (ANOVA), confidence intervals, and hypothesis tests.
Numerous theoretical and applied exercises are provided in each chapter, and answers to selected exercises are included at the end of the book. An appendix features three case studies that illustrate the challenges often encountered in real-world experiments, such as randomization, unbalanced data, and outliers. Minitab® software is used to perform analyses throughout the book, and an accompanying FTP site houses additional exercises and data sets.
With its breadth of real-world examples and accessible treatment of both theory and applications, Statistical Analysis of Designed Experiments is a valuable book for experimental design courses at the upper-undergraduate and graduate levels. It is also an indispensable reference for practicing statisticians, engineers, and scientists who would like to further their knowledge of DOE.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 920
Veröffentlichungsjahr: 2012
Ähnliche
Contents
Cover
Half Title page
Title page
Copyright page
Dedication
Preface
Abbreviations
Chapter 1: Introduction
1.1 Observational Studies and Experiments
1.2 Brief Historical Remarks
1.3 Basic Terminology and Concepts of Experimentation
1.4 Basic Principles of Experimentation
1.5 Chapter Summary
Exercises
Chapter 2: Review of Elementary Statistics
2.1 Experiments for a Single Treatment
2.2 Experiments for Comparing Two Treatments
2.3 Linear Regression
2.4 Chapter Summary
Exercises
Chapter 3: Single Factor Experiments: Completely Randomized Designs
3.1 Summary Statistics and Graphical Displays
3.2 Model
3.3 Statistical Analysis
3.4 Model Diagnostics
3.5 Data Transformations
3.6 Power of F-Test and Sample Size Determination
3.7 Quantitative Treatment Factors
3.8 One-Way Analysis of Covariance
3.9 Chapter Notes
3.10 Chapter Summary
Exercises
Chapter 4: Single-Factor Experiments: Multiple Comparison and Selection Procedures
4.1 Basic Concepts of Multiple Comparisons
4.2 Pairwise Comparisons
4.3 Comparisons with a Control
4.4 General Contrasts
4.5 Ranking and Selection Procedures
4.6 Chapter Summary
Exercises
Chapter 5: Randomized Block Designs and Extensions
5.1 Randomized Block Designs
5.2 Balanced Incomplete Block Designs
5.3 Youden Square Designs
5.4 Latin Square Designs
5.5 Chapter Notes
5.6 Chapter Summary
Exercises
Chapter 6: General Factorial Experiments
6.1 Factorial Versus One-Factor-at-a-Time Experiments
6.2 Balanced Two-Way Layouts
6.3 Unbalanced Two-Way Layouts
6.4 Chapter Notes
6.5 Chapter Summary
Exercises
Chapter 7: Two-Level Factorial Experiments
7.1 Estimation of Main Effects and Interactions
7.2 Statistical Analysis
7.3 Single-Replicate Case
7.4 2p Factorial Designs in Incomplete Blocks: Confounding of Effects
7.5 Chapter Notes
7.6 Chapter Summary
Exercises
Chapter 8: Two-Level Fractional Factorial Experiments
8.1 2p-q Fractional Factorial Designs
8.2 Plackett-Burman Designs
8.3 Hadamard Designs
8.4 Supersaturated Designs
8.5 Orthogonal Arrays
8.6 Sequential Assemblies of Fractional Factorials
8.7 Chapter Summary
Exercises
Chapter 9: Three-Level and Mixed-Level Factorial Experiments
9.1 Three-Level Full Factorial Designs
9.2 Three-Level Fractional Factorial Designs
9.3 Mixed-Level Factorial Designs
9.4 Chapter Notes
9.5 Chapter Summary
Exercises
Chapter 10: Experiments for Response Optimization
10.1 Response Surface Methodology
10.2 Mixture Experiments
10.3 Taguchi Method of Quality Improvement
10.4 Chapter Summary
Exercises
Chapter 11: Random and Mixed Crossed-Factors Experiments
11.1 One-Way Layouts
11.2 Two-Way Layouts
11.3 Three-Way Layouts
11.4 Chapter Notes
11.5 Chapter Summary
Exercises
Chapter 12: Nested, Crossed–Nested, and Split-Plot Experiments
12.1 Two-Stage Nested Designs
12.2 Three-Stage Nested Designs
12.3 Crossed and Nested Designs
12.4 Split-Plot Designs
12.5 Chapter Notes
12.6 Chapter Summary
Exercises
Chapter 13: Repeated Measures Experiments
13.1 Univariate Approach
13.2 Multivariate Approach
13.3 Chapter Notes
13.4 Chapter Summary
Exercises
Chapter 14: Theory of Linear Models with Fixed Effects
14.1 Basic Linear Model and Least Squares Estimation
14.2 Confidence Intervals and Hypothesis Tests
14.3 Power of F-Test
14.4 Chapter Notes
14.5 Chapter Summary
Exercises
Appendix A: Vector-Valued Random Variables and Some Distribution Theory
A.1 Mean Vector and Covariance Matrix of Random Vector
A.2 Covariance Matrix of Linear Transformation of Random Vector
A.3 Multivariate Normal Distribution
A.4 Chi-Square, F-, and t-Distributions
A.5 Distributions of Quadratic Forms
A.6 Multivariate t-Distribution
A.7 Multivariate Normal Sampling Distribution Theory
Appendix B: Case Studies
B.1 Case Study 1: Effects of Field Strength and Flip Angle on MRI Contrast
B.2 Case Study 2: Growing Stem Cells for Bone Implants
B.3 Case Study 3: Router Bit Experiment
Appendix C: Statistical Tables
Answers to Selected Exercises
References
Index
Statistical Analysis of Designed Experiments
WILEY SERIES IN PROBABILITY AND STATISTICS
ESTABLISHED BY WALTER A. SHEWHART AND SAMUEL S. WILKS
Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg
Editors Emeriti: Vic Barnett, J. Stuart Hunter, Jozef L. Teugels
The Wiley Series in Probability and Statistics is well established and authoritative. It covers many topics of current research interest in both pure and applied statistics and probability theory. Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods.
Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches.
This series provides essential and invaluable reading for all statisticians, whether in academia, industry, government, or research.
