To construct a generalized linear model, you must select response and explanatory variables for your data. You then must choose an appropriate link function and probability distribution for your response. Explanatory variables can be any combination of continuous variables, classification variables, and interactions. Some common examples of generalized linear models are listed in Table 12.1.
Model 
Response Variable 
Distribution 
Default Link Function 

Traditional Linear Model 
continuous 
Normal 
identity, g(μ) = μ 
Logistic Regression 
a count or a binary random variable 
Binomial 
logit, 
Poisson Regression in Log Linear Model 
a count 
Poisson 
log, g(μ) = log(μ) 
Exponential Regression 
positive continuous 
Exponential 
The platform fits a generalized linear model to the data by maximum likelihood estimation of the parameter vector. In general, there is no closedform solution for the maximum likelihood estimates of the parameters. Therefore, the platform estimates the parameters of the model numerically through an iterative fitting process using a technique pioneered by Nelder and Wedderburn (1972). The overdispersion parameter φ is estimated by dividing the Pearson goodnessoffit 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 the Generalized Linear Model personality of the Fit Model platform. Table 12.2 lists the builtin link functions.
Link Function Name 
Link Function Formula 

Identity 
g(μ) = μ 
Logit 

Probit 
g(μ) = Φ1(μ), where Φ is the standard normal cumulative distribution function 
Log 
g(μ) = log(μ) 
Reciprocal 
g(μ) = 
Power 

Comp LogLog 
g(m) = log(–log(1 – μ)) 
When you select the Power link function, a number box appears that enables you to enter the desired power.
Table 12.3 lists the variance functions associated with the available distributions for the response variable.
Distribution 
Variance Function 

Normal 
V(μ) = 1 
Binomial 
V(μ) = μ(1 – μ) 
Poisson 
V(μ) = μ 
Exponential 
V(μ) = μ2 