Say I have one regression equation $y_1 = a_1x + b_1$ And another $y_2 = a_2x+b_2$ Can I subtract these two equations to get a new $y_3 = (a_1-a_2)x + b_1-b_2$ Then randomly sample some points from (how many?), then test that for significance of a slope greater than 0?
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$\begingroup$ The two lines could also be different if just the intercepts differ. $\endgroup$– Michael R. ChernickCommented Aug 18, 2017 at 0:32
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$\begingroup$ @MichaelChernick True, but I'm only interested in the slopes, I should've made that more clear. $\endgroup$– Reed OeiCommented Aug 18, 2017 at 0:38
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$\begingroup$ Assuming the assumptions for the two regression lines hold, you should be able to fit the difference by least squares and test that the slope of the difference of the regression differs from 0. The error term for the difference of the two lines will have mean 0 since both lines are assumed to have 0 mean and constant variance. $\endgroup$– Michael R. ChernickCommented Aug 18, 2017 at 0:59
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Your notion is misplaced; it seems to be based on some misunderstandings.
What makes two estimated lines potentially fail to be significantly different is that the estimated coefficients have sampling variation that can make the estimates different even if the populations the samples were drawn from are identical. Specifically if they're not more different than you'd reasonably expect to see from that random variation with the same population, you won't reject the null.
We estimate this variability by using the samples from which the lines were estimated because it's there that the uncertainty arises.
Your scheme is treating the estimated lines as population lines and then drawing new samples. This is essentially pointless, because instead of telling you about the uncertainty in your estimated lines, what it would do is tell you either that those estimated lines are not exactly the same (if you did a large enough sample of your simulated values from the difference) or it would fail to tell you that (typically because the simulation size wasn't large enough to give you power very close to 1), but you can already tell the estimates are not identical simply by looking at them.
You might instead do some kind of parametric bootstrap, but it's not going to tell you anything that simply testing the two lines for equality wouldn't tell you (we have posts on site that discuss ways to do that).
