Modeling and Multivariate Methods - SAS

Chapter 6 Fitting Generalized Linear Models 181
Examples of Generalized Linear Models
Table 6.1 Examples of Generalized Linear Models
Model Response Variable Distribution Canonical Link
Function
Traditional Linear
Model
continuous Normal identity, g(μ) = μ
Logistic Regression
a count or a binary
random variable
Binomial
logit,
g( μ)
=
log -----------
μ
1 – μ
Poisson Regression in
Log Linear Model
a count Poisson log, g(μ) = log(μ)
Exponential
Regression
positive continuous
Exponential
--
1
μ
JMP fits a generalized linear model to the data by maximum likelihood estimation of the parameter vector.
There is, in general, no closed form solution for the maximum likelihood estimates of the parameters. JMP
estimates the parameters of the model numerically through an iterative fitting process. The dispersion
parameter φ is also estimated by dividing the Pearson goodness-of-fit statistic by its degrees of freedom.
Covariances, standard errors, and confidence limits are computed for the estimated parameters based on the
asymptotic normality of maximum likelihood estimators.
A number of link functions and probability distributions are available in JMP. The built-in link functions
are
identity: g(μ) = μ
logit:
g( μ)
=
log -----------
μ
1 – μ
probit: g(μ) = Φ -1 (μ), where Φ is the standard normal cumulative distribution function
log: g(μ) = log(μ)
1
reciprocal: g(μ) = --
μ
power: g( μ)
μ λ =
if ( λ ≠ 0)
log( μ)
if λ = 0
complementary log-log: g(m) = log(–log(1 – μ))
When you select the Power link function, a number box appears enabling you to enter the desired power.
The available distributions and associated variance functions are
normal: V(μ) = 1
binomial (proportion): V(μ) = μ(1 – μ)