† ABRAHAM and LEDOLTER · Statistical Methods for Forecasting
AGRESTI · Analysis of Ordinal Categorical Data
AGRESTI · An Introduction to Categorical Data Analysis, Second Edition
AGRESTI · Categorical Data Analysis, Second Edition
ALTMAN, GILL, and MCDONALD · Numerical Issues in Statistical Computing for the Social Scientist
AMARATUNGA and CABRERA · Exploration and Analysis of DNA Microarray and Protein Array Data
ANDL · Mathematics of Chance
ANDERSON · An Introduction to Multivariate Statistical Analysis, Third Edition
* ANDERSON · The Statistical Analysis of Time Series
ANDERSON, AUQUIER, HAUCK, OAKES, VANDAELE, and WEISBERG · Statistical Methods for Comparative Studies
ANDERSON and LOYNES · The Teaching of Practical Statistics
ARMITAGE and DAVID (editors) · Advances in Biometry
ARNOLD, BALAKRISHNAN, and NAGARAJA · Records
* ARTHANARI and DODGE · Mathematical Programming in Statistics
* BAILEY · The Elements of Stochastic Processes with Applications to the Natural Sciences
BALAKRISHNAN and KOUTRAS · Runs and Scans with Applications
BALAKRISHNAN and NG · Precedence-Type Tests and Applications
BARNETT · Comparative Statistical Inference, Third Edition
BARNETT · Environmental Statistics
BARNETT and LEWIS · Outliers in Statistical Data, Third Edition
BARTOSZYNSKI and NIEWIADOMSKA-BUGAJ · Probability and Statistical Inference
BASILEVSKY · Statistical Factor Analysis and Related Methods: Theory and Applications
BASU and RIGDON · Statistical Methods for the Reliability of Repairable Systems
BATES and WATTS · Nonlinear Regression Analysis and Its Applications
BECHHOFER, SANTNER, and GOLDSMAN · Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons
BELSLEY · Conditioning Diagnostics: Collinearity and Weak Data in Regression
† BELSLEY, KUH, and WELSCH · Regression Diagnostics: Identifying Influential Data and Sources of Collinearity
BENDAT and PIERSOL · Random Data: Analysis and Measurement Procedures, Third Edition
BERRY, CHALONER, and GEWEKE · Bayesian Analysis in Statistics and Econometrics: Essays in Honor of Arnold Zellner
BERNARDO and SMITH · Bayesian Theory
BHAT and MILLER · Elements of Applied Stochastic Processes, Third Edition
BHATTACHARYA and WAYMIRE · Stochastic Processes with Applications
BILLINGSLEY · Convergence of Probability Measures, Second Edition
BILLINGSLEY · Probability and Measure, Third Editio
BIRKES and DODGE · Alternative Methods of Regression
BISWAS, DATTA, FINE, and SEGAL · Statistical Advances in the Biomedical Sciences: Clinical Trials, Epidemiology, Survival Analysis, and Bioinformatics
BLISCHKE AND MURTHY (editors) · Case Studies in Reliability and Maintenance
BLISCHKE AND MURTHY · Reliability: Modeling, Prediction, and Optimization
BLOOMFIELD · Fourier Analysis of Time Series: An Introduction, Second Edition
BOLLEN · Structural Equations with Latent Variables
BOLLEN and CURRAN · Latent Curve Models: A Structural Equation Perspective
BOROVKOV · Ergodicity and Stability of Stochastic Processes
BOULEAU · Numerical Methods for Stochastic Processes
BOX · Bayesian Inference in Statistical Analysis
BOX · R. A. Fisher, the Life of a Scientist
BOX and DRAPER · Response Surfaces, Mixtures, and Ridge Analyses, Second Edition
* BOX and DRAPER · Evolutionary Operation: A Statistical Method for Process Improvement
BOX and FRIENDS · Improving Almost Anything, Revised Edition
BOX, HUNTER, and HUNTER · Statistics for Experimenters: Design, Innovation, and Discovery, Second Editon
BOX, JENKINS, and REINSEL · Time Series Analysis: Forcasting and Control, Fourth Edition
BOX and LUCENO · Statistical Control by Monitoring and Feedback Adjustment
BRANDIMARTE · Numerical Methods in Finance: A MATLAB-Based Introduction
† BROWN and HOLLANDER · Statistics: A Biomedical Introduction
BRUNNER, DOMHOF, and LANGER · Nonparametric Analysis of Longitudinal Data in Factorial Experiments
BUCKLEW · Large Deviation Techniques in Decision, Simulation, and Estimation
CAIROLI and DALANG · Sequential Stochastic Optimization
CASTILLO, HADI, BALAKRISHNAN, and SARABIA · Extreme Value and Related Models with Applications in Engineering and Science
CHAN · Time Series: Applications to Finance
CHARALAMBIDES · Combinatorial Methods in Discrete Distributions
CHATTERJEE and HADI · Regression Analysis by Example, Fourth Edition
CHATTERJEE and HADI · Sensitivity Analysis in Linear Regression
CHERNICK · Bootstrap Methods: A Guide for Practitioners and Researchers, Second Edition
CHERNICK and FRIIS · Introductory Biostatistics for the Health Sciences
CHILÈS and DELFINER · Geostatistics: Modeling Spatial Uncertainty
CHOW and LIU · Design and Analysis of Clinical Trials: Concepts and Methodologies, Second Edition
CLARKE · Linear Models: The Theory and Application of Analysis of Variance
CLARKE and DISNEY · Probability and Random Processes: A First Course with Applications, Second Edition
* COCHRAN and COX · Experimental Designs, Second Edition
CONGDON · Applied Bayesian Modelling
CONGDON · Bayesian Models for Categorical Data
CONGDON · Bayesian Statistical Modelling
CONOVER · Practical Nonparametric Statistics, Third Edition
COOK · Regression Graphics
COOK and WEISBERG · Applied Regression Including Computing and Graphics
COOK and WEISBERG · An Introduction to Regression Graphics
CORNELL · Experiments with Mixtures, Designs, Models, and the Analysis of Mixture Data, Third Edition
COVER and THOMAS · Elements of Information Theory
COX · A Handbook of Introductory Statistical Methods
* COX · Planning of Experiments
CRESSIE · Statistics for Spatial Data, Revised Edition
CSÖRGÖ and HORVÁTH · Limit Theorems in Change Point Analysis
DANIEL · Applications of Statistics to Industrial Experimentation
DANIEL · Biostatistics: A Foundation for Analysis in the Health Sciences, Eighth Edition
* DANIEL · Fitting Equations to Data: Computer Analysis of Multifactor Data, Second Edition
DASU and JOHNSON · Exploratory Data Mining and Data Cleaning
DAVID and NAGARAJA · Order Statistics, Third Edition
* DEGROOT, FIENBERG, and KADANE · Statistics and the Law
DEL CASTILLO · Statistical Process Adjustment for Quality Control
DEMARIS · Regression with Social Data: Modeling Continuous and Limited Response Variables
DEMIDENKO · Mixed Models: Theory and Applications
DENISON, HOLMES, MALLICK and SMITH · Bayesian Methods for Nonlinear Classification and Regression
DETTE and STUDDEN · The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis
DEY and MUKERJEE · Fractional Factorial Plans
DILLON and GOLDSTEIN · Multivariate Analysis: Methods and Applications
DODGE · Alternative Methods of Regression
* DODGE and ROMIG · Sampling Inspection Tables, Second Edition
* DOOB · Stochastic Processes
DOWDY, WEARDEN, and CHILKO · Statistics for Research, Third Edition
DRAPER and SMITH · Applied Regression Analysis, Third Edition
DRYDEN and MARDIA · Statistical Shape Analysis
DUDEWICZ and MISHRA · Modern Mathematical Statistics
