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The Four Types of Estimable Functions

Introduction to Reduction Notation

Reduction notation can be used to represent differences in Sums of Squares 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;
   run;

and

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

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
and
y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \epsilon
With PROC REG, this is the difference in Model SS for the models produced by

   model y=x1 x2 x3 x4;

and

   model y=x1 x2;

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