# 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?

• Are these the same statistical units that are measured at the two time points? – chl Dec 28 '10 at 17:55

## 1 Answer

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.

• (+1) I won't post my answer since it's basically your 2nd § (less well explained) + some digressions about an additional test for equal intercept (if we cannot reject the null for the test of the equality of slopes, compute a common slope for both regression lines and ask whether the new lines are parallel or identical)--but, it's always under the assumption that the sampled units are independent at the two time points. – chl Dec 28 '10 at 18:12
• @chl I refrained from commenting on the option of computing a common slope because that implicitly assumes homoscedasticity in the combined year 1 + year 2 datasets. Given that the slope appears to have changed substantially I would want to check for a change in residual variance, which seems easiest to do by conducting a separate regression for each year--which the OP already has done. The advantage of modeling the combined datasets comes from the additional power it offers of evaluating serial correlation of the residuals. – whuber Dec 28 '10 at 18:27
• @chl You bring up an interesting question concerning equal intercepts. However, an intercept is an artifact of the origin of the x coordinate system. We can handle that problem by comparing two models in which a pair of linear splines (plus one common intercept) is nested within two slopes+two intercepts. The splines would of course have a "knot" at the transition from year 1 to year 2. Assuming the splines aren't a significantly worse fit than the bigger model, we can then test for equality of their coefficients to see whether the slope changed. – whuber Dec 28 '10 at 18:33
• Thanks for the clarification and added comment. I'm afraid to say that my initial ideas were not that advanced, but I definitively have to think about this last suggestion. – chl Dec 28 '10 at 20:35
• Awesome, thanks, all. In summary, calculate t where $t=\frac{b1-b2}{se_{b1-b2}}$. To estimate the denominator, you can calculate this as $\sqrt{se_{b1}^2+se_{b2}^2}$. You can also use the sums of squares and a different formula if you like, but you'll get the same result. Then just pull out the df, and voila, answer. – jebyrnes Dec 28 '10 at 20:52