DUNN and CLARK · Basic Statistics: A Primer for the Biomedical Sciences, Third Edition
DUPUIS and ELLIS · A Weak Convergence Approach to the Theory of Large Deviations
EDLER and KITSOS · Recent Advances in Quantitative Methods in Cancer and Human Health Risk Assessment
* ELANDT-JOHNSON and JOHNSON · Survival Models and Data Analysis
ENDERS · Applied Econometric Time Series
† ETHIER and KURTZ · Markov Processes: Characterization and Convergence
EVANS, HASTINGS, and PEACOCK · Statistical Distributions, Third Edition
FELLER · An Introduction to Probability Theory and Its Applications, Volume I, Third Edition, Revised; Volume II, Second Edition
FISHER and VAN BELLE · Biostatistics: A Methodology for the Health Sciences
FITZMAURICE, LAIRD, and WARE · Applied Longitudinal Analysis
* FLEISS · The Design and Analysis of Clinical Experiments
FLEISS · Statistical Methods for Rates and Proportions, Third Edition
† FLEMING and HARRINGTON · Counting Processes and Survival Analysis
FULLER · Introduction to Statistical Time Series, Second Edition
† FULLER · Measurement Error Models
GALLANT · Nonlinear Statistical Models
GEISSER · Modes of Parametric Statistical Inference
GELMAN and MENG · Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives
GEWEKE · Contemporary Bayesian Econometrics and Statistics
GHOSH, MUKHOPADHYAY, and SEN · Sequential Estimation
GIESBRECHT and GUMPERTZ · Planning, Construction, and Statistical Analysis of Comparative Experiments
GIFI · Nonlinear Multivariate Analysis
GIVENS and HOETING · Computational Statistics
GLASSERMAN and YAO · Monotone Structure in Discrete-Event Systems
GNANADESIKAN · Methods for Statistical Data Analysis of Multivariate Observations, Second Edition
GOLDSTEIN and LEWIS · Assessment: Problems, Development, and Statistical Issues
GREENWOOD and NIKULIN · A Guide to Chi-Squared Testing
GROSS, SHORTLE, THOMPSON, and HARRIS · Fundamentals of Queueing Theory, Fourth Edition
GROSS, SHORTLE, THOMPSON, and HARRIS · Solutions Manual to Accompany Fundamentals of Queueing Theory, Fourth Edition
* HAHN and SHAPIRO · Statistical Models in Engineering
HAHN and MEEKER · Statistical Intervals: A Guide for Practitioners
HALD · A History of Probability and Statistics and their Applications Before 1750
HALD · A History of Mathematical Statistics from 1750 to 1930
† HAMPEL · Robust Statistics: The Approach Based on Influence Functions
HANNAN and DEISTLER · The Statistical Theory of Linear Systems
HARTUNG, KNAPP, and SINHA · Statistical Meta-Analysis with Applications
HEIBERGER · Computation for the Analysis of Designed Experiments
HEDAYAT and SINHA · Design and Inference in Finite Population Sampling
HEDEKER and GIBBONS · Longitudinal Data Analysis
HELLER · MACSYMA for Statisticians
HINKELMANN and KEMPTHORNE · Design and Analysis of Experiments, Volume 1: Introduction to Experimental Design, Second Edition
HINKELMANN and KEMPTHORNE · Design and Analysis of Experiments, Volume 2: Advanced Experimental Design
HOAGLIN, MOSTELLER, and TUKEY · Fundamentals of Exploratory Analysis of Variance
* HOAGLIN, MOSTELLER, and TUKEY · Exploring Data Tables, Trends and Shapes
* HOAGLIN, MOSTELLER, and TUKEY · Understanding Robust and Exploratory Data Analysis
HOCHBERG and TAMHANE · Multiple Comparison Procedures
HOCKING · Methods and Applications of Linear Models: Regression and the Analysis of Variance, Second Edition
HOEL · Introduction to Mathematical Statistics, Fifth Edition
HOGG and KLUGMAN · Loss Distributions
HOLLANDER and WOLFE · Nonparametric Statistical Methods, Second Edition
HOSMER and LEMESHOW · Applied Logistic Regression, Second Edition
HOSMER, LEMESHOW, and MAY · Applied Survival Analysis: Regression Modeling of Time-to-Event Data, Second Edition
† HUBER and RONCHETTI · Robust Statistics, Second Edition
HUBERTY · Applied Discriminant Analysis
HUBERTY and OLEJNIK · Applied MANOVA and Discriminant Analysis, Second Edition
HUNT and KENNEDY · Financial Derivatives in Theory and Practice, Revised Edition
HURD and MIAMEE · Periodically Correlated Random Sequences: Spectral Theory and Practice
HUSKOVA, BERAN, and DUPAC · Collected Works of Jaroslav Hajek—with Commentary
HUZURBAZAR · Flowgraph Models for Multistate Time-to-Event Data
IMAN and CONOVER · A Modern Approach to Statistics
† JACKSON · A User’s Guide to Principle Components
JOHN · Statistical Methods in Engineering and Quality Assurance
JOHNSON · Multivariate Statistical Simulation
JOHNSON and BALAKRISHNAN · Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz
JOHNSON and BHATTACHARYYA · Statistics: Principles and Methods, Fifth Edition
JOHNSON and KOTZ · Distributions in Statistics
JOHNSON and KOTZ (editors) · Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present
JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 1, Second Edition
JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 2, Second Edition
JOHNSON, KOTZ, and BALAKRISHNAN · Discrete Multivariate Distributions
JOHNSON, KEMP, and KOTZ · Univariate Discrete Distributions, Third Edition
JUDGE, GRIFFITHS, HILL, LÜTKEPOHL, and LEE · The Theory and Practice of Econometrics, Second Edition
JUREKOVÁ and SEN · Robust Statistical Procedures: Aymptotics and Interrelations
JUREK and MASON · Operator-Limit Distributions in Probability Theory
KADANE · Bayesian Methods and Ethics in a Clinical Trial Design
KADANE AND SCHUM · A Probabilistic Analysis of the Sacco and Vanzetti Evidence
KALBFLEISCH and PRENTICE · The Statistical Analysis of Failure Time Data, Second Edition
KARIYA and KURATA · Generalized Least Squares
KASS and VOS · Geometrical Foundations of Asymptotic Inference
† KAUFMAN and ROUSSEEUW · Finding Groups in Data: An Introduction to Cluster Analysis
KEDEM and FOKIANOS · Regression Models for Time Series Analysis
KENDALL, BARDEN, CARNE, and LE · Shape and Shape Theory
KHURI · Advanced Calculus with Applications in Statistics, Second Edition
KHURI, MATHEW, and SINHA · Statistical Tests for Mixed Linear Models
KLEIBER and KOTZ · Statistical Size Distributions in Economics and Actuarial Sciences
KLUGMAN, PANJER, and WILLMOT · Loss Models: From Data to Decisions, Third Edition
KLUGMAN, PANJER, and WILLMOT · Solutions Manual to Accompany Loss Models: From Data to Decisions, Third Edition
KOTZ, BALAKRISHNAN, and JOHNSON · Continuous Multivariate Distributions, Volume 1, Second Edition
KOVALENKO, KUZNETZOV, and PEGG · Mathematical Theory of Reliability of Time-Dependent Systems with Practical Applications
KOWALSKI and TU · Modern Applied U-Statistics
KROONENBERG · Applied Multiway Data Analysis
KVAM and VIDAKOVIC · Nonparametric Statistics with Applications to Science and Engineering
LACHIN · Biostatistical Methods: The Assessment of Relative Risks
LAD · Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction
LAMPERTI · Probability: A Survey of the Mathematical Theory, Second Edition
LANGE, RYAN, BILLARD, BRILLINGER, CONQUEST, and GREENHOUSE · Case Studies in Biometry
