Introduction to the Design and Analysis of Experiments Geoffrey M. Clarke, MA,Dip.stats.,c.stat. Honorary Reader in Applied Statistics, University of Kent at Canterbury and Consultant to the Applied Statistics Research Unit Robert E. Kempson, B.SC, M.SC, Ph.D., c.stat. formerly of the Applied Statistics Research Unit, University of Kent at Canterbury and of Wye College, University of London SUB Gdttlngen 8 393 382 98 A755 A member of the Hodder Headline Group LONDON SYDNEY AUCKLAND Copublished in North, Central and South America by John Wiley & Sons, Inc., New York Toronto
Contents Preface (Collecting data by experiments. Introduction.2 Experiments.3 Measurements of yield or response.4 Natural variation in data.5 Initial data analysis.6 General applications of experimentation.7 Exercises 2 Basic statistical methods: the normal distribution 2. Statistical inference for one sample of normally distributed data 2.2 Hypothesis test 2.3 Comparison of two samples of normally distributed data 2.4 The F-test for comparing two estimated variances 2.5 Confidence interval for the difference between two means 2.6 'Paired data' f-test when samples are not independent 2.7 Linear functions of normally distributed variables 2.8 Linear models including normal random variation 2.9 Exercises 3 Principles of experimental design 3. 3.2 3.3 3.4 Introduction Treatment structure Changing background conditions - the need for comparison Replication * 3.5 Randomization 3.6 Blocking 3.7 3.8 3.9 Sources of variation Planning the size of an experiment Exercises 4 The 4. 4.2 i.3 analysis of data from orthogonal designs Introduction Comparing, treatments Confidence intervals vii 3 4 5 7 8 9 9 0 4 5 5 7 8 8 22 22 23 26 27 29 30 32 32 38 40
iv Contents 4.4 Homogeneity of variance 40 4.5 The randomized complete block 42 4.6 Duncan's multiple range test 46 4.7 Extra replication of important treatments 47 4.8 Contrasts among treatments 47 4.9 Latin squares and other orthogonal designs 52 4.0 Graeco-Latin squares 58 4. Two fallacies 6 4.2 Assumptions in analysis: using residuals to examine them 63 4.3 Transformations 67 4.4 Theory of variance stabilization 67 4.5 Missing data in block designs 69 4.6 Exercises 7 Appendix 4A Cochran's Theorem on Quadratic Forms 78 5 Factorial experiments 80 5. Introduction 80 5.2 Notation for factors at two levels 82 5.3 Definition of main effect and interaction 82 5.4 Three factors each at two levels 87 5.5 A single factor at more than two levels 90 5.6 General method for computing coefficients for orthogonal polynomials 96 5.7 Exercises 98 6 Experiments with many factors: confounding and fractional replication 04 6. Introduction 04 6.2 The principal block in confounding 07 6.3 Single replicate 09 6.4 Small experiments: partial confounding 2 6.5 Very large experiments: fractional replication 4 6.6 Replicates smaller than half size 7 6.7 Confounding with fractional replication 8 6.8 Confounding three-level factors * 2 6.9 Fractional replication in 3-level experiments - 25 6.0 Exercises 26 Appendix 6A Methods of confounding in V factorial experiments 3 7 Confounding main effects - split-plot designs 33 7. Introduction 33 7.2 Linear model and analysis 34 7.3 Studying interactions 38 7.4 Repeated splitting 39 7.5 Confounding in split-plot experiments 39 7.6 Other designs for main plots 4 7.7 Criss-cross design 4 7.8 Exercises 44
Contents v 8 Industrial experimentation 48 8. Introduction 48 8.2 Taguchi methods in statistical quality control 48 8.3 Loss functions 49 8.4 Sources s>f variation 52 8.5 Orthogonal arrays 54 8.6 Choice of design 57 9 Response surfaces and mixture designs 66 9. Introduction 66 9.2 Are experimental conditions 'constant'? 66 9.3 Response surfaces 67 9.4 Experiments with three factors, x x, x 2 and x 3 69 9.5 Second-order surfaces 74 9.6 Contour diagrams in analysis 77 9.7 Transformations 79 9.8 Mixture designs 80 9.9 Other types of response surface 87 9.0 Exercises 9 0 The analysis of covariance 94 0. Introduction 94 0.2 Analysis for a design in randomized blocks: general theory 95 0.3 Individual contrasts 2 0.4 Dummy covariates 3 0.5 Systematic trend not removed by blocking 5 0.6 Accidents in recording 6 0.7 Assumptions in covariance analysis 6 0.8 Missing values 7 0.9 Double covariance 8 0.0 Exercises 8 Balanced incomplete blocks and general non-orthogonal block designs 22. Introduction < 22.2 Definition and existence of a balanced incomplete block ; ^ 22.3 Methods of construction 24.4 Linear model and analysis - 25.5 Row and column design: the Youden square 28.6 General block designs 29.7 Linear model and analysis 222.8 Generalized inverses 224.9 Application to designs with special patterns 227.0 Exercises 230 Appendix A Generalized inverse matrix by spectral decomposition 238 Appendix B Natural contrasts and effective replication 240
vi Contents 2 More advanced designs 24 2. Introduction 24 2.2 Crossover designs 24 2.3 Lattices 243 2.4 Alpha designs 245 2.5 Partially balanced incomplete blocks (PBIBs) 247 3 Random effects models: variance components and sampling schemes 250 3. Introduction 250 3.2 Two stages of sampling: between and within units 250 3.3 Assessing alternative sampling schemes 253 3.4 Using variance components in planning when sampling costs are given 254 3.5 Three levels of variation 256 3.6 Costs in a three-stage scheme 259 3.7 Example where one estimate is negative 26 3.8 Exercises 263 4 Computer output using SAS 266 Bibliography and references 332 Tables 336 Index 343