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L1592_frame_C40 Page 357 Tuesday, December 18, 2001 3:24 PM

Slopes Diffferent Slopes Equal
Intercepts y =(α +β )+(α +β )x +e y =(α +β )+α x + e
Different i 0 0 1 1 i i i 0 0 1 i i
Intercepts y =α +(α +β )x +e y =α +α x +e
Equal i 0 1 1 i i i 0 1 i i

FIGURE 40.2 Four possible models to fit a straight line to data in two categories.

Complete model
y=(α +β )+(α +β )x+e
1
0
0
1
Category 2:
β 0 y = (α +β )+α x+e
0
1
0
α 1
α + β slope = α 1
0 0 for both lines
α Category 1:
0 α 1
y = α +α x+e
0
1
FIGURE 40.3 Model with two categories having different intercepts but equal slopes.
For the second category, Z = 1 and:
y i = ( α 0 + β 0 ) + ( α 1 + β 1 )x i + e i

The regression might estimate either β 0 or β 1 as zero, or both as zero. If β 0 = 0, the two lines have the
same intercept. If β 1 = 0, the two lines have the same slope. If both β 1 and β 0 equal zero, a single straight
line fits all the data. Figure 40.2 shows the four possible outcomes. Figure 40.3 shows the particular
case where the slopes are equal and the intercepts are different.
If simplification seems indicated, a simplified version is fitted to the data. We show later how the full
model and simplified model are compared to check whether the simplification is justified.
To deal with three categories, two categorical variables are defined:

Category 1: Z 1 = 1 and Z 2 = 0
Category 2: Z 1 = 0 and Z 2 = 1

This implies Z 1 = 0 and Z 2 = 0 for category 3.
The model is:
(
(
y i = ( α 0 + α 1 x i ) + Z 1 β 0 + β 1 x i ) + Z 2 γ 0 + γ 1 x i ) + e i
The parameters with subscript 0 estimate the intercept and those with subscript 1 estimate the slopes.
This can be rearranged to give:

y i = α 0 + β 0 Z 1 + γ 0 Z 2 + α 1 x i + β 1 Z 1 x i + γ 1 Z 2 x i + e i

The six parameters are estimated by fitting the original independent variable x i plus the four created
variables Z 1 , Z 2 , Z 1 x i , and Z 2 x i .
Any of the parameters might be estimated as zero by the regression analysis. A couple of examples
explain how the simpler models can be identified. In the simplest possible case, the regression would
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