how to interpret this interaction in this regression?

Question

(this is simulated data without error variance)

So, I have model:

y = x + gender(categorical variable, effect coded) + interaction(x and gender)

x and gender and interaction were all significant

and the plot y and x and gender.

[3

But if I see the plot, I really don't get why the interaction is significant. you can see the line is almost parallel.

would anyone help me with this?

Answer 1

I think it is simply because the lines are not plotted correctly. For gender=1, we get the equation

$$\hat y=.99x$$

This is exactly the line that would go through the blue points.

For gender=-1, we would get a different equation which would seem to be the least squares line for the red points.

$$\hat y=3.8+.2x$$

The interaction effect represents the difference in effect of x between the two genders.

Answer 2

1. The blue line looks incorrect, since the blue dots lie on a perfect straight line.

2. You have quite large sample (n = 500), and almost no variance for the blue dots. Small variance and large sample size will make even small effect significant, since the p.value decreases in the pace of $\sqrt n$ under $H_1$.