# How to test (and accept) that a coefficient in a linear regression model equals zero

I understand that in a linear regression model like:

$y_i = b_0 + b_1 x_i + \epsilon_i$

I can have a null and an alternative hypothesis:

$H_0: b_1 = 0$ and $H_1: b_1 \neq 0$.

And then I can reject $H_0$ or fail to reject $H_0$. But what if I want to accept that $b_1 = 0$?

• Look into equivalence testing. – Peter Flom - Reinstate Monica Oct 14 '13 at 22:00
• You can't show that it's 0, since it can be arbitrarily close to zero while being unequal to it. e.g. if $b_1 = 0.0000001$ then $b_1\neq 0$ - and you'd ideally reject that point null, yet with reasonable values for and moderate sample size (and for the disposition of the $x$'s I guess), you can't. Peter's suggestion to consider equivalence testing is a good one (but it's showing something a bit different from what you're asking). – Glen_b -Reinstate Monica Oct 14 '13 at 22:04
• Thanks @PeterFlom, equivalence testing is what I was looking for. If you write it as an answer I will gladly accept it. – papirrin Oct 14 '13 at 22:49

Look into equivalence testing. See this search for lots of threads. Also see Esteban & Nowacki

You cannot. "Accept that $b_1=0$" is the same as "reject that $b_1\ne 0$". But on what basis you could do this? No matter how many observations you have, you cannot distinguish between 0 and sufficiently small value of $b_1$. You can only accept that $|b_1|<\epsilon$ (the smaller $\epsilon$ the more observations you need).

• I agree with that. But that is from the perspective of significance testing. I was looking for a technique specifically designed to test the hypothesis of whether two groups are equal (or a coefficient is zero). – papirrin Oct 15 '13 at 6:13
• As user31264 and also glen_b said: You can't. Equivalence tests are only able to show that a parameter is close to a certain value but not that it has exactly this value. – Michael M Oct 15 '13 at 10:05