Generalized Linear Models

202. Generalized Linear Models

Three components:

Specifies the distribution of the response variable $𝑌$ .

$𝑌$ can come from the exponential family of distributions (which includes Normal, Binomial, Poisson, Gamma, etc.)

Specifies the linear predictor, a linear combination of explanatory variables:

𝜂 = 𝛽_{0} + 𝛽_{1} 𝑥_{1} + 𝛽_{2} 𝑥_{2} + \dots + 𝛽_{𝑝} 𝑥_{𝑝}

Connects the expected value of $𝑌$ , denoted $𝜇 = 𝐸 [𝑌]$ , to the linear predictor $𝜂$ :

𝑔 (𝜇) - 𝜂

Where $𝑔 (\cdot)$ is called the link function

Example

Logistic Regression

Fit linear model

𝜂 = 𝛽_{0} + 𝛽_{1} 𝑥

\begin{aligned} 𝛽_{0} & = −0.4 \\ 𝛽_{1} & = 0.257 \end{aligned}

Logit link function:

𝑔 (𝑝) = log \frac{𝑝}{1 - 𝑝} = 𝜂

Model

log \frac{𝑝}{1 - 𝑝} = 𝛽_{0} + 𝛽_{1} 𝑥

Equivalently

𝑝 = \frac{1}{1 + 𝑒^{- (𝛽_{0} + 𝑏_{1} 𝑥)}}