Day 32 - 11/04/2024

Generalized linear model (GLM)

Extending our understanding of how we construct statistical models

Model equation form

• \(y_{ij} = \mu + \tau_i + \varepsilon_{ij}, \ \varepsilon_{ij}\sim N(0,\sigma^2)\)

Probability distribution form

• \(y_{ij}\sim N(\mu_{ij},\sigma^2), \ \mu_{ij} = \mu + \tau_i\)
• \(y_{ij}\sim N(\mu + \tau_i,\sigma^2)\)

Necessary steps:

• Identify the probability distribution of \(y\)
  
• • Define what your model is focusing on (usually the expected value \(E(y)\))
    
• State the linear predictor \(\eta\)
  
• Identify a link function that connects \(E(y)\) to the linear predictor

Model equation versus Probability distribution forms

• Model equation is much more restrictive for non-normal distributions
  • What is the residual even?
    
  • \(Var(y)\) in other distributions
    
  • ‘If the data are not Gaussian, we must make them “act Gaussian”’, essentially amounts to the modeling version of the “when you have a hammer, try to make every problem look like a nail” (Stroup et al., 2024)

The general linear model as a special case of the GLM

• Identify the probability distribution of \(y\)
  • \(y_i \sim N(\mu_i, \sigma^2)\)
• Define what your model is focusing on: the expected value \(E(y)\)
  
• State the linear predictor \(\eta\)
  • \(\eta_i = \eta + \tau_i\)
    
• Identify a link function that connects \(E(y)\) to the linear predictor
  • “identity function”: \(\eta_i = \mu_i\)

Applied examples and implications

On the whiteboard:

• Modeling proportions
  
• Modeling disease tolerance in plants
  
• Modeling counts of insects