Introduction to Statistical Modelling and Inference

Description

The complexity of large-scale data sets (“Big Data”) has stimulated the development of advanced computational methods for analysing them. There are two different kinds of methods to aid this. The model-based method uses probability models and likelihood and Bayesian theory, while the model-free method does not require a probability model, likelihood or Bayesian theory. These two approaches are based on different philosophical principles of probability theory, espoused by the famous statisticians Ronald Fisher and Jerzy Neyman. Introduction to Statistical Modelling and Inference covers simple experimental and survey designs, and probability models up to and including generalised linear (regression) models and some extensions of these, including finite mixtures. A wide range of examples from different application fields are also discussed and analysed. No special software is used, beyond that needed for maximum likelihood analysis of generalised linear models. Students are expected to have a basic mathematical background in algebra, coordinate geometry and calculus. Features • Probability models are developed from the shape of the sample empirical cumulative distribution function (cdf) or a transformation of it. • Bounds for the value of the population cumulative distribution function are obtained from the Beta distribution at each point of the empirical cdf. • Bayes’s theorem is developed from the properties of the screening test for a rare condition. • The multinomial distribution provides an always-true model for any randomly sampled data. • The model-free bootstrap method for finding the precision of a sample estimate has a model-based parallel – the Bayesian bootstrap – based on the always-true multinomial distribution. • The Bayesian posterior distributions of model parameters can be obtained from the maximum likelihood analysis of the model. This book is aimed at students in a wide range of disciplines including Data Science. The book is based on the model-based theory, used widely by scientists in many fields, and compares it, in less detail, with the model-free theory, popular in computer science, machine learning and official survey analysis. The development of the model-based theory is accelerated by recent developments in Bayesian analysis. Cover Half Title Title Page Copyright Page Table of Contents Preface Chapter 1 Introduction 1.1 What is Statistical Modelling? 1.2 What is Statistical Analysis? 1.3 What is Statistical Inference? Chapter 2 What is (or are) Big Data? Chapter 3 Data and research studies 3.1 Lifetimes of radio transceivers 3.2 Clustering of V1 missile hits in South London 3.3 Court case on vaccination risk 3.4 Clinical trial of Depepsen for the treatment of duodenal ulcers 3.5 Effectiveness of treatments for respiratory distress in newborn babies 3.6 Vitamin K 3.7 Species counts 3.8 Toxicology in small animal experiments 3.9 Incidence of Down’s syndrome in four regions 3.10 Fish species in lakes 3.11 Absence from school 3.12 Hostility in husbands of suicide attempters 3.13 Tolerance of racial intermarriage 3.14 Hospital bed use 3.15 Dugong growth 3.16 Simulated motorcycle collision 3.17 Global warming 3.18 Social group membership Chapter 4 The StatLab database 4.1 Types of variables 4.2 StatLab population questions Chapter 5 Sample surveys – should we believe what we read? 5.1 Women and Love 5.2 Would you have children? 5.3 Representative sampling 5.4 Bias in the Newsday sample 5.5 Bias in the Women and Love sample Chapter 6 Probability 6.1 Relative frequency 6.2 Degree of belief 6.3 StatLab dice sampling 6.4 Computer sampling 6.4.1 Natural random processes 6.5 Probability for sampling 6.5.1 Extrasensory perception 6.5.2 Representative sampling 6.6 Probability axioms 6.6.1 Dice example 6.6.2 Coin tossing 6.7 Screening tests and Bayes’s theorem 6.8 The misuse of probability in the Sally Clark case 6.9 Random variables and their probability distributions 6.9.1 Definitions 6.10 Sums of independent random variables Chapter 7 Statistical inference I – discrete distributions 7.1 Evidence-based policy 7.2 The basis of statistical inference 7.3 The survey sampling approach 7.4 Model-based inference theories 7.5 The likelihood function 7.6 Binomial distribution 7.6.1 The binomial likelihood function 7.6.1.1 Sufficient and ancillary statistics 7.6.1.2 The maximum likelihood estimate (MLE) 7.7 Frequentist theory 7.7.1 Parameter transformations 7.7.2 Ambiguity of notation 7.8 Bayesian theory 7.8.1 Bayes’s theorem 7.8.2 Summaries of the posterior distribution 7.8.3 Conjugate prior distributions 7.8.4 Improving frequentist interval coverage 7.8.5 The bootstrap 7.8.6 Non-informative prior rules 7.8.7 Frequentist objections to flat priors 7.8.8 General prior specifications 7.8.9 Are parameters really just random variables? 