Introduction to the Design and Analysis of Experiments

Transcription

1 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 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

2 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 Introduction Treatment structure Changing background conditions - the need for comparison Replication * 3.5 Randomization 3.6 Blocking Sources of variation Planning the size of an experiment Exercises 4 The i.3 analysis of data from orthogonal designs Introduction Comparing, treatments Confidence intervals vii

3 iv Contents 4.4 Homogeneity of variance The randomized complete block Duncan's multiple range test Extra replication of important treatments Contrasts among treatments Latin squares and other orthogonal designs Graeco-Latin squares Two fallacies Assumptions in analysis: using residuals to examine them Transformations Theory of variance stabilization Missing data in block designs Exercises 7 Appendix 4A Cochran's Theorem on Quadratic Forms 78 5 Factorial experiments Introduction Notation for factors at two levels Definition of main effect and interaction Three factors each at two levels A single factor at more than two levels General method for computing coefficients for orthogonal polynomials Exercises 98 6 Experiments with many factors: confounding and fractional replication Introduction The principal block in confounding Single replicate Small experiments: partial confounding Very large experiments: fractional replication Replicates smaller than half size Confounding with fractional replication Confounding three-level factors * Fractional replication in 3-level experiments Exercises 26 Appendix 6A Methods of confounding in V factorial experiments 3 7 Confounding main effects - split-plot designs Introduction Linear model and analysis Studying interactions Repeated splitting Confounding in split-plot experiments Other designs for main plots Criss-cross design Exercises 44

4 Contents v 8 Industrial experimentation Introduction Taguchi methods in statistical quality control Loss functions Sources s>f variation Orthogonal arrays Choice of design 57 9 Response surfaces and mixture designs Introduction Are experimental conditions 'constant'? Response surfaces Experiments with three factors, x x, x 2 and x Second-order surfaces Contour diagrams in analysis Transformations Mixture designs Other types of response surface Exercises 9 0 The analysis of covariance Introduction Analysis for a design in randomized blocks: general theory Individual contrasts Dummy covariates Systematic trend not removed by blocking Accidents in recording Assumptions in covariance analysis Missing values Double covariance 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 Row and column design: the Youden square 28.6 General block designs 29.7 Linear model and analysis Generalized inverses Application to designs with special patterns Exercises 230 Appendix A Generalized inverse matrix by spectral decomposition 238 Appendix B Natural contrasts and effective replication 240

5 vi Contents 2 More advanced designs Introduction Crossover designs Lattices Alpha designs Partially balanced incomplete blocks (PBIBs) Random effects models: variance components and sampling schemes Introduction Two stages of sampling: between and within units Assessing alternative sampling schemes Using variance components in planning when sampling costs are given Three levels of variation Costs in a three-stage scheme Example where one estimate is negative Exercises Computer output using SAS 266 Bibliography and references 332 Tables 336 Index 343