Generalized Linear Models (GLM)

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# Generalized Linear Models (GLM) ## 8.1 Components of a GLM 1. **Random component:** Distribution of $y$ (exponential family) 2.

Random component: Distribution of $y$ (exponential family)
Systematic component: Linear predictor $\eta = \mathbf{X}\boldsymbol{\beta}$
Link function: $g(\mu) = \eta$ where $\mu = E[y]$

For binary

y

\log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k

Interpretation:

e^{\beta_j}

= odds ratio for a one-unit increase in

x_j

For count

y

\log(\lambda) = \beta_0 + \beta_1 x_1 + \cdots + \beta_k x_k

where

y \sim \text{Pois}(\lambda)