7.9 Inferences from posterior sampling 7.9.1 The precision of posterior draws 7.10 Sample design 7.11 Parameter transformations 7.12 The Poisson distribution 7.12.1 Poisson likelihood and ML 7.12.2 Bayesian inference 7.12.3 Prediction of a new Poisson value 7.12.4 Side effect risk 7.12.4.1 Frequentist analysis 7.12.4.2 Bayesian analysis 7.12.5 A two-parameter binomial distribution 7.12.5.1 Frequentist analysis 7.12.5.2 Bayesian analysis 7.13 Categorical variables 7.13.1 The multinomial distribution 7.14 Maximum likelihood 7.15 Bayesian analysis 7.15.1 Posterior sampling 7.15.2 Sampling without replacement Chapter 8 Comparison of binomials 8.1 Definition 8.2 Example – RCT of Depepsen for the treatment of duodenal ulcers 8.2.1 Frequentist analysis: confidence interval 8.2.2 Bayesian analysis: credible interval 8.3 Monte Carlo simulation 8.4 RCT continued 8.5 Bayesian hypothesis testing/model comparison 8.5.1 The null and alternative hypotheses, and the two models 8.6 Other measures of treatment difference 8.6.1 Frequentist analysis: hypothesis testing 8.6.2 How are the hypothetical samples to be drawn? 8.6.3 Conditional testing 8.7 The ECMO trials 8.7.1 The first trial 8.7.2 Frequentist analysis 8.7.3 The likelihood 8.7.3.1 Bayesian Analysis 8.7.4 The second ECMO study Chapter 9 Data visualisation 9.1 The histogram 9.2 The empirical mass and cumulative distribution functions 9.3 Probability models for continuous variables Chapter 10 Statistical inference II – the continuous exponential, Gaussian and uniform distributions 10.1 The exponential distribution 10.2 The exponential likelihood 10.3 Frequentist theory 10.3.1 Parameter transformations 10.3.2 Frequentist asymptotics 10.4 Bayesian theory 10.4.1 Conjugate priors 10.5 The Gaussian distribution 10.6 The Gaussian likelihood function 10.7 Frequentist inference 10.8 Bayesian inference 10.8.1 Prior arguments 10.9 Hypothesis testing 10.10 Frequentist hypothesis testing 10.10.1 µ1 vs µ2 10.10.2 µ0 vs µ .= µ0 10.11 Bayesian hypothesis testing 10.11.1 µ1 vs µ2 10.11.2 µ0 vs µ .= µ0 10.11.2.1 Use the credible interval 10.11.2.2 Use the likelihood ratio 10.11.2.3 The integrated likelihood 10.12 Pivotal functions 10.13 Conjugate priors 10.14 The uniform distribution 10.14.1 The location-shifted uniform distribution Chapter 11 Statistical Inference III – two-parameter continuous distributions 11.1 The Gaussian distribution 11.2 Frequentist analysis 11.3 Bayesian analysis 11.3.1 Inference for σ 11.3.2 Inference for μ 11.3.2.1 Simulation marginalisation 11.3.3 Parametric functions 11.3.4 Prediction of a new observation 11.4 The lognormal distribution 11.4.1 The lognormal density 11.5 The Weibull distribution 11.5.1 The Weibull likelihood 11.5.2 Frequentist analysis 11.5.3 Bayesian analysis 11.5.4 The extreme value distribution 11.5.5 Median Rank Regression (MRR) 11.5.6 Censoring 11.6 The gamma distribution 11.7 The gamma likelihood 11.7.1 Frequentist analysis 11.7.2 Bayesian analysis Chapter 12 Model assessment 12.1 Gaussian model assessment 12.2 Lognormal model assessment 12.3 Exponential model assessment 12.4 Weibull model assessment 12.5 Gamma model assessment Chapter 13 The multinomial distribution 13.1 The multinomial likelihood 13.2 Frequentist analysis 13.3 Bayesian analysis 13.4 Criticisms of the Haldane prior 13.4.1 The Dirichlet process prior 13.4.2 Posterior sampling 13.5 Inference for multinomial quantiles 13.6 Dirichlet posterior weighting 13.7 The frequentist bootstrap 13.7.1 Two-category sample 13.8 Stratified sampling and weighting Chapter 14 Model comparison and model averaging 14.1 Comparison of two fully specified models 14.2 General model comparison 14.2.1 Known parameters 