Generalised Linear Models

MATH38171 Generalised Linear Models 20 Semester 1 Undergraduate Level 3 Madhuchhanda Bhattacharjee Unit coordinator Department of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Last part of a Bachelors ' European Credit Transfer & Accumulation System Rating : 10.0 As an important modelling strategy Linear Models is concerned with investigating whether, and how, one or more so-called explanatory variables, such as age, sex, blood pressure, etc., influence a response variable, such as a patient's diagnosis, by taking random variations of data into account. In Linear Models, linear regression technique and Normal distribution are used to explore the possible linear relation between a continuous response and one or more explanatory variables. In this course unit we depart from linearity and normality, the very strict limitation in Linear Models. We study the extension of linearity to non-linearity and normality to a commonly encountered distribution family, called the exponential family of distributions. This extension forms Generalized Linear Models (GLM). The GLM, on the one hand, unifies linear and non-linear models in terms of statistical modelling. On the other hand, it can be used to analyze discrete data, including binary, binomial, counted and categorical data that arise very often in biomedical and industrial applications. In MATH20282 Statistical Methods and MATH38141, the basic idea of regression modeling was developed for a normally distributed response variable. However in many real-world application areas, including science, engineering, and finance, the response variable is often non-normal. Common examples include binary or count responses, e.g.: • whether a mechanical component fails (binary response) • whether a consumer defaults on a loan (binary response) • population size of an endangered species (count response) • the number of motor insurance claims in a particular category (count response). In this module, we introduce a flexible family of regression models that enables a wide variety of non-normal response distributions to be fitted.  • To introduce the theory and application of generalized linear models, including parameter estimation via numerical maximization of the likelihood function, confidence intervals, hypothesis testing, model selection, model diagnostics, and use of R.   After successful completion of the course, students will be able to: 1. Formulate appropriate generalized linear models and statistical hypotheses to investigate real-world questions involving non-normal response variables; 2.  apply generalized linear model techniques to make inferences and predictions about the relationship between covariates and a non-normal response variable, and to shed light on real-world questions; 3. check whether the assumptions underpinning such analyses are justified; 4. explain and interpret the resulting models and conclusions, with reference to the original real-world questions where appropriate, and also explain key underpinning ideas, assumptions, procedures, and theoretical results; 5. prove underpinning mathematical and theoretical results; 6. use R to apply the methods and to conduct appropriate simulation studies to evaluate the performance of the methods studied. Introduction [6]Motivation for GLMs: binary and count data, brief introduction to logistic and Poisson regression.  Review of linear models, likelihood theory and R programming. Categorical factors and interactions. Generalized linear models and their estimation [5]Exponential dispersion family. General form of a GLM including response distribution, link function and linear predictor. Iterative numerical methods for maximum likelihood estimation of the regression coefficients. Comparison of methods for estimation of the dispersion parameter. Inference, model selection and diagnostics [6]Wald tests and confidence intervals. Likelihood ratio tests, score tests, and analysis of deviance. Goodness-of-fit tests. AIC and BIC. Stepwise selection. Pearson, deviance and quantile residuals; diagnostic plots. Applications and extensions [5]Binary and binomial data: particular issues, separation, goodness-of-fit. Contingency tables, log-linear models. Gamma GLMs. Insurance claims. Credit scoring. Time permitting, a brief introduction to a more advanced extension. 0 Other 20% 1 Written exam 80% <ul> <li>Coursework: 20%</li> <li>End of semester examination: weighting 80%</li></ul> - Tutorials, labs and office hours give the opportunity to receive individual feedback MATH11022 Linear Algebra Pre-Requisite Compulsory MATH27720 Probability and Statistics 2 Pre-Requisite Compulsory MATH27711 Linear Regression Models Pre-Requisite Compulsory MATH10202 Linear Algebra A Pre-Requisite Compulsory MATH10212 Linear Algebra B Pre-Requisite Compulsory MATH20701 Probability 2 Pre-Requisite Compulsory MATH20802 Statistical Methods Pre-Requisite Compulsory MATH38171 Pre-Requisites: MATH11022, MATH27720 and MATH27711 N • Dunn, P.K. and Smyth, G.K. (2018). Generalized linear models with examples in R. Springer. • Agresti, A. (2015). Foundations of linear and generalized linear models. Wiley. • McCullagh, P. and Nelder, J.A. (1983). Generalized linear models. Chapman and Hall. • Faraway, J.J. (2016). Extending the linear model with R: generalized linear, mixed effects and nonparametric regression models. CRC press. Lectures 22 Practical classes & workshops 4 Tutorials 6 0 168