How to tell if the slope of a line is 0 or there is just no relationship? I am attempting to examine the change in slope between a predictor and response over two years.  In year 1, it is definitely positive. (Linear regression, the 95% CI of the slope doesn't overlap 0).  In year 2, the point estimate of the slope is close to 0 (0.002) and the CI overlaps 0.  This is what I would expect if the slope was, well, actually 0.  And given that any test of the slope will suggest that I cannot reject that it is 0 - great!  Although, yes, I know, that doesn't confirm that the slope is 0.  Not falling into that trap.  But, with a slope value very close to 0, and a CI that overlaps 0...it all seems like the slope has declined.
However, I now have a colleague who thinks I cannot say that the value of the slope declined between year 1 and year 2.  Rather, they think that I can only say that there is no relationship between the predictor and response in year 2.
Am I correct in interpreting my results that the slope of the year 2 relationship is indeed ~0?  Or is there a problem with my interpretation?
 A: Well, your colleague definitely is incorrect, as many standard examples will show.  For example, for the data $(x, x(1-x))$ with values of $x$ equal to $0, 1/n, 2/n, \ldots, 1$, the slope is exactly zero but obviously there's a strong (quadratic) relationship between "predictor" ($x$) and "response" ($x(1-x)$).  Your colleague would be correct to say "there is no significant evidence of a linear relationship in year 2."  But you can do more than that, as you know.  You can assess the change between the two years.
Asserting that the "slope has declined" compares two estimates to each other.  As such, you need to account for the uncertainty in both estimates, not just in the one for year two.  From the mere fact that one slope is significantly different from zero and the other slope is not, you cannot legitimately conclude their difference is significant.  However, a valid comparison is simple to do: associated with each slope is a standard error (a routine part of regression software output) and a degree of freedom (usually equal to the number of data values minus the number of parameters).  Assuming no temporal correlation in the error terms, you can compute the standard error for the difference in slopes as usual: take the root of the sum of squares of their SEs.  Refer the t-statistic (i.e., the ratio of the slope difference to the SE of their difference) to Student's t distribution.  The degree of freedom to use equals the sum of the DFs of the individual slopes.
The usual caveats and assumptions about tests hold, of course.  But presumably, because you're already using linear regressions in each of the years separately, you have checked that suitable assumptions are reasonable for these data.  About the only additional thing you might want to check is that the variances of the residuals are approximately the same in the two years.  (You could use an F-test for this.)  If not, that's worth exploring further in order to understand just how the change came about.  Assuming it was abrupt and there's no sign of temporal trends in the residuals (no heteroscedasticity), you might consider using a t-test for unequal variances instead of the simpler test described above.
