# How to test for parallelism for two linear models?

I'm taking a graduate course in regression analysis and I'm suck on a particular homework question that should be very simple to me!

I have the following model:

E(y) = B0 + B1x1 + B2x2 + B3x3 + B4x1x3 + B5x2x3


x3 is coded as 1 if "smoker" and 0 if "non-smoker".

Therefore the regression equations are:

x3 = 1: E(y) = (B3 + B0) + (B1 + B4)x1 + (B2 + B5)x2
x3 = 0: E(y*) = B0 + B1x1 + B2x2


Now I know how to test for parallelism if x2 is absent in the models:

H0: B4 = 0
H1: B4 != 0


But I'm lost as to what to do with the inclusion of the x2 variable. Parallelism is obviously testing for slope, but I'm not sure where to find the "slope" coefficient.

I was thinking about using an F-Test but then I realized I don't actually want to test the whole model, just the parallelism.

Could someone please point me into the right direction? Even hints would be sufficient.

## 1 Answer

I am bit unsure what exactly you mean by 'parallelism' but perhaps you mean that you want to test if the interaction terms are significant or not in which case you would do a joint test that B4=0 and B5=0.

• Right. Rewriting the x3=1 case as E[y] = B0' + B1'x1 + B2'x2 shows that we're really fitting two models (with a common error variance): one for non-smokers with parameters B0, B1, B2 and another for smokers with parameters B0', B1', B2'. "Parallelism" means B1 = B1' and B2 = B2' (which says, geometrically, that plots of the fits are parallel planes). In terms of the original variables these hypotheses are B4 = 0 and B5 = 0, exactly as Srikant has proposed. – whuber Nov 7 '10 at 21:42
• @Srikant, @whuber - Yeah this is what I was leaning towards (doing the B1=B1' and B2=B2' but wasn't sure where to go, but now I've got a good idea. Thank you both. – TheCloudlessSky Nov 7 '10 at 21:48