Tangent: GLMs

• Generalised Linear Models are a set of statistical models which have a dependent variable which is a function of a linear combination of variables and parameters, with an error component. Normal regresson is the simplest example; loglinear models are fitted using `poisson regression'.
  
  $$f(y_i) = \alpha + \mathbf{\beta x_i} + e_i$$
  
  
  • The program GLIM used to be used extensively for loglinear modelling because it was the first package to do Generalised Linear Models.
  • Most statistics packages now have GLM modules so even if they don't have explicit loglinear commands they can fit loglinear models.
  • A loglinear model takes this form:
    
    $$\log f_{ijk} = \lambda_0 + \lambda_iI + \lambda_jJ + \lambda_kK + e_p$$
    
    
    or
    
    $$\log F_{ijk} = \lambda_0 + \lambda_iI + \lambda_jJ + \lambda_kK$$
    
    
    where $F_{ijk}=f_{ijk}+e_p$. This is exactly equivalent to
    
    $$F_{ijk} = \tau_0 \tau_i^I \tau_j^J \tau_k^K$$
    
    
    where $\lambda = \log \tau$.