Modeling and Multivariate Methods - SAS

Chapter 3 Fitting Standard Least Squares Models 127
Examples with Statistical Details
– x 2i is the level of the second predictor variable in the i th trial
– β 0 , β 1 , and β 2 are parameters for the intercept, the first predictor variable, and the second predictor
variable, respectively
– ε ι are the independent and normally distributed error terms in the i th trial
As shown here, the first dummy variable denotes that Drug=a contributes a value 1 and Drug=f contributes
a value –1 to the dummy variable:
1 a
x 1i
= 0 d
–1 f
The second dummy variable is given the following values:
0 a
x 2i
= 1 d
–1 f
The last level does not need a dummy variable because in this model, its level is found by subtracting all the
other parameters. Therefore, the coefficients sum to zero across all the levels.
The estimates of the means for the three levels in terms of this parameterization are as follows:
μ 1
= β 0
+ β 1
μ 2
= β 0
+ β 2
μ 3
= β 0
– β 1
– β 2
Solving for β i yields the following:
( μ 1
+ μ 2
+ μ 3
)
β 0
= ----------------------------------- = μ
3
(the average over levels)
β 1
= μ 1
– μ
β 2
= μ 2
– μ
β 3
= β 1
– β 2
= μ 3
– μ
Therefore, if regressor variables are coded as indicators for each level minus the indicator for the last level,
then the parameter for a level is interpreted as the difference between that level’s response and the average
response across all levels. See the appendix “Statistical Details” on page 651 for additional information about
the interpretation of the parameters for nominal factors.
Figure 3.52 shows the Parameter Estimates and the Effect Tests reports from the one-way analysis of the
drug data. Figure 3.1, at the beginning of the chapter, shows the Least Squares Means report and LS Means
Plot for the Drug effect.