LARSON · Introduction to Probability Theory and Statistical Inference, Third Edition
LAWLESS · Statistical Models and Methods for Lifetime Data, Second Edition
LAWSON · Statistical Methods in Spatial Epidemiology
LE · Applied Categorical Data Analysis
LE · Applied Survival Analysis
LEE and WANG · Statistical Methods for Survival Data Analysis, Third Edition
LEPAGE and BILLARD · Exploring the Limits of Bootstrap
LEYLAND and GOLDSTEIN (editors) · Multilevel Modelling of Health Statistics
LIAO · Statistical Group Comparison
LINDVALL · Lectures on the Coupling Method
LIN · Introductory Stochastic Analysis for Finance and Insurance
LINHART and ZUCCHINI · Model Selection
LITTLE and RUBIN · Statistical Analysis with Missing Data, Second Edition
LLOYD · The Statistical Analysis of Categorical Data
LOWEN and TEICH · Fractal-Based Point Processes
MAGNUS and NEUDECKER · Matrix Differential Calculus with Applications in Statistics and Econometrics, Revised Edition
MALLER and ZHOU · Survival Analysis with Long Term Survivors
MALLOWS · Design, Data, and Analysis by Some Friends of Cuthbert Daniel
MANN, SCHAFER, and SINGPURWALLA · Methods for Statistical Analysis of Reliability and Life Data
MANTON, WOODBURY, and TOLLEY · Statistical Applications Using Fuzzy Sets
MARCHETTE · Random Graphs for Statistical Pattern Recognition
MARDIA and JUPP · Directional Statistics
MASON, GUNST, and HESS · Statistical Design and Analysis of Experiments with Applications to Engineering and Science, Second Edition
McCULLOCH, SEARLE, and NEUHAUS · Generalized, Linear, and Mixed Models, Second Edition
McFADDEN · Management of Data in Clinical Trials, Second Edition
* McLACHLAN · Discriminant Analysis and Statistical Pattern Recognition
McLACHLAN, DO, and AMBROISE · Analyzing Microarray Gene Expression Data
McLACHLAN and KRISHNAN · The EM Algorithm and Extensions, Second Edition
McLACHLAN and PEEL · Finite Mixture Models
McNEIL · Epidemiological Research Methods
MEEKER and ESCOBAR · Statistical Methods for Reliability Data
MEERSCHAERT and SCHEFFLER · Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice
MICKEY, DUNN, and CLARK · Applied Statistics: Analysis of Variance and Regression, Third Edition
* MILLER · Survival Analysis, Second Edition
MONTGOMERY, JENNINGS, and KULAHCI · Introduction to Time Series Analysis and Forecasting
MONTGOMERY, PECK, and VINING · Introduction to Linear Regression Analysis, Fourth Edition
MORGENTHALER and TUKEY · Configural Polysampling: A Route to Practical Robustness
MUIRHEAD · Aspects of Multivariate Statistical Theory
MULLER and STOYAN · Comparison Methods for Stochastic Models and Risks
MURRAY · X-STAT 2.0 Statistical Experimentation, Design Data Analysis, and Nonlinear Optimization
MURTHY, XIE, and JIANG · Weibull Models
MYERS, MONTGOMERY, and ANDERSON-COOK · Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Third Edition
MYERS, MONTGOMERY, and VINING · Generalized Linear Models. With Applications in Engineering and the Sciences
† NELSON · Accelerated Testing, Statistical Models, Test Plans, and Data Analyses
† NELSON · Applied Life Data Analysis
NEWMAN · Biostatistical Methods in Epidemiology
OCHI · Applied Probability and Stochastic Processes in Engineering and Physical Sciences
OKABE, BOOTS, SUGIHARA, and CHIU · Spatial Tesselations: Concepts and Applications of Voronoi Diagrams, Second Edition
OLIVER and SMITH · Influence Diagrams, Belief Nets and Decision Analysis
PALTA · Quantitative Methods in Population Health: Extensions of Ordinary Regressions
PANJER · Operational Risk: Modeling and Analytics
PANKRATZ · Forecasting with Dynamic Regression Models
PANKRATZ · Forecasting with Univariate Box-Jenkins Models: Concepts and Cases
* PARZEN · Modern Probability Theory and Its Applications
PEÑTA, TIAO, and TSAY · A Course in Time Series Analysis
PIANTADOSI · Clinical Trials: A Methodologic Perspective
PORT · Theoretical Probability for Applications
POURAHMADI · Foundations of Time Series Analysis and Prediction Theory
POWELL · Approximate Dynamic Programming: Solving the Curses of Dimensionality
PRESS · Bayesian Statistics: Principles, Models, and Applications
PRESS · Subjective and Objective Bayesian Statistics, Second Edition
PRESS and TANUR · The Subjectivity of Scientists and the Bayesian Approach
PUKELSHEIM · Optimal Experimental Design
PURI, VILAPLANA, and WERTZ · New Perspectives in Theoretical and Applied Statistics
† PUTERMAN · Markov Decision Processes: Discrete Stochastic Dynamic Programming
QIU · Image Processing and Jump Regression Analysis
* RAO · Linear Statistical Inference and Its Applications, Second Edition
RAUSAND and HØYLAND · System Reliability Theory: Models, Statistical Methods, and Applications, Second Edition
RENCHER · Linear Models in Statistics
RENCHER · Methods of Multivariate Analysis, Second Edition
RENCHER · Multivariate Statistical Inference with Applications
* RIPLEY · Spatial Statistics
* RIPLEY · Stochastic Simulation
ROBINSON · Practical Strategies for Experimenting
ROHATGI and SALEH · An Introduction to Probability and Statistics, Second Edition
ROLSKI, SCHMIDLI, SCHMIDT, and TEUGELS · Stochastic Processes for Insurance and Finance
ROSENBERGER and LACHIN · Randomization in Clinical Trials: Theory and Practice
ROSS · Introduction to Probability and Statistics for Engineers and Scientists
ROSSI, ALLENBY, and McCULLOCH · Bayesian Statistics and Marketing
† ROUSSEEUW and LEROY · Robust Regression and Outlier Detection
* RUBIN · Multiple Imputation for Nonresponse in Surveys
RUBINSTEIN and KROESE · Simulation and the Monte Carlo Method, Second Edition
RUBINSTEIN and MELAMED · Modem Simulation and Modeling
RYAN · Modern Engineering Statistics
RYAN · Modern Experimental Design
RYAN · Modern Regression Methods, Second Edition
RYAN · Statistical Methods for Quality Improvement, Second Edition
SALEH · Theory of Preliminary Test and Stein-Type Estimation with Applications
* SCHEFFE · The Analysis of Variance
SCHIMEK · Smoothing and Regression: Approaches, Computation, and Application
SCHOTT · Matrix Analysis for Statistics, Second Edition
SCHOUTENS · Levy Processes in Finance: Pricing Financial Derivatives
SCHUSS · Theory and Applications of Stochastic Differential Equations
SCOTT · Multivariate Density Estimation: Theory, Practice, and Visualization
† SEARLE · Linear Models for Unbalanced Data
† SEARLE · Matrix Algebra Useful for Statistics
† SEARLE, CASELLA, and McCULLOCH · Variance Components
SEARLE and WILLETT · Matrix Algebra for Applied Economics
SEBER · A Matrix Handbook For Statisticians
† SEBER · Multivariate Observations
SEBER and LEE · Linear Regression Analysis, Second Edition
† SEBER and WILD · Nonlinear Regression
SENNOTT · Stochastic Dynamic Programming and the Control of Queueing Systems
* SERFLING · Approximation Theorems of Mathematical Statistics
SHAFER and VOVK · Probability and Finance: It’s Only a Game!
