An introduction to generalized linear models

5.4.1 Example: Maximum likelihood estimators for the Normal linear model 5.5 Log-likelihood ratio statistic 5.6 Sampling distribution for the deviance 5.6.1 Example: Deviance for a Binomial model 5.6.2 Example: Deviance for a Normal linear model 5.6.3 Example: Deviance for a Poisson model 5.7 Hypothesis testing 5.7.1 Example: Hypothesis testing for a Normal linear model 5.8 Exercises 6 Normal Linear Models 6.1 Introduction 6.2 Basic results 6.2.1 Maximum likelihood estimation 6.2.2 Least squares estimation 6.2.3 Deviance 6.2.4 Hypothesis testing 6.2.5 Orthogonality 6.2.6 Residuals 6.2.7 Other diagnostics 6.3 Multiple linear regression 6.3.1 Example: Carbohydrate diet 6.3.2 Coefficient of determination, R 2 6.3.3 Model selection 6.3.4 Collinearity 6.4 Analysis of variance 6.4.1 One-factor analysis of variance 6.4.2 Two-factor analysis of variance 6.5 Analysis of covariance 6.6 General linear models 6.7 Non-linear associations 6.7.1 PLOS Medicine journal data 6.8 Fractional polynomials 6.9 Exercises x 7 Binary Variables and Logistic Regression 7.1 Probability distributions 7.2 Generalized linear models 7.3 Dose response models 7.3.1 Example: Beetle mortality 7.4 General logistic regression model 7.4.1 Example: Embryogenic anthers 7.5 Goodness of fit statistics 7.6 Residuals 7.7 Other diagnostics 7.8 Example: Senility and WAIS 7.9 Odds ratios and prevalence ratios 7.10 Exercises 8 Nominal and Ordinal Logistic Regression 8.1 Introduction 8.2 Multinomial distribution 8.3 Nominal logistic regression 8.3.1 Example: Car preferences 8.4 Ordinal logistic regression 8.4.1 Cumulative logit model 8.4.2 Proportional odds model 8.4.3 Adjacent categories logit model 8.4.4 Continuation ratio logit model 8.4.5 Comments 8.4.6 Example: Car preferences 8.5 General comments 8.6 Exercises 9 Poisson Regression and Log-Linear Models 9.1 Introduction 9.2 Poisson regression 9.2.1 Example of Poisson regression: British doctors' smoking and coronary death 9.3 Examples of contingency tables 9.3.1 Example: Cross-sectional study of malignant melanoma 9.3.2 Example: Randomized controlled trial of influenza vaccine xi 9.3.3 Example: Case-control study of gastric and duodenal ulcers and aspirin use 9.4 Probability models for contingency tables 9.4.1 Poisson model 9.4.2 Multinomial model 9.4.3 Product multinomial models 9.5 Log-linear models 9.6 Inference for log-linear models 9.7 Numerical examples 9.7.1 Cross-sectional study of malignant melanoma 9.7.2 Case-control study of gastric and duodenal ulcer and aspirin use 9.8 Remarks 9.9 Exercises 10 Survival Analysis 10.1 Introduction 10.2 Survivor functions and hazard functions 10.2.1 Exponential distribution 10.2.2 Proportional hazards models 10.2.3 Weibull distribution 10.3 Empirical survivor function 10.3.1 Example: Remission times 10.4 Estimation 10.4.1 Example: Exponential model 10.4.2 Example: Weibull model 10.5 Inference 10.6 Model checking 10.7 Example: Remission times 10.8 Exercises 11 Clustered and Longitudinal Data 11.1 Introduction 11.2 Example: Recovery from stroke 11.3 Repeated measures models for Normal data 11.4 Repeated measures models for non-Normal data 11.5 Multilevel models 11.6 Stroke example continued 11.7 Comments 11.8 Exercises xii 12 Bayesian Analysis 12.1 Frequentist and Bayesian paradigms 12.1.1 Alternative definitions of p-values and confidence intervals 12.1.2 Bayes' equation 12.1.3 Parameter space 12.1.4 Example: Schistosoma japonicum 12.2 Priors 12.2.1 Informative priors 12.2.2 Example: Sceptical prior 12.2.3 Example: Overdoses amongst released prisoners 12.3 Distributions and hierarchies in Bayesian analysis 12.4 WinBUGS software for Bayesian analysis 12.5 Exercises 13 Markov Chain Monte Carlo Methods 13.1 Why standard inference fails 13.2 Monte Carlo integration 13.3 Markov chains 13.3.1 The Metropolis-Hastings sampler 13.3.2 The Gibbs sampler 13.3.3 Comparing a Markov chain to classical maximum likelihood estimation 13.3.4 Importance of parameterization 13.4 Bayesian inference 13.5 Diagnostics of chain convergence 13.5.1 Chain history 13.5.2 Chain autocorrelation 13.5.3 Multiple chains 13.6 Bayesian model fit: the deviance information criterion 13.7 Exercises 14 Example Bayesian Analyses 14.1 Introduction 14.2 Binary variables and logistic regression 14.2.1 Prevalence ratios for logistic regression 14.3 Nominal logistic regression 14.4 Latent variable model 14.5 Survival analysis 14.6 Random effects xiii 14.7 Longitudinal data analysis 14.8 Bayesian model averaging 14.8.1 Example: Stroke recovery 14.