Type III Tests
Type III tests examine the significance of each
partial effect, that is, the significance of an
effect with all the other effects in the model.
They are computed by constructing a type III
hypothesis matrix L and then computing statistics
associated with the hypothesis L = 0.
Refer to the chapter titled "The Four Types of Estimable Functions," in the
SAS/STAT User's Guide
for the construction of the matrix L.
For linear models, the type III or partial sum of squares

(Lb)' (L (X'X)^{1} L')^{1} (Lb)
is used to test the hypothesis L = 0.
The Type III Tests table for linear models,
as illustrated by Figure 39.15,
includes the following:
 Source
 is the name for each effect.
 DF
 is the degrees of freedom associated with each effect.
 Sum of Squares
 is the partial sum of squares for each effect in the model.
 Mean Square
 is the sum of squares divided by its
associated degrees of freedom.
 F Stat
 is the F statistic for testing the null hypothesis
that the linear combinations of parameters described
previously for the hypothesis matrix L are 0.
This is formed by dividing the mean square for the
hypothesis matrix L by the mean square
for error from the complete model.
 Pr > F
 is the probability of obtaining a greater F statistic
than that observed if the null hypothesis is true.
Figure 39.15: Type III Tests Table for Linear Models
For generalized linear models, either the Wald
statistic or the likelihoodratio statistic can
be used to test the hypothesis L = 0.
For the linear model, the two tests are equivalent.
The Wald statistic is given by
where is the estimated
covariance matrix of the parameters.
The likelihoodratio statistic is computed as twice
the difference between the maximum loglikelihood
achievable under the unconstrained model and the
maximum loglikelihood for the model under the
restriction or constraint L = 0.
Both the Wald statistic and the likelihoodratio statistic
have an asymptotic distribution.
The Type III (Wald) Tests and Type III (LR) Tests
tables, as illustrated by Figure 39.16,
include the following:
 Source
 is the name for each effect.
 DF
 is the degrees of freedom associated with each effect.
 ChiSq
 is the Wald statistic for the Wald tests or the likelihoodratio
statistic for the LR tests of the null hypothesis
that the parameters for the effect are 0.
This has an asymptotic distribution.
 Pr > ChiSq
 is the probability of obtaining a greater statistic than that observed, if the null hypothesis is true.
Figure 39.16: Type III Tests Tables for Generalized Linear Models
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.