SILVAPULLE and SEN · Constrained Statistical Inference: Inequality, Order, and Shape Restrictions
SMALL and McLEISH · Hilbert Space Methods in Probability and Statistical Inference
SRIVASTAVA · Methods of Multivariate Statistics
STAPLETON · Linear Statistical Models
STAPLETON · Models for Probability and Statistical Inference: Theory and Applications
STAUDTE and SHEATHER · Robust Estimation and Testing
STOYAN, KENDALL, and MECKE · Stochastic Geometry and Its Applications, Second Edition
STOYAN and STOYAN · Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics
STREET and BURGESS · The Construction of Optimal Stated Choice Experiments: Theory and Methods
STYAN · The Collected Papers of T. W. Anderson: 1943–1985
SUTTON, ABRAMS, JONES, SHELDON, and SONG · Methods for Meta-Analysis in Medical Research
TAKEZAWA · Introduction to Nonparametric Regression
TAMHANE · Statistical Analysis of Designed Experiments: Theory and Applications
TANAKA · Time Series Analysis: Nonstationary and Noninvertible Distribution Theory
THOMPSON · Empirical Model Building
THOMPSON · Sampling, Second Edition
THOMPSON · Simulation: A Modeler’s Approach
THOMPSON and SEBER · Adaptive Sampling
THOMPSON, WILLIAMS, and FINDLAY · Models for Investors in Real World Markets
TIAO, BISGAARD, HILL, PEÑNTA, and STIGLER (editors) · Box on Quality and Discovery: with Design, Control, and Robustness
TIERNEY · LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics
TSAY · Analysis of Financial Time Series, Second Edition
UPTON and FINGLETON · Spatial Data Analysis by Example, Volume II: Categorical and Directional Data
† VAN BELLE · Statistical Rules of Thumb, Second Edition
VAN BELLE, FISHER, HEAGERTY, and LUMLEY · Biostatistics: A Methodology for the Health Sciences, Second Edition
VESTRUP · The Theory of Measures and Integration
VIDAKOVIC · Statistical Modeling by Wavelets
VINOD and REAGLE · Preparing for the Worst: Incorporating Downside Risk in Stock Market Investments
WALLER and GOTWAY · Applied Spatial Statistics for Public Health Data
WEERAHANDI · Generalized Inference in Repeated Measures: Exact Methods in MANOVA and Mixed Models
WEISBERG · Applied Linear Regression, Third Edition
WELSH · Aspects of Statistical Inference
WESTFALL and YOUNG · Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment
WHITTAKER · Graphical Models in Applied Multivariate Statistics
WINKER · Optimization Heuristics in Economics: Applications of Threshold Accepting
WONNACOTT and WONNACOTT · Econometrics, Second Edition
WOODING · Planning Pharmaceutical Clinical Trials: Basic Statistical Principles
WOODWORTH · Biostatistics: A Bayesian Introduction
WOOLSON and CLARKE · Statistical Methods for the Analysis of Biomedical Data, Second Edition
WU and HAMADA · Experiments: Planning, Analysis, and Parameter Design Optimization
WU and ZHANG · Nonparametric Regression Methods for Longitudinal Data Analysis
YANG · The Construction Theory of Denumerable Markov Processes
YOUNG, VALERO-MORA, and FRIENDLY · Visual Statistics: Seeing Data with Dynamic Interactive Graphics
ZACKS · Stage-Wise Adaptive Designs
ZELTERMAN · Discrete Distributions—Applications in the Health Sciences
* ZELLNER · An Introduction to Bayesian Inference in Econometrics
ZHOU, OBUCHOWSKI, and McCLISH · Statistical Methods in Diagnostic Medicine
* Now available in a lower priced paperback edition in the Wiley Classics Library.
† Now available in a lower priced paperback edition in the Wiley-Interscience Paperback Series.
Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at http://www.wiley.com/go/permission.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at 877-762-2974, outside the United States at 317-572-3993 or fax 317-572-4002.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.
Library of Congress Cataloging-in-Publication Data:
Tamhane, Ajit C.Statistical analysis of designed experiments : theory and applications / Ajit C. Tamhane.p. cm.Includes bibliographical references and index.ISBN 978-0-471-75043-7 (cloth)1. Experimental design. I. Title.
QA279.T36 2008
519.5’7—dc22
2008009432
To All My Teachers — From Grade School to Grad School
Preface
There are many excellent books on design and analysis of experiments, for example, Box, Hunter, and Hunter (2005), Montgomery (2005), and Wu and Hamada (2000), so one may ask, why another book? The answer is largely personal. An instructor who teaches any subject over many years necessarily develops his or her own perspective of how the subject should be taught. Specifically, in my teaching of DOE (a popular abbreviation for design of experiments that I will use below for convenience), I have felt it necessary to put equal emphasis on theory and applications. Also, I have tried to motivate the subject by using real data examples and exercises drawn from a range of disciplines, not just from engineering or medicine, for example, since, after all, the principles of DOE are applicable everywhere. Therefore I wanted to present the subject according to my personal preferences and because this mode of presentation has worked for students in my classes over the years. Accordingly, the primary goal of this book is to provide a balanced coverage of the underlying theory and applications using real data. The secondary goal is to demonstrate the versatility of the DOE methodology by showing applications to wide-ranging areas, including agriculture, biology, education, engineering, marketing, medicine, and psychology. The book is mainly intended for seniors and first-year graduate students in statistics and those in applied disciplines with the necessary mathematical and statistical prerequisites (calculus and linear algebra and a course in statistical methods covering distribution theory, confidence intervals and hypothesis tests, and simple and multiple linear regression). It can also serve as a reference for practitioners who are interested in understanding the “whys” of the designs and analyses that they use and not just the “hows.”
As the title indicates, the main focus of the book is on the analysis; design and planning, although equally if not more important, require discussion of many practical issues, some of which are application-specific, and hence are not emphasized to the same degree. The book provides an in-depth coverage of most of the standard topics in a first course on DOE. Many advanced topics such as nonnormal responses, generalized linear models, unbalanced or missing data, complex aliasing, and optimal designs are not covered. An extensive coverage of these topics would require a separate volume. The readers interested in these topics are referred to a more advanced book such as Wu and Hamada (2000). However, in the future I hope to add short reviews of these topics as supplementary materials on the book’s website. Additional test problems as well as data sets for all the examples and exercises in the book will also be posted on this website.
A model-based approach is followed in the book. Discussion of each new design and its analysis begins with the specification of the underlying model and assumptions. This is followed by inference methods, for example, analysis of variance (ANOVA), confidence intervals and hypothesis tests, and residual analyses for model diagnostics. Derivations of the more important formulas and technical results are given in Chapter Notes at the end of each chapter. All designs are illustrated by fully worked out real data examples. Appropriate graphics accompany each analysis. The importance of using a statistical package to perform computations and for graphics cannot be overemphasized, but some calculations are worked out by hand as, in my experience, they help to enhance understanding of the methods. Minitab® is the main package used for performing analyses as it is one of the easiest to use and thus allows a student to focus on understanding the statistical concepts rather than learning the intricacies of its use. However, any other package would work equally well if the instructor and students are familiar with it. Because of the emphasis on using a statistical package, I have not provided catalogs of designs since many of the standard designs are now available in these packages, particularly those that specialize in DOE.
