The Four Types of Estimable Functions

Introduction to Reduction Notation

Reduction notation can be used to represent differences in sums of squares (SS) for two models. The notation $R(\mu , A, B, C)$ denotes the complete main-effects model for effects A, B, and C. The notation

\[ R(A~ |~ \mu ,B,C) \]

denotes the difference between the model SS for the complete main-effects model containing A, B, and C and the model SS for the reduced model containing only B and C.

In other words, this notation represents the differences in model SS produced by

proc glm;
   class a b c;
   model y = a b c;


proc glm;
   class b c;
   model y = b c;

As another example, consider a regression equation with four independent variables. The notation $R(\beta _3, \beta _4~ |~ \beta _1, \beta _2)$ denotes the differences in model SS between

\[ y = \beta _0 + \beta _1 x_1 + \beta _2 x_2 + \beta _3 x_3 + \beta _4 x_4 + \epsilon \]


\[ y = \beta _0 + \beta _1 x_1 + \beta _2 x_2 + \epsilon \]

This is the difference in the model SS for the models produced, respectively, by

model y = x1 x2 x3 x4;


model y = x1 x2;

The following examples demonstrate the ability to manipulate the symbolic representation of a generating set. Note that any operations performed on the symbolic notation have corresponding row operations that are performed on the generating set itself.