Ways of comparing linear regression intercepts and slopes?

Question

I'm a little bit confused about this, so any help would be appreciated!

Let's say I have a repeated-measures design in which participants take part in a task where they have to rate the attractiveness of 50 different faces in 4 different conditions of facial expression type. Condition A is a baseline with neutral expression, condition B is happy, condition C sad and condition D angry.

I've run individual linear regressions with conditions B, C and D as the dependent variable and condition A as a predictor, generating 3 different regression equations. I would like to find a way to statistically compare these equations in terms of their slope and intercept.

Specifically, the theory I am trying to test predicts there will be no change in slope between them, but condition B should raise the intercept, condition D should lower it and there should be no change in C. In other words, the different conditions should result in a uniform shift up or down in terms of attractiveness.

Is there any way to test this statistically?

Answer 1

You can compare slopes and intercepts using dummy variables. You need one less dummy variable that the number of regressions you are comparing. If you have two regression lines the dummy variable $d1$ has values of $1$ or $0$. Assuming you have multiple sample $y$ and $x$, some of them will be from one group, lets call it $Group\space A$, the others will be from another $Group\space B$. To the set of $x$ - $y$ variables you would add $d1$ with value $0$ for observations from $Group\space A$ and value $1$ for observations from $Group\space B$. You now fit a model where $y$ ~ $x,\space d1$ and $x*d1$. Multiple regression will give you coefficients $b0, b1, b2$ and $b3$ for the $intercept$, $x$, $d1$ and $d1*x$. The equation just fitted is

$y = b0 + b1*x + b2*d1 + b3*x*d1$

For $Group\space A$ where $d1 = 0$ the equation is

$y = b0 + b1*x$ (I)

For $Group\space B$ where $d1 = 1$ the equation is

$y = b0 + b1*x + b2*d1 + b3*d1*x = (b0 + b2) + (b1 + b3)*x$ (II)

The intercept is $(b0 + b2)$, the slope is $(b1 + b3)$

If the coefficient for $d1$ is statistically significant then the slopes and intercepts are calculated as in (II), otherwise there is no differences and the model is (I).

The process can be extended to as many groups just adding more dummy variables $d1,\space d2...$, etc.

If you have 3 groups as you describe this is the table for the dummy variables values:

$Group\space A: d1 = 0;\space d2 = 0$
$Group\space B: d1 = 1;\space d2 = 0$
$Group\space C: d1 = 1;\space d2 = 1$

The equation to be fitted is

$y$ ~ $x\space d1\space d2\space x*d1\space x*d2$

The regression equation will be:

$y = b0 + b1*x + b2*d1 + b3*d2 + b4*x*d1 + b5*x*d2$

$Group\space A,\space d1 = 0, d2 = 0$
$y = b0 + b1*x$

$Group\space B,\space d1 = 1, d2 = 0$
$y = (b0 + b2) + (b1 + b4)*x*d1$
$(b0 + b2)$ is the slope; $(b1 + b4)$ is the intercept.

$Group\space C,\space d1 = 1, d2 = 1$
$y = (b0 + b2 + b3) + (b1 + b4 + b5)*x*d1*d2$
$(b0 + b2 + b3)$ is the slope; $(b1 + b4 + b5)$ is the intercept.

Hope this helps. Is quite straightforward.