14.2.2 Unknown parameters 14.3 Posterior distribution of the likelihood 14.4 The deviance 14.5 Asymptotic distribution of the deviance 14.6 Nested models 14.7 Model choice and model averaging Chapter 15 Gaussian linear regression models 15.1 Simple linear regression 15.1.1 Vitamin K 15.2 Model assessment through residual examination 15.3 Likelihood for the simple linear regression model 15.4 Maximum likelihood 15.4.1 Vitamin K example 15.5 Bayesian and frequentist inferences 15.6 Model-robust analysis 15.6.1 The robust variance estimate 15.7 Correlation and prediction 15.7.1 Correlation 15.7.2 Prediction 15.7.3 Example 15.7.4 Prediction as a model assessment tool 15.8 Probability model assessment 15.9 “Dummy variable” regression 15.10 Two-variable models 15.11 Model assumptions 15.12 The p-variable linear model 15.13 The Gaussian multiple regression likelihood 15.13.1 Absence from school 15.14 Interactions 15.14.1 ANOVA, ANCOVA and MR 15.14.1.1 ANOVA 15.14.1.2 Backward elimination 15.14.1.3 ANCOVA 15.15 Ridge regression, the Lasso and the “elastic net" 15.16 Modelling boy birthweights 15.17 Modelling girl intelligence at age ten and family income 15.18 Modelling of the hostility data 15.18.1 Data structure 15.18.1.1 Replication and variance heterogeneity 15.19 Principal component regression Chapter 16 Incomplete data and their analysis with the EM and DA algorithms 16.1 The general incomplete data model 16.2 The EM algorithm 16.3 Missingness 16.4 Lost data 16.5 Censoring in the exponential distribution 16.6 Randomly missing Gaussian observations 16.7 Missing responses and/or covariates in simple and multiple regression 16.7.1 Missing values in the single covariate in simple linear regression 16.7.2 Modelling the covariate distribution – Gaussian 16.7.3 Modelling the covariate distribution – multinomial 16.7.4 Multiple covariates missing 16.8 Mixture distributions 16.8.1 The two-component Gaussian mixture model 16.9 Bayesian analysis and the Data Augmentation algorithm 16.9.1 The galaxy recession velocity study 16.9.2 The Dirichlet process prior Chapter 17 Generalised linear models (GLMs) 17.1 The exponential family 17.2 Maximum likelihood 17.3 The GLM algorithm 17.4 Bayesian package development 17.5 Bayesian analysis from ML 17.6 Binary response models 17.6.1 Probit or logit analysis? 17.6.2 Other binomial link functions and their origins 17.6.3 The Racine data 17.6.4 Maximum likelihood 17.6.5 Bayesian analysis 17.6.6 The beetle data 17.7 The menarche data 17.7.1 Down’s syndrome analysis 17.7.1.1 BC analysis 17.7.1.2 Four regions analysis 17.7.2 The Finney vasoconstriction data 17.7.3 Cross-classifications with binary data 17.7.3.1 Region 1 17.7.3.2 Region 2 17.7.3.3 Region 3 17.7.3.4 Region 4 17.7.3.5 Observed and (fitted) proportions, all regions 17.8 Poisson regression – fish species frequency 17.8.1 Gaussian approximation 17.8.2 The Bayesian bootstrap and posterior weighting 17.8.3 Omitted variables, overdispersion and the negative binomial model 17.8.4 Conjugate W 17.9 Gamma regression Chapter 18 Extensions of GLMs 18.1 Double GLMs 18.2 Maximum likelihood 18.3 Bayesian analysis 18.3.1 Hospital beds and patients 18.3.2 The absence from school data 18.3.3 The fish species data 18.3.4 Sea temperatures 18.3.5 Model assessment 18.4 Segmented or broken-stick regressions 18.4.1 Nile flood volumes 18.4.2 Modelling the break 18.4.3 Down’s syndrome 18.5 Heterogeneous regressions 18.6 Highly non-linear functions 18.7 Neural networks 18.8 Social networks and social group membership 18.8.1 History of network structures 18.8.2 The Natchez women network 18.8.3 Statistical models 18.8.3.1 The “null” random graph model 18.8.3.2 The “saturated” model 18.8.3.3 The Rasch model 18.8.4 The Exponential Random Graph Model (ERGM) 18.8.4.1 The latent class or mixed Rasch model 18.9 The motorcycle data Chapter 19 Appendix 1 – length-biased sampling Chapter 20 Appendix 2 – two-component Gaussian mixture Chapter 21 Appendix 3 – StatLab variables 21.1 Child variables 21.2 Family variables 21.3 Mother variables 21.4 Father variables Chapter 22 Appendix 4 – a short history of statistics from 1890 22.1 Karl Pearson (1857–1936) 22.2 Ronald Fisher (1890–1962) 22.3 Jerzy Neyman (1894–1981) 22.4 Harold Jeffreys (1891–1989) Chapter 23 References Index