Modeling and Multivariate Methods - SAS

Chapter 3 Fitting Standard Least Squares Models 131
Examples with Statistical Details
Analysis of Covariance with Separate Slopes
This example is a continuation of the Drug.jmp example presented in the previous section. The example
uses data from Snedecor and Cochran (1967, p. 422). A one-way analysis of variance for a variable called
Drug, shows a difference in the mean response among the levels a, d, and f, with a significance probability of
0.03.
The lack of fit test for the model with main effect Drug and covariate x is not significant. However, for the
sake of illustration, this example includes the main effects and the Drug*x effect. This model tests whether
the regression on the covariate has separate slopes for different Drug levels.
This specification adds two columns to the linear model (call them x 4i and x 5i ) that allow the slopes for the
covariate to be different for each Drug level. The new variables are formed by multiplying the dummy
variables for Drug by the covariate values, giving the following formula:
y i
= β 0
+ β 1
x 1i
+ β 2
x 2i
+ β 3
x 3i
+ β 4
x 4i
+ β 5
x 5i
+ ε i
Table 3.17 on page 131, shows the coding of this Analysis of Covariance with Separate Slopes. (The mean
of X is 10.7333.)
Table 3.17 Coding of Analysis of Covariance with Separate Slopes
Regressor Effect Values
x 1 Drug[a] +1 if a, 0 if d, –1 if f
x 2 Drug[d] 0 if a, +1 if d, –1 if f
x 3 x the values of x
x 4 Drug[a]*(x - 10.733) +x – 10.7333 if a, 0 if d, –(x – 10.7333) if f
x 5 Drug[d]*(x - 10.733) 0 if a, +x – 10.7333 if d, –(x – 10.7333) if f
A portion of the report is shown in Figure 3.56. The Regression Plot shows fitted lines with different slopes.
The Effect Tests report gives a p-value for the interaction of 0.56. This is not significant statistically,
indicating the model does not need to have different slopes.