Actuarial Notes

An Introduction to Generalized Linear Models

By Annette J. Dobson and Adrian G. Barnett · CRC Press · 2018

Study guide: Exam MAS-I

A textbook presenting generalized linear models as a unified framework for statistical modelling, accessible to undergraduates and researchers in other fields. It sets out the theoretical background — the Normal, Poisson and binomial distributions, estimation, model fitting and inference — before methods for particular kinds of data: multiple linear regression, analysis of variance, logistic regression, log-linear models, survival analysis, multilevel models, and Bayesian models fitted by Markov chain Monte Carlo, with code for Stata, R and WinBUGS. The fourth edition adds sections on model selection and non-linear associations and a Postface on good statistical practice.

On the syllabus

  • Exam MAS-I — objectives C1–C9; Chapters 1–9, excluding 6.3.3, 6.7, 6.8 and 7.9.

1 Introduction

  • 1.1 Background
  • 1.2 Scope
  • 1.3 Notation
  • 1.4 Distributions related to the Normal distribution
    • 1.4.1 Normal distributions
    • 1.4.2 Chi-squared distribution
    • 1.4.3 t-distribution
    • 1.4.4 F-distribution
    • 1.4.5 Some relationships between distributions
  • 1.5 Quadratic forms
  • 1.6 Estimation
    • 1.6.1 Maximum likelihood estimation
    • 1.6.2 Example: Poisson distribution
    • 1.6.3 Least squares estimation
    • 1.6.4 Comments on estimation
    • 1.6.5 Example: Tropical cyclones
  • 1.7 Exercises

2 Model Fitting

  • 2.1 Introduction
  • 2.2 Examples
    • 2.2.1 Chronic medical conditions
    • 2.2.2 Example: Birthweight and gestational age
  • 2.3 Some principles of statistical modelling
    • 2.3.1 Exploratory data analysis
    • 2.3.2 Model formulation
    • 2.3.3 Parameter estimation
    • 2.3.4 Residuals and model checking
    • 2.3.5 Inference and interpretation
    • 2.3.6 Further reading
  • 2.4 Notation and coding for explanatory variables
    • 2.4.1 Example: Means for two groups
    • 2.4.2 Example: Simple linear regression for two groups
    • 2.4.3 Example: Alternative formulations for comparing the means of two groups
    • 2.4.4 Example: Ordinal explanatory variables
  • 2.5 Exercises

3 Exponential Family and Generalized Linear Models

  • 3.1 Introduction
  • 3.2 Exponential family of distributions
    • 3.2.1 Poisson distribution
    • 3.2.2 Normal distribution
    • 3.2.3 Binomial distribution
  • 3.3 Properties of distributions in the exponential family
  • 3.4 Generalized linear models
  • 3.5 Examples
    • 3.5.1 Normal linear model
    • 3.5.2 Historical linguistics
    • 3.5.3 Mortality rates
  • 3.6 Exercises

4 Estimation

  • 4.1 Introduction
  • 4.2 Example: Failure times for pressure vessels
  • 4.3 Maximum likelihood estimation
  • 4.4 Poisson regression example
  • 4.5 Exercises

5 Inference

  • 5.1 Introduction
  • 5.2 Sampling distribution for score statistics
    • 5.2.1 Example: Score statistic for the Normal distribution
    • 5.2.2 Example: Score statistic for the Binomial distribution
  • 5.3 Taylor series approximations
  • 5.4 Sampling distribution for maximum likelihood estimators
    • 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
  • 6.4 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

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
    • 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

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
  • 14.7 Longitudinal data analysis
  • 14.8 Bayesian model averaging
    • 14.8.1 Example: Stroke recovery
    • 14.8.2 Example: PLOS Medicine journal data
  • 14.9 Some practical tips for WinBUGS
  • 14.10 Exercises

Postface

Appendix

Software

Related readings

  • Generalized Linear Models (Larsen - 2015) — assigned with it for Domain C (Extended Linear Models) of the MAS-I outline; the note supplements this book's Sections 3.2, 7.7 and 9.8
  • An Introduction to Statistical Learning (James et al. - 2021) — assigned with it for Domain C of the MAS-I outline
  • Introduction to Mathematical Statistics (Hogg et al. - 2018) — assigned with it for Domain C of the MAS-I outline

Sources