Modeling and Multivariate Methods - SAS

186 Fitting Generalized Linear Models Chapter 6
Examples
Second, goodness-of-fit statistics are presented. Analogous to lack-of-fit tests, they test for adequacy of the
model. Low p-values for the ChiSquare goodness-of-fit statistics indicate that you may need to add
higher-order terms to the model, add more covariates, change the distribution, or (in Poisson and binomial
cases especially) consider adding an overdispersion parameter. AICc is also included and is the corrected
Akaike’s Information Criterion, where
AICc = -2loglikelihood + 2k( k+
1 ))
2k + ------------------------- n – k – 1
and k is the number of estimated parameters in the model and n is the number of observations in the data
set. This value may be compared with other models to determine the best-fitting model for the data. The
model having the smallest value, as discussed in Akaike (1974), is usually the preferred model.
The Effect Tests table shows joint tests that all the parameters for an individual effect are zero. If an effect
has only one parameter, as with simple regressors, then the tests are no different from the tests in the
Parameter Estimates table.
The Parameter Estimates table shows the estimates of the parameters in the model and a test for the
hypothesis that each parameter is zero. Simple continuous regressors have only one parameter. Models with
complex classification effects have a parameter for each anticipated degree of freedom. Confidence limits are
also displayed.
Poisson Regression with Offset
The sample data table Ship Damage.JMP is adapted from one found in McCullugh and Nelder (1983). It
contains information on a certain type of damage caused by waves to the forward section of the hull. Hull
construction engineers are interested in the risk of damage associated with three variables: ship Type, the
year the ship was constructed (Yr Made) and the block of years the ship saw service (Yr Used).
In this analysis we use the variable Service, the log of the aggregate months of service, as an offset variable.
An offset variable is one that is treated like a regression covariate whose parameter is fixed to be 1.0.
These are most often used to scale the modeling of the mean in Poisson regression situations with log link.
In this example, we use log(months of service) since one would expect that the number of repairs be
proportional to the number of months in service. To see how this works, assume the linear component of
the GLM is called eta. Then with a log link function, the model of the mean with the offset included is:
exp[Log(months of service) + eta] = [(months of service) * exp(eta)].
To run this example, assign
• Generalized Linear Model as the Personality
• Poisson as the Distribution, which automatically selects the Log link function
• N to Y
• Service to Offset
• Type, Yr Made, Yr Used as effects in the model
• Overdispersion Tests and Intervals with a check mark