Ten Projects in Applied Statistics

Description

Preface Goals Computation Organization Acknowledgments Contents 1 Rat Surgery 1.1 Healing of Surgical Wounds 1.2 An Elementary Analysis 1.3 Two Incorrect Analyses 1.4 Model Formulae 1.5 A More Appropriate Formal Analysis 1.6 Further Issues 1.6.1 Exclusions 1.6.2 Missing Components 1.6.3 Back-Transformation 1.7 Summary of Statistical Concepts 1.8 Exercises 2 Chain Saws 2.1 Efficiency of Chain Saws 2.2 Covariate and Treatment Factors 2.3 Goals of Statistical Analysis 2.4 Formal Models 2.5 REML and Likelihood Ratios 2.6 Summary of Conclusions 2.7 Exercises 3 Fruit Flies 3.1 Diet and Mating Preferences 3.2 Initial Analyses 3.2.1 Assortative Mating 3.2.2 Initial Questions and Exercises 3.3 Refractory Effects 3.3.1 More Specific Mating Counts 3.3.2 Follow-Up Analyses 3.3.3 Lexis Dispersion 3.3.4 Is Under-Dispersion Possible? 3.3.5 Independence 3.3.6 Acknowledgement 3.4 Technical Points 3.4.1 Hypergeometric Simulation by Random Matching 3.4.2 Pearson's Statistic 3.5 Further Drosophila Project 3.6 Exercises 4 Growth Curves 4.1 Plant Growth: Data Description 4.2 Growth Curve Models 4.3 Technical Points 4.3.1 Non-linear Model with Variance Components 4.3.2 Fitted Versus Predicted Values 4.4 Modelling Strategies 4.5 Miscellaneous R Functions 4.6 Exercises 5 Louse Evolution 5.1 Evolution of Lice on Captive Pigeons 5.1.1 Background 5.1.2 Experimental Design 5.1.3 Deconstruction of the Experimental Design 5.2 Data Analysis 5.2.1 Role of Tables and Graphs 5.2.2 Trends in Mean Squares 5.2.3 Initial Values and Factorial Subspaces 5.2.4 A Simple Variance-Components Model 5.2.5 Conformity with Randomization 5.3 Critique of Published Claims 5.4 Further Remarks 5.4.1 Role of Louse Sex 5.4.2 Persistence of Initial Patterns 5.4.3 Observational Units 5.5 Follow-Up 5.5.1 New Design Information 5.5.2 Modifications to Analyses 5.5.3 Further Remarks 5.6 Exercises 6 Time Series I 6.1 A Meteorological Temperature Series 6.2 Seasonal Cycles 6.2.1 Means and Variances 6.2.2 Skewness and Kurtosis 6.3 Annual Statistics 6.3.1 Means and Variances 6.3.2 Variance of Block Averages 6.3.3 Variogram at Short and Long Lags 6.4 Stochastic Models for the Seasonal Cycle 6.4.1 Structure of Observational Units 6.4.2 Seasonal Structure 6.4.3 Stationary Periodic Processes 6.5 Estimation of Secular Trend 6.5.1 Gaussian Estimation and Prediction 6.5.2 Application to Trend Estimation 6.5.3 Matérn Models 6.5.4 Statistical Tests and Likelihood Ratios 6.5.5 Rough Paths Versus Smooth Paths 6.5.6 Smooth Versus Ultra-Smooth Paths 6.6 Exercises 7 Time Series II 7.1 Frequency-Domain Analyses 7.1.1 Fourier Transformation 7.1.2 Anova Decomposition by Frequency 7.2 Temperature Spectrum 7.2.1 Spectral Plots 7.2.2 A Parametric Spectral Model 7.3 Stationary Temporal Processes 7.3.1 Stationarity 7.3.2 Visualization of Trajectories 7.3.3 Whittle Likelihood 7.4 Exercises 8 Out of Africa 8.1 Linguistic Diversity 8.2 Phoneme Inventory 8.3 Distances 8.4 Maps and Scatterplots 8.5 Point Estimates and Confidence Regions 8.5.1 Simple Version Shortcomings 8.5.2 Accommodating Correlations Three Points of Clarification Shortcomings 8.6 Matters for Further Consideration 8.6.1 Phoneme Inventory as Response 8.6.2 Vowels, Consonants and Tones 8.6.3 Granularity 8.7 Follow-Up Project 8.7.1 Extended Data Frame 8.7.2 An Elementary Misconception 8.8 Exercises 9 Environmental Projects 9.1 Effects of Atmospheric Warming 9.1.1 The Experiment 9.1.2 The Data 9.1.3 Exercises 9.2 The Plight of the Bumblebee 9.2.1 Introduction 9.2.2 Risk of Infection 9.2.3 Mixed Models 