5.1 Parallel lines

First case: a single dichotomous variable. Our example: Pre 1980 vs Post 1980.

The way JMP does it (contrast = sum in S-Plus): model

$$Av(Y\vert x) = \beta_0 + \beta_1 x + \beta_2 z$$

where z = 1 if observation is in the first group and -1 if observation is in the second group.

Check to understand the model: plug in z = 1 and -1.

Group 1 model

$$Av(Y\vert x,z=1) = \beta_0 + \beta_1 x + \beta_2 \times 1.$$

$$Av(Y\vert x,z=1) = \beta_0 + \beta_1 x + \beta_2.$$

$$Av(Y\vert x,z=1) = (\beta_0 + \beta_2) + \beta_1 x.$$

Group 2 model

$$Av(Y\vert x,z=-1) = \beta_0 + \beta_1 x + \beta_2 \times -1.$$

$$Av(Y\vert x,z=-1) = \beta_0 + \beta_1 x - \beta_2.$$

$$Av(Y\vert x,z=-1) = (\beta_0 - \beta_2) + \beta_1 x.$$

Compare Group 1 and Group 2.

Av(Y|x,z=1) - Av(Y|x,z=-1) is the difference in height between the two regression lines.

Notes.

Both groups have the same slopes ($\beta_1$) - parallel lines.

The difference in heights is $2 \beta_2$.

Note that $\beta_1$ represents a comparison against the "norm".