The book is organized as follows. Chapter 1 introduces the basic concepts and a brief history of DOE, Chapter 2 gives a review of elementary statistical methods through multiple regression. This background is assumed as a prerequisite in the course that I teach, although some instructors may want to go over this material, especially multiple regression using the matrix approach. Chapter 3 discusses the simplest single-factor experiments (one-way layouts) using completely randomized designs with and without covariates. Chapter 4 introduces multiple comparison and selection procedures for one-way layouts. These procedures provide practical alternatives to ANOVA F-tests of equality of treatment means. Chapter 5 returns to the single-factor setup but with randomization restrictions necessitated by blocking over secondary (noise) factors in order to evaluate the robustness of the effects of primary (treatment) factors of interest or to eliminate the biasing effects of secondary factors. This gives rise to randomized block designs (including balanced incomplete block designs), Latin squares, Youden squares and Graeco-Latin squares. Chapter 6 covers two-factor and three-factor experiments. Chapter 7 covers 2p factorial experiments in which each of the p ≥ 2 factors is at two levels. These designs are intended for screening purposes, but become impractically large very quickly as the number of runs increases exponentially with p. To economize on the number of runs without sacrificing the ability to estimate the important main effects and interactions of the factors, 2p – q fractional factorial designs are used which are studied in Chapter 8. This chapter also discusses other fractional factorial designs including Plackett-Burman and Hadamard designs. A common thread among these designs is their orthogonality property. Orthogonal arrays, which provide a general mathematical framework for these designs, are also covered in this chapter. Chapter 9 discusses three-level and mixed-level full and fractional factorial experiments using orthogonal arrays. Response optimization using DOE is discussed in Chapter 10. The methodologies covered include response surface exploration and optimization, mixture experiments and the Taguchi method for robust design of products and processes. In Chapter 11 random and mixed effects models are introduced for single-factor and crossed factors designs. These are extended to nested and crossed-nested factors designs in Chapter 12. This chapter also discusses split-plot factorial designs. These designs are commonly employed in practice because a complete randomization with respect to all factors is not possible since some factors are harder to change than others, and so randomization is done in stages. Chapter 13 introduces repeated measures designs in which observations are taken over time on the same experimental units given different treatments. In addition to the effects due to time trends, the time-series nature of the data introduces correlations among observations which must be taken into account in the analysis. Both univariate and multivariate analysis methods are presented. Finally, Chapter 14 gives a review of the theory of fixed-effects linear models, which underlies the designs discussed in Chapters 3 through 10. Chapters 11 through 13 cover designs that involve random factors, but their general theory is not covered in this book. The reader is referred to the book by Searle, Casella and McCulloch (1992) for this theory.
There are three appendices. Appendix A gives a summary of results about vector-valued random variables and distribution theory of quadratic forms under multivariate normality. This appendix supplements the theory of linear models covered in Chapter 14. Appendix B gives three case studies. Two of these case studies are taken from student projects. These case studies illustrate the level of complexity encountered in real-life experiments that students taking a DOE course based on this book may be expected to deal with. They also illustrate two interesting modern applications in medical imaging and stem cell research. Appendix C contains some useful statistical tables.
Some of the exercises are also based on student projects. The exercises in each chapter are divided by sections and within each section are further divided into theoretical and applied. It is hoped that this will enable both the instructor and the student to choose exercises to suit their theoretical/applied needs. Answers to selected exercises are included at the end. A solution manual will be made available to the instructors from the publisher upon adopting the book in their courses.
Some of the unique features of the book are as follows. Chapter 4 gives a modern introduction to multiple comparison procedures and also to ranking and selection procedures, which are not covered in most DOE texts. Chapter 13 discusses repeated-measures designs, a topic of significant practical importance that is not covered in many texts. Chapter 14 gives a survey of linear model theory (along with the associated distribution theory in Appendix A) that can serve as a concise introduction to the topic in a more theoretically oriented course. Finally, the case studies discussed in Appendix B should give students a taste of complexities of practical experiments, including constraints on randomization, unbalanced data, and outliers.
There is obviously far more material in the book than can be covered in a term-long course. Therefore the instructor must pick and choose the topics. Chapter 1 must, of course, be covered in any course. Chapter 2 is mainly for review and reference; the sections on simple and multiple regression using matrix notation may be covered if students do not have this background. In a two-term graduate course on linear models and DOE, this material can be supplemented with Chapter 14 at a more mathematical depth but also at a greater investment of time. From the remaining chapters, for a one-term course, I suggest Chapters 3, 4, 5, 6, 7, 8, and 11. For a two-term course, Chapters 9, 10, 12, and 13 can be added in the second term. Not all sections from each chapter can be covered in the limited time, so choices will need to be made by the instructor.
As mentioned at the beginning, there are several excellent books on DOE which I have used over the years and from which I have learned a lot. Another book that I have found very stimulating and useful for providing insights into various aspects of DOE is the collection of short articles written for practitioners (many from the Quality Quandaries column in Quality Engineering) by Box and Friends (2006). I want to acknowledge the influence of these books on the present volume. Most examples and exercises use data sets from published sources, which I have tried to cite wherever possible. I am grateful to all publishers who gave permission to use the data sets or figures from their copyrighted publications. I am especially grateful to Pearson Education, Inc. (Prentice Hall) for giving permission without fee for reproducing large portions of material from my book Statistics and Data Analysis: From Elementary to Intermediate with Dorothy Dunlop. Unfortunately, in a few cases, I have lost the original references and I offer my apologies for my inability to cite them. I have acknowledged the students whose projects are used in exercises and case studies individually in appropriate places.
I am grateful to three anonymous reviewers of the book who pointed out many errors and suggested improvements in earlier drafts of the book. I especially want to thank one reviewer who offered very detailed comments, criticisms, and suggestions on the pre–final draft of the book which led to significant revision and rearrangement of some chapters. This reviewer’s comments on the practical aspects of design, analysis, and interpretations of the data sets in examples were particularly useful and resulted in substantial rewriting.
I want to thank Professor Bruce Ankenman of my department at Northwestern for helpful discussions and clarifications about some subtle points. I am also grateful to my following graduate students who helped with collection of data sets for examples and exercises, drawing figures and carrying out many computations: Kunyang Shi, Xin (Cindy) Wang, Jiaxiao Shi, Dingxi Qiu, Lingyun Liu, and Lili Yao. Several generations of students in my DOE classes struggled through early drafts of the manuscript and pointed out many errors, ambiguous explanations, and so on; I thank them all. Any remaining errors are my responsibility.
Finally, I take this opportunity to express my indebtedness to all my teachers — from grade school to grad school — who taught me the value of inquiry and knowledge. This book is dedicated to all of them.
AJIT C. TAMHANE
Department of Industrial Engineering & Management SciencesNorthwestern University, Evanston, IL
Abbreviations
ANCOVA
Analysis of covariance
ANOVA
Analysis of variance
BB
Box–Behnken
BIB
Balanced incomplete block
BLUE
Best linear unbiased estimator
BTIB
Balanced-treatment incomplete block
CC
Central composite
c.d.f.
Cumulative distribution function
CI
Confidence interval
CR
Completely randomized
CWE
Comparisonwise error rate
d.f.
Degrees of freedom
E(MS)
Expected mean square
FCC
Face-centered cube
FDR
False discovery rate
FWE
Familywise error rate
GLS
Generalized least squares
GLSQ
Graeco–Latin square
iff
If and only if
i.i.d.
