## Parameterization Used in PROC GENMOD

*Design Matrix*

The linear predictor
part of a generalized linear model is

where is an unknown parameter vector
and **X** is a known design matrix.
By default, all models automatically contain an
intercept term; that is, the first column of **X** contains
all 1s. Additional columns of **X** are generated
for classification variables, regression variables, and
any interaction terms included in the model.
PROC GENMOD
parameterizes main effects and interaction terms using the same
ordering rules that PROC GLM uses.
This
is important to understand when you want to construct
likelihood ratios for custom contrasts
using the CONTRAST statement.
See Chapter 30, "The GLM Procedure,"
for more details on model parameterization.
Some columns of **X** can be linearly dependent
on other columns due to specifying an overparameterized model.
For example, when you specify a model consisting of an intercept
term and a class variable, the column
corresponding to any one of the levels of the class variable
is linearly dependent on the other columns of **X**.
PROC GENMOD handles this in the same manner as PROC GLM.
The columns of **X'X** are checked
in the order in which the model is specified
for dependence on preceding columns.
If a dependency is found, the parameter corresponding
to the dependent column is set to 0 along with its
standard error to indicate that it is not estimated.
The order in which the levels of a class variable
are checked for dependencies can be set by the ORDER=
option in the PROC GENMOD statement.

You can exclude the intercept term from the model by
specifying the NOINT option in the MODEL statement.

*Missing Level Combinations*

All levels of interaction terms involving classification
variables may not be represented in the data.
In that case, PROC GENMOD does not include
parameters in the model for the missing levels.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.