Generalized linear models

最后发布时间 : 2022-08-17 21:36:37 浏览量 :

linear models assume that the response variable is

Normally distributed
Constant variance
Independent

There are many situations where these assumptions are inappropriate

The response is either binary(0,1) or a count
The response is continuous, but non-normal

Generalized linear models(GLMs): Response distribution is a member of the exponential family(normal, exponential, gammma, binomial, Poisson)

GLMs are simple models; include linear regression and OLS as aa special case
Parameter estimation is by maximum likelihood(assume that the response distribution is known)
Inference on parameters is based on large-sample or asymptotic theory

Random component: the distribution of $y_i$ id from the exponential family:

$f(y_i;\theta_i,\phi)=exp\left \{\frac{y_i\theta_i-b(\theta_i)}{a(\theta)}+h(y_i,\theta) \right \}$

Normal, binomial, mutinomial, Poisson, exponential and gamma are all form exponential family.

Systematic component: linear predictor

$\eta_i = \beta_0 + \beta_1x_{i1}+\dots +\beta_Kx_{iK}$

Linkage:
$g[E(y_i)]=g(\mu_i)=\eta_i$

g(.) = link function = monotone and differentiable