Generalized Linear Models

Syllabus Details

• Unit I:(Exponential Family and Generalized Linear Models) Introduction: background, review of linear models in matrix notation, model assessment, some pre-required knowledge; the exponential family of distributions: definition and examples, mean and variance, variance function and scale parameter; generalized linear models (GLM): linear predictor, link function, canonical link, maximum likelihood estimation, iterative reweighted least squares and Fisher scoring algorithms, significance of parameter estimates, bias and consistency in GLMs, covariance matrices of linear combinations, sampling distribution of parameter estimators, estimating the dispersion parameter, confidence intervals.
• Unit II:(Assessing the Model) Deviance, distribution of the deviance, deviance residuals, examples, partial residual/regression plots; model goodness of fit, goodness of fit of nested models with known dispersion, goodness of fit of nested models with unknown dispersion; Akaike’s Information Criterion (AIC); collinearity and its detection, variance inflation factors (VIFs); testing factor variables; interactions.
• Unit III:(Regression Models for Different Types of Response) Normal linear regression models: least squares, analysis of variance, orthogonality of parameters, factors, interactions between factors; fitting multiple regression; binary and binomial data analysis: distribution and models, logistic regression models, odds ratio, one- and two-way logistic regression analysis; multinomial regression and interpretation; Poisson count data analysis: Poisson regression models with offset, two-dimensional contingency tables, log-linear models; Gamma GLMs, alternate models, Cox-PH model.
• Unit IV:(Quasi-likelihood and Extensions) Other GLMs: quasi likelihoods; extending GLMs: random effects models, mixed effect models.

Course Learning Objective

1. Discuss linear and non-linear models, apply data transformations, and appreciate the need and uses of generalized linear models.
2. Discuss the exponential family of distributions and derive likelihood based inference for such a family of distributions.
3. Apply logistic, multinomial, Poisson and Gamma regression models.
4. Discuss the concept of deviance, analysis of deviance, lack-of-fit tests in logistic and Poisson regression, and the concept of overdispersion.
5. Apply the concepts of generalized linear models in real life problems.
6. Discuss and apply quasi likelihood.

Recommended Books

1. Dobson, A. J. An Introduction to Generalized Linear Models. Chapman & Hall, 2002
2. Krzanowski, W. An Introduction to Statistical Modelling. Edward Arnold, 1998
3. McCullagh, P. and Nelder, J. A. Generalized Linear Models. Chapman & Hall, 1990
4. Faraway, J. Extending the Linear Model with R. Chapman & Hall/CRC, 2006
5. Agresti, A. Foundations of Linear and Generalized Linear Models. Wiley, 2015