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

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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
[tex]
y_{gtk}=\mu+\alpha_g+\beta_t +(\alpha\beta)_{gt}+\epsilon_{gtk}

[/tex]

I know that to see if genes have any connection at all with the disease, I just fit the reduced model without the [tex] (\alpha\beta)_{gt} [/tex] 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 [tex] H_0 : (\alpha\beta)_{25,t}, \textrm{ equal for all } t[/tex], 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 [tex] H_0 : (\alpha\beta)_{25,1}-(\alpha\beta)_{25,2}=0 [/tex], but how!?

Homework Statement





Homework Equations





The Attempt at a Solution

 

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