Linear statistical model: inference about interaction coefficients, two-factor

[solar42]

Homework Statement

I have measurements of some response of a gene, and two factors: the gene, g=1...G and whether the patient/subject has a certain disease, t=1,2.

the full model is
$$y_{gtk}=\mu+\alpha_g+\beta_t +(\alpha\beta)_{gt}+\epsilon_{gtk}$$

I know that to see if genes have any connection at all with the disease, I just fit the reduced model without the $$(\alpha\beta)_{gt}$$ interaction and compare the two, but if I want to see if, say, gene number g=25 has anything to do with the disease... I know that the null hypothesis is $$H_0 : (\alpha\beta)_{25,t}, \textrm{ equal for all } t$$, but how do I test this hypothesis? I am confused at what to do when I can't drop the whole factor and compare.
I don't want to know how to do this in R or something, but how to do it by hand.

Homework Equations

The Attempt at a Solution

Well, there are only two levels t=1,2 , so we can basically test $$H_0 : (\alpha\beta)_{25,1}-(\alpha\beta)_{25,2}=0$$, but how!?

 

[lippyka]

I have tried looking on the internet but all I can find is the ANOVA F test which doesn't seem to apply here. I feel like it should be simple, but I'm completely lost.