9.2.4 Exchangeability 9.2.5 Role of GLMs and GLMMs 9.3 Two Further Projects 9.4 Exercises 10 Fulmar Fitness 10.1 The Eynhallow Colony 10.1.1 Background 10.1.2 The Eynhallow Breeding Record 10.1.3 The Breeding Sequence 10.1.4 Averages for Cohorts 10.1.5 Averages for Disjoint Subsets 10.1.6 Resolution of a Paradox 10.2 Formal Models 10.2.1 A Linear Gaussian Model 10.2.2 Prediction 10.2.3 Model Adequacy 10.3 Mark-Recapture Designs 10.4 Further References 10.5 Exercises 11 Basic Concepts 11.1 Stochastic Processes 11.1.1 Process 11.1.2 Probability 11.1.3 Self-consistency 11.1.4 Statistical Model 11.2 Samples 11.2.1 Baseline 11.2.2 Observational Unit 11.2.3 Population 11.2.4 Biological Populations 11.2.5 Samples and Sub-samples 11.2.6 Illustrations 11.3 Variables 11.3.1 Ordinary Variables Quantitative Variable Qualitative Variable Response Covariate Treatment External Variable 11.3.2 Relationship Block Factor 11.3.3 External Variable 11.4 Comparative Studies 11.4.1 Randomization 11.4.2 Experimental Unit 11.4.3 Covariate and Treatment Effects 11.4.4 Additivity 11.4.5 Design 11.4.6 Replication 11.4.7 Independence 11.4.8 Interference 11.4.9 State Space 11.4.10 State-Space Evolution 11.4.11 Longitudinal Study 11.4.12 Cemetery State 11.5 Non-comparative Studies 11.5.1 Examples 11.5.2 Stratified Population 11.5.3 Heterogeneity 11.5.4 Random Sample 11.5.5 Stratified Random Sample 11.5.6 Accessibility 11.5.7 Population Averages 11.5.8 Target of Estimation I 11.5.9 Inverse Probability Weighting 11.5.10 Target of Estimation II 11.6 Interpretations of Variability 11.6.1 A Tale of Two Variances 11.6.2 Which Variance Is Appropriate? 11.7 Exercises 12 Principles 12.1 Sampling Consistency 12.2 Adequacy for the Application 12.3 Likelihood Principle 12.4 Attitudes 12.5 Exercises 13 Initial Values 13.1 Randomization Protocols 13.2 Four Gaussian Models 13.2.1 Distribution and Likelihood 13.2.2 Numerical Comparison of Estimates 13.2.3 Initial Values Versus Covariates 13.2.4 Initial Values in an Observational Study 13.3 Exercises 14 Probability Distributions 14.1 Exchangeable Processes 14.1.1 Unconditional Exchangeability 14.1.2 Regression Processes 14.1.3 Block Exchangeability 14.1.4 Stationarity 14.1.5 Exchangeability 14.1.6 Axiomatic Point 14.1.7 Block Randomization 14.2 Families with Independent Components 14.2.1 Parametric Models 14.2.2 IID Model I 14.2.3 IID Model II 14.3 Non-i.d. Models 14.3.1 Classification Factor 14.3.2 Treatment 14.3.3 Classification Factor Plus Treatment 14.3.4 Quantitative Covariate Plus Treatment 14.3.5 Random Coefficient Models 14.4 Examples of Treatment Effects 14.4.1 Simple Gaussian Model Without Interaction 14.4.2 Additive Interaction 14.4.3 Survival Models Hazard Multiplication Temporal Dilation Non-constant Hazard Multiplication Classification Factor Plus Treatment 14.5 Incomplete Processes 14.5.1 Gosset Process 14.5.2 Factual and Counterfactual Processes 14.5.3 Limitations of Incomplete Processes 14.6 Exercises 15 Gaussian Distributions 15.1 Real Gaussian Distribution 15.1.1 Density and Moments 15.1.2 Gaussian Distribution on Rn 15.2 Complex Gaussian Distribution 15.2.1 One-Dimensional Distribution 15.2.2 Gaussian Distribution on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper C Superscript n) /StPNE pdfmark [/StBMC pdfmarkCnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 15.2.3 Moments 15.3 Gaussian Hilbert Space 15.3.1 Euclidean Structure 15.3.2 Cautionary Remarks 15.3.3 Projections Specification by Image Specification by Kernel Self-adjointness Identity Mixed Products Trace and Rank Rank