# 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

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.
• 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