I am performing a t-test to compare two slopes, each fit to independent X,Y data. Does the nature of this test change depending on the range of slopes being compared? For example, is comparing a slope of 10 to a slope of 1 different than comparing a slope of 0.01 to 0.001, even though both tests compares slopes that are an order of magnitude different?
My calculations follow the math outlined here (http://www.real-statistics.com/regression/hypothesis-testing-significance-regression-line-slope/comparing-slopes-two-independent-samples/) and described in [P. Cohen, S. West and L. Aiken, ``Applied multiple regression/correlation analysis for the behavioral sciences'', Routledge Press, 2014.]
To demonstrate my concern, one can plug in values to this online calculator: https://www.danielsoper.com/statcalc/calculator.aspx?id=103. Fix both sample sizes to 100 and both standard errors to 0.005. If slope 1 is set to 0.1 and slope 2 set to 0.01, the t-test returns (t=12.7, p=0.0) indicating the slopes are different. If slope 1 is set to 0.001 and slope 2 is set to 0.0001, the t-test returns (t=0.13, p=0.9) indicating the slopes are the same. In both cases the slopes are an order of magnitude different, and the amount of data and standard errors are the same, so why does the t-test result change so much?
EDIT: I have learned that the problem is that this test compares the difference between two slopes rather than their ratio. If slope1 is 1.0 and slope2 is 0.1, the difference is 0.9 units. If slope3 is 0.001 and slope4 is 0.0001, the difference is 0.0009 units. Even though both pairs of slopes are an order of magnitude different, the difference between the pairs of slopes is very different. This is why the test changes depending on the range of slopes. I have not yet found a better test, but I recommend anyone else finding this question avoid using the above test unless they are specifically comparing slopes with values near 1.