Degeneracy 15.3.4 Dual Space of Linear Combinations 15.4 Statistical Interpretations 15.4.1 Canonical Norm 15.4.2 Independence Cochran's Theorem 15.4.3 Prediction and Conditional Expectation Partitioned Matrix Representation Example: Exchangeable Gaussian Process 15.4.4 Eddington's Formula Scalar Signal Estimation Isotropic Vector Signal Estimation Spectral Moments for Matrix Reconstruction 15.4.5 Linear Regression 15.4.6 Linear Regression and Prediction Notation for Component-Wise Transformation Fiducial Prediction 15.5 Additivity 15.5.1 1DOFNA Algorithm 15.5.2 1DOFNA Theory 15.5.3 Scope and Rationale 15.6 Exercises 16 Space-Time Processes 16.1 Gaussian Processes 16.2 Stationarity and Isotropy 16.2.1 Definitions 16.2.2 Stationarity on Increments 16.2.3 Stationary Process on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper Z left parenthesis mod k right parenthesis) /StPNE pdfmark [/StBMC pdfmarkZ 8mu(mod6muk)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 16.3 Stationary Gaussian Time Series 16.3.1 Spectral Representation 16.3.2 Matérn Class 16.4 Stationary Spatial Process 16.4.1 Spectral Decomposition 16.4.2 Matérn Spatial Class Spectral Convolution Frequency Translation Decomposition of Spectral Measure Domain Restriction 16.4.3 Illustration by Simulation 16.5 Covariance Products 16.5.1 Hadamard Product 16.5.2 Separable Products and Tensor Products 16.6 Real Spatio-Temporal Process 16.6.1 Covariance Products 16.6.2 Examples of Covariance Products Patterned Covariance Matrices Complex Moments Complex Covariance Product A 3D Real Process 16.6.3 Travelling Wave 16.6.4 Perturbation Theory 16.7 Hydrodynamic Processes 16.7.1 Frame of Reference 16.7.2 Rotation and Group Action 16.7.3 Action on Matrices 16.7.4 Borrowed Products 16.7.5 Hydrodynamic Symmetry 16.8 Summer Cloud Cover in Illinois 16.9 More on Gaussian Processes 16.9.1 White Noise 16.9.2 Limit Processes Existence of a Limit Process Existence of a Limit Distribution Limit of Conditional Distributions Conditional Distributions for the Limit Process Limit Process as a Markov Kernel 16.10 Exercises 17 Likelihood 17.1 Introduction 17.1.1 Non-Bayesian Model 17.1.2 Bayesian Resolution 17.2 Likelihood Function 17.2.1 Definition 17.2.2 Bartlett Identities 17.2.3 Implications for Estimation 17.2.4 Likelihood-Ratio Statistic I 17.2.5 Profile Likelihood 17.2.6 Two Worked Examples Example 1: Treatment Effect Estimation Example 2: Inference for the LD90 17.3 Generalized Linear Models 17.4 Variance-Components Models 17.5 Mixture Models 17.5.1 Two-Component Mixtures 17.5.2 Likelihood-Ratio Statistic 17.5.3 Sparse Signal Detection 17.6 Inferential Compromises 17.7 Exercises 18 Residual Likelihood 18.1 Background 18.2 Simple Linear Regression 18.3 The REML Likelihood 18.3.1 Projections 18.3.2 Determinants 18.3.3 Marginal Likelihood with Arbitrary Kernel 18.3.4 Likelihood Ratios 18.4 Computation 18.4.1 Software Options 18.4.2 Likelihood-Ratios 18.4.3 Testing for Interaction 18.4.4 Singular Models 18.5 Exercises 19 Response Transformation 19.1 Likelihood for Gaussian Models 19.2 Box-Cox Transformation 19.2.1 Power Transformation 19.2.2 Re-scaled Power Transformation 19.2.3 Worked Example 19.2.4 Transformation and Residual Likelihood 19.3 Quantile-Matching Transformation 19.4 Exercises 20 Presentations and Reports 20.1 Coaching Tips I 20.2 Coaching Tips II 20.3 Exercises 21 Q & A 21.1 Scientific Investigations 21.1.1 Observational Unit 21.1.2 Clinical Trials 21.1.3 Agricultural Field Trials 21.1.4 Covariates 21.1.5 Matched Design 21.1.6 The Effect of Treatment References Index