Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 657
The Factor Models
Nominal Factors
Nominal factors are transformed into indicator variables for the design matrix. SAS GLM constructs an
indicator column for each nominal level. JMP constructs the same indicator columns for each nominal level
except the last level. When the last nominal level occurs, a one is subtracted from all the other columns of
the factor. For example, consider a nominal factor A with three levels coded for GLM and for JMP as shown
below.
Table A.1 Nominal Factor A
GLM
JMP
A A1 A2 A3 A13 A23
A1 1 0 0 1 0
A2 0 1 0 0 1
A3 0 0 1 –1 –1
In GLM, the linear model design matrix has linear dependencies among the columns, and the least squares
solution employs a generalized inverse. The solution chosen happens to be such that the A3 parameter is set
to zero.
In JMP, the linear model design matrix is coded so that it achieves full rank unless there are missing cells or
other incidental collinearity. The parameter for the A effect for the last level is the negative sum of the other
levels, which makes the parameters sum to zero over all the effect levels.
Interpretation of Parameters
Note: The parameter for a nominal level is interpreted as the differences in the predicted response for that
level from the average predicted response over all levels.
The design column for a factor level is constructed as the zero-one indicator of that factor level minus the
indicator of the last level. This is the coding that leads to the parameter interpretation above.
Table A.2 Interpreting Parameters
JMP Parameter Report How to Interpret Design Column Coding
Intercept mean over all levels 1´
A[1]
A[2]
α 1
– 1 ⁄ 3( α 1
+ α 2
+ α 3
)
α 2
– 1 ⁄ 3( α 1
+ α 2
+ α 3
)
(A==1) – (A==3)
(A==2) – (A==3)