Generalized Linear Models

• syllabus
• literature

  
  

  Purpose

  Regression analysis playes a central role in statistics being one of its most powerful and commonly used techniques. The standard linear regression models assume that the response variable is normal (or at least can be transformed to a normal one). However, unfortunately/fortunately (?) it is not always the case. A wide variety of models with a categorical or counting response is typical (althgough not the only ones!) examples, where the assumption of normality cannot be accepted as reasonable. In this course we study generalized linear models, where the response variables are allowed to be non-normal. We start from the general theory of generalazied linear models, extending the corresponding results for standard linear regression, and then consider the most useful particular cases in more details.

  Topics:

  1. Introduction
     • Standard (normal) linear regression model
     • Generalized linear regression model
  2. Theory of Generalized Linear Models
     • Model components
       • exponential family and its properties
       • link functions
     • Maximum likelihood estimation
       • Newton-Raphson method
       • iteratively reweighted least squares
     • Goodness-of-fit
       • analysis of deviance
       • Pearson statistic
       • analysis of residuals
     • Model selection
  3. Particular Models
     • Binary data
     • Binomial data
     • Multinomial data
     • Poisson data
  4. Overdispersion & Quasi-Likelihood Models
  5. Nonparametric GLM
  6. Normal linear models with heterogeneous variance and GLM
  7. Generalized linear mixed effects models

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  Literature

  • Dobson, A.J. An Introduction to Generalized Linear Models.
  • McCullagh, P. and Nelder, J.A. Generalized Linear Models.
  • Wood, S.N. Generalized Additive Models. An Introduction with R (Chapter 2).
  • Myers, R.H. and Montgomery, D.C. A tutorial on Generalized Linear Models. Journal of Quality Technology, 29, 274-291.
  • Chapter 5: Green, P.G. and Silverman B.W. Nonparametric Regression and Generalized Linear Models.

• Aitkin, M., Francis, B., Hinde, J. and Darnell, R. Statistical Modelling in R.
• Faraway, J.J. Extending the Linear Models with R (Chapters 1-10).
• Venables, W.N. and Ripley, B.D. Modern Applied Statistics with S-Plus (Chapter 7).

Some specific S-Plus and R notes you may find useful for generalized linear models:

• To fit a generalized linear model you will generally use the function glm:
  >glm(formula, family=...(link=...),...)
  The glm function creates an object of class glm that contains most of information you need. See help(glm.object) for details.

  For some data the convergence of the iteratively reweighted least squares algorithm is slow and does not occur in (default) 10 iterations. It may happen, for example, in binomial models with a lot of empty cells. S-Plus gives you a "Warning". Don't panic! You can increase the number of iterations by the parameter maxit:

  >glm(formula,...,maxit=...)

  To evaluate the fitted model at some new values of the predictors use the function predict:

  >predict(glm.object, type=...,se=T)

  The output will contain, in particlular, a vector of estimated response (depending on type) and a vector of standard errors for constructing confidence intervals for the mean response.