Modeling and Multivariate Methods - SAS

656 Statistical Details Appendix A
The Factor Models
Fitting Principle For Ordinal Response
Base Model
The maximum likelihood fitting principle for an ordinal response model is the same as for a nominal
response model. It estimates the parameters such that the joint probability for all the responses that occur is
the greatest obtainable by the model. It uses an iterative method that is faster and uses less memory than
nominal fitting.
The simplest model for an ordinal response, like a nominal response, is a set of response probabilities fitted
as the occurrence rates of the response in the whole data table.
The Factor Models
The way the x-variables (factors) are modeled to predict an expected value or probability is the subject of the
factor side of the model.
The factors enter the prediction equation as a linear combination of x values and the parameters to be
estimated. For a continuous response model, where i indexes the observations and j indexes the parameters,
the assumed model for a typical observation, y i , is written
y i
= β 0
+ β 1
x 1i
+ … + β k
x ki
+ ε i
where
y i
x ij
ε i
is the response
are functions of the data
is an unobservable realization of the random error
β j are unknown parameters to be estimated.
The way the x’s in the linear model are formed from the factor terms is different for each modeling type.
The linear model x’s can also be complex effects such as interactions or nested effects. Complex effects are
discussed in detail later.
Continuous Factors
Continuous factors are placed directly into the design matrix as regressors. If a column is a linear function of
other columns, then the parameter for this column is marked zeroed or nonestimable. Continuous factors are
centered by their mean when they are crossed with other factors (interactions and polynomial terms).
Centering is suppressed if the factor has a Column Property of Mixture or Coding, or if the centered
polynomials option is turned off when specifying the model. If there is a coding column property, the factor
is coded before fitting.