Independent and identically distributed
IQR
Interquartile range
LFC
Least favorable configuration
LS
Least squares
LSD
Least significant difference
LSQ
Latin square
MANOVA
Multivariate analysis of variance
MCP
Multiple comparison procedure
ML
Maximum likelihood
MOLSQ
Mutually orthogonal Latin square
MS
Mean square
MVN
Multivariate normal
n.c.p.
Noncentrality parameter
OA
Orthogonal array
OC
Operating characteristic
OME
Orthogonal main effect
PB
Plackett–Burman
P(CS)
Probability of correct selection
p.d.f
Probability distribution function
PI
Prediction interval
PSE
Pseudo standard error
Quantile–quantile
RB
Randomized block
REML
Restricted maximum likelihood
RM
Repeated measures
R&R
Reproducibility and repeatability
RSM
Response surface methodology
RSP
Ranking and selection procedure
r.v.
Random variable
SCC
Simultaneous confidence coefficient
SCI
Simultaneous confidence interval
SD
Standard deviation
SE
Standard error
SPC
Statistical process control
SS
Sum of squares
s.t.
Such that
STP
Simultaneous test procedure
UI
Union–intersection
WLS
Weighted least squares
w.r.t.
With respect to
YSQ
Youden square
CHAPTER 1
Introduction
Humans have always been curious about nature. Since prehistoric times, they have tried to understand how the universe around them operates. Their curiosity and ingenuity have led to innumerable scientific discoveries that have fundamentally changed our lives for the better. This progress has been achieved primarily through careful observation and experimentation. Even in cases of serendipity, for example, Alexander Fleming’s discovery of penicillin when a petri dish in which he was growing cultures of bacteria had a clear area (because the bacteria were killed) where a bit of mold had accidentally fallen (Roberts, 1989, pp. 160–161) or Charles Goodyear’s discovery of vulcanization of rubber when he inadvertently allowed a mixture of rubber and sulfur to touch a hot stove (Roberts, 1989, p. 53), experimental confirmation of a discovery is a must. This book is about how to design experiments and analyze the data obtained from them to draw useful conclusions. In this chapter we introduce the basic terminology and concepts of experimentation.
The outline of the chapter is as follows. Section 1.1 contrasts observational studies with experimental studies. Section 1.2 gives a brief history of the subject. Section 1.3 defines the basic terminology and concepts followed by a discussion of principles in Section 1.4. Section 1.5 gives a summary of the chapter.
1.1 OBSERVATIONAL STUDIES AND EXPERIMENTS
Observational studies and experiments are the two primary methods of scientific inquiry. In an observational study the researcher is a passive observer who records variables of interest (often categorized as independent/explanatory variables or factors and dependent/response variables) and draws conclusions about associations between them. In an experiment the researcher actively manipulates the factors and evaluates their effects on the response variables.
For example, an observational study may find that people who exercise regularly live healthier lives. But is it the exercise that makes people healthy or is it something else that makes people exercise regularly and also makes them healthy? After all, there are many other variables such as diet, sleep, and use of medication that can affect a person’s health. People who exercise regularly are likely to be more disciplined in their dietary and sleep habits and hence may be healthy. These variables are not controlled in an observational study and hence may confound the outcome. Only a controlled experiment in which people are randomly assigned to different exercise regimens can establish the effect of exercise on health.
An observational study can only show association, not causation, between the factors of interest (referred to as treatment factors) and the response variable. This is because of possible confounding caused by all other factors that are not controlled (referred to as noise factors) and are often not even recognized to be important to be observed (hence referred to as lurking variables). Any conclusion about cause–effect relationships is further complicated by the fact that some noise factors may affect not only the response variable but also the treatment factors. For example, lack of good diet or sleep may cause a person to get tired quickly and hence not exercise.
Epidemiological studies are an important class of observational studies. In these studies the suspected risk factors of a disease are the treatment factors, and the objective is to find out whether they are associated with the disease. These studies are of two types. In prospective studies, subjects with and without risk factors are followed forward in time and their disease outcome (yes or no) is recorded. In retrospective studies (also called case–control studies), subjects with and without disease are followed backward in time and their exposure to suspected risk factors (yes or no) is recorded. Retrospective studies are practically easier, but their results are more likely to be invalidated or at least more open to question because of uncontrolled lurking variables. This is also a problem in prospective studies, but to a lesser extent. For instance, if a study establishes association between obesity and hypertension, one could argue that both may be caused by a common gene rather than obesity causing hypertension. This general phenomenon of a lurking variable influencing both the predictor variable and the response variable is depicted diagrammatically in Figure 1.1. An even more perplexing possibility is that the roles of “cause” and “effect” may be reversed. For example, a person may choose not to exercise because of poor health.
Figure 1.1 Lurking variable z influences both predictor variable x and response variable y.
On the other hand, an experiment can establish causation, that is, a cause–effect relationship between the treatment factors that are actively changed and the response variable. This is because the treatment factors are controlled by the investigator and so cannot be affected by uncontrolled and possibly unobserved noise factors. Furthermore, selected noise factors may be controlled for experimental purposes to remove their confounding effects, and the effects of the others can be averaged out using randomization; see Section 1.4.
In addition to establishing causation, another advantage of experimentation is that by active intervention in the causal system we can try to improve its performance rather than wait for serendipity to act. Even if an improvement occurs due to serendipity, we are left to guess as to which input variables actually caused the improvement.
The general goal of any experiment is knowledge and discovery about the phenomenon under study. By knowledge we mean a better understanding of the phenomenon; for example, which are the key factors that affect the outcomes of the phenomenon and how. This knowledge can then be used to discover how to make improvements by tuning the key design factors. This process is often iterative or sequential since, as our knowledge base expands, we can make additional adjustments and improvements. The sequential nature of experimentation is discussed in Section 1.4.2.
Some specific goals of an experiment include the following:
Statistics plays a crucial role in the design and analysis of experiments and of observational studies. The design of an experiment involves many practical considerations. Statistics is especially useful in determining the appropriate combinations of factor settings and the necessary sample sizes. This book focuses mainly on the statistical analyses of data collected from designed experiments. Often the same methods of data analysis are used for observational studies and experiments, but as explained above, stronger conclusions are possible from experiments.
1.2 BRIEF HISTORICAL REMARKS
The field of statistical design and analysis of experiments was founded by Sir Ronald A. Fisher (1890–1962) in the 1920s and 1930s while he was working at the Rothamsted Agricultural Experimental Station in England. Fisher was an intellectual giant who made seminal contributions to statistics and genetics. In design of experiments he invented many important basic ideas (e.g., randomization), experimental designs (e.g., Latin squares), and methods of analysis (e.g., analysis of variance) and wrote the first book on the subject (Fisher 1935). Figure 1.2 shows a picture of Fisher in his younger days taken from his excellent biography by his daughter, Joan Fisher-Box (1978). Fisher was followed in his position at Rothamsted by Sir Frank Yates (1902–1994), who proposed novel block designs and factorial designs and their methods of analysis.
Figure 1.2 Sir Ronald A. Fisher in 1924.
(Source: Fisher-Box, 1978, Plate 4).
In the 1940s and 1950s, Sir George Box, while working at the Imperial Chemical Industry, developed response surface methodology (Box and Wilson, 1953) as a statistical method for process optimization. There are some crucial differences between agricultural experimentation, the original setting of the subject, and industrial experimentation, the setting in which Box and his co-workers extended the subject in new directions:
In the 1950s, a mathematical theory of construction of experimental designs based on combinatorial analysis and group theory was developed by Raj Chandra Bose (1901–1987) and others. Later a theory of optimal designs was proposed by Jack Kiefer (1923–1980).
