How to tell if two linear regression slopes (one theoretical, one observed) are significantly different?

Question

I have created a linear regression model for a set of data. I have calculated the slope and intercept of the line.

If given a theoretical slope for this kind of data, how can I test to see if the slope that I calculated and the theoretical slope are significantly different?

For example, the slope I calculated is 0.02 and the theoretical slope is 0.01, are they significantly different? How can I tell? I am working in R.

Answer 1

People are usually interested in testing if a parameter estimate is equal to $0$ because that represents no relationship. (In other contexts, 'no relationship' can be other values, such as a ratio of $1$.) However, there is no reason you need to use that specific value as your null. If you have a theoretical basis for wanting to determine if the observed slope is consistent with $.01$, you very much can use that as your null. You will just have to do that manually, since it isn't what software will output by default. The standard formula for a $t$-test is:
$$ t = \frac{(\hat\beta_j - \beta_{j\text{ NULL}})}{SE} $$ Since $\beta_{j\text{ NULL}}$ is taken by default to be $0$, that drops out and software just provides your estimate divided by the standard error. Here, you will just need to do the arithmetic yourself. Then you compare your computed $t$-statistic to the relevant $t$-distribution with the same degrees of freedom as the original test.