Around the same time, A. Bradford Hill (1897–1991) promoted randomized assignments of patients in clinical trials. Psychology, education, marketing, and other disciplines also witnessed applications of designed experiments. A random assignment of human subjects is not always ethical and sometimes not even practical in social and medical experiments. This led to the development of quasi-experiments in the fields of psychology and education by Donald Campbell (1916–1996) and Julian Stanley.
The most recent infusion of new ideas in design of experiments came from engineering applications, in particular designing quality into manufactured products. The person primarily responsible for this renaissance is the Japanese engineer Genichi Taguchi, who proposed that a product or a process should be designed so that its performance is insensitive to factors that are not easily controlled, such as variations in manufacturing conditions or field operating conditions. The resulting methodology of planning and analysis of experiments is called robust design.
1.3 BASIC TERMINOLOGY AND CONCEPTS OF EXPERIMENTATION
In designed experiments the factors whose effects on the response variable are of primary interest are referred to as treatment factors or design factors. The different settings of a treatment factor are called its levels. Because the experimenter can set the levels of the treatment factors, they are said to be controllable factors. In the health–exercise example, exercise (yes or no) is the treatment factor, whose effect on the subjects’ health is evaluated by comparing a group that follows a prescribed exercise regimen with another group that does not exercise. The other factors that may also possibly affect the response variable can be broadly divided into two categories: noise factors and blocking factors. These are discussed in more detail later.
In this book we restrict discussion to a single response variable but possibly multiple treatment factors. A qualitative factor has categorical (nominal or ordinal) levels, while a quantitative factor has numerical levels. For example, the type of a drug (e.g., three analgesics: aspirin, tylenol, and ibuprofen) is a qualitative factor, while the dose of a drug is a quantitative factor. A particular combination of factor levels is called a treatment combination or simply a treatment. (If there is a single factor, then its levels are the treatments.)
The treatments are generally applied to physical entities (e.g., subjects, items, animals, plots of land) whose responses are then observed. An entity receiving an independent application of a treatment is called an experimental unit. An experimental run is the process of “applying” a particular treatment combination to an experimental unit and recording its response. A replicate is an independent run carried out on a different experimental unit under the same conditions. The importance of independent application of a treatment is worth emphasizing for estimation of replication error (see the next section for a discussion of different errors). If an experimental unit is further subdivided into smaller units on which measurements are made, then they do not constitute replicates, and the sample variance among those measurements does not provide an estimate of replication error. As an example, if a batch of cookie dough is made according to a certain recipe (treatment) from which many cookies are made and are scored by tasters, then the batch would be an experimental unit—not the cookies. To obtain another replicate, another batch of dough must be prepared following the same recipe.
A repeat measurement is another measurement of the same response of a given experimental unit; it is not an independent replicate. Taste scores on cookies made from the same batch and assigned by the same taster can be viewed as repeat measurements assuming that the cookies are fairly homogeneous and the only variation is caused by variation in the taster’s perception of the taste. Sample variance among repeat measurements estimates measurement error—not the replication error that is needed to compare the differences between different recipes for dough. The measurement error is generally smaller than the replication error (as can be seen from the cookie example). If it is incorrectly used to compare recipes, then it may falsely find nonexisting or negligible differences between recipes as significant.
All experimental units receiving the same treatment form a treatment group. Often, an experiment includes a standard or a control treatment, which is used as a benchmark for comparison with other, so-called test treatments. For example, in a clinical trial a new therapy is compared to a standard therapy (called an active control) if one exists or a therapy that contains no medically active ingredient, called a placebo or a passive control (e.g., the proverbial “sugar pill”). All experimental units receiving a control treatment form a control group, which forms a basis for comparison for the treatment group.
Let us now turn to noise and blocking factors. These factors differ from the treatment factors in that they represent intrinsic attributes of the experimental units or the conditions of the experiment and are not externally “applied.” For example, in the exercise experiment the age of a subject (young or old) may be an important factor, as well as diet, medications, and amount of sleep that a subject gets. The noise factors are not controlled or generally not even measured in observational studies. On the other hand, blocking factors are controlled in an experiment because their effects and especially their interactions with the treatment factors (e.g., consistency or lack thereof of the effects of the treatment factors across different categories of experimental units) are of interest since they determine the scope and robustness of the applicability of the treatments. For example, different varieties of a crop (treatment factor) may be compared in an agricultural experiment across different fields (blocking factor) having different growing conditions (soils, weather, etc.) to see whether there is a universal winner with the highest yield in all growing conditions. In designed experiments some noise factors may be controlled and used as blocking factors mainly for providing uniform conditions for comparing different treatments. This use of blocking to reduce the variation or bias caused by noise factors is discussed in the next section.
Example 1.1 (Heat Treatment of Steel: Treatment and Noise Factors)
Suppose that 20 steel samples are available for experimentation. In order to regard them as experimental units, each sample must receive an independent application of furnace heating followed by a quench bath, and the temperatures of each should be independently set in a random order (subject to the condition that all four treatments are replicated five times to have a balanced design). But this may not be feasible in practice. If the engineer can assure us that the furnace and quench bath temperatures are perfectly controllable, then a simpler experiment can be conducted in which 10 samples are heated together in the furnace at one temperature followed by the remaining 10 samples at the other temperature. Each group of 10 samples is then randomly divided into two subgroups of five samples each, which are then quenched at two different temperatures. If replication error is estimated from the samples, it will underestimate the true replication error if the assumption of perfect controllability of furnace and quench bath temperatures is not correct. In this case different methods of analyses are required.
As mentioned above, often multiple treatment factors are studied in a single experiment. An experiment in which the factors are simultaneously varied (in a random order) is called a factorial experiment. In contrast, in a one-factor-at-a-time experiment only one factor is varied at a time, keeping the levels of the other factors fixed. In a full factorial experiment all factor-level combinations are studied, while in a fractional factorial experiment only a subset of them are studied.
In a factorial experiment each factor can be classified as fixed or random. The levels of a fixed factor are chosen because of specific a priori interest in comparing them. For example, consider a clinical trial to compare three different therapies to treat breast cancer: mastectomy, chemotherapy, and radiation therapy. The therapy is then a fixed factor. The levels of a random factor are chosen at random from the population of all levels of that factor. The purpose generally is not to compare the specific levels chosen but rather (i) to estimate the variability of the responses over the population of all levels and (ii) to assess the generalizability of the results to that population. For example, consider an experiment to compare the mean assembly times using two types of fixtures. Suppose three different operators, Tom, Dick, and Harry, are chosen to participate in the experiment. Clearly, the fixture is a fixed factor. The operator would be a fixed factor if Tom, Dick, and Harry are chosen because the experimenter was specifically interested in comparing them or because they are the only operators in the factory. If there are many operators in the factory from whom these three are chosen at random, then the operator would be a random factor. In this latter case, there would be less interest in comparing Tom, Dick, and Harry with each other since they simply happened to be chosen. However, the variability among these three can be used to estimate the variability that could be expected across all operators in the factory. In practice, however, comparisons will and should be made between the chosen operators if there are large differences between them to determine the causes for the differences.
The parameters that quantify how the mean response depends on the levels of a factor are called its effects. For a fixed factor, the effects are fixed quantities and are called fixed effects
