Combining parameters for Question 10.13

[wzlee12]

Hi everyone!

I understood the reason behind alpha_i + beta_j, and the sum of the linear predictors before they combine. I do not understand the logic behind the combinations (as in why we are doing these combinations in the solutions given). Also I would like to ask why the "combination" of gamma_0 and beta_j is not considered? I thought they can be considered too! Please help! Thank you!

Regards,
Wei Zhe

 

[John Lee]

The reason we combine is because we can't estimate all the parameters otherwise. So every time we add a new covariate will lose one parameter (either think of it being set to zero and other values adjusted, or combined).

For example age + sex = \(\gamma_0 + \gamma_1 x + \alpha_i\) has 4 unknowns - the two gammas, and the constant for male and the constant for female.

Since there's 4 unknowns surely 4 pieces of information should be able to solve it. So here you go:

male age 20 is 61.5
female age 20 is 66.5
male age 30 is 86.5
female age 30 is 91.5

 

[wzlee12]

Sir,

Thanks for your reply here. I now understand why we combine the parameters. Though what I really want to know is why we do not include \[\gamma_{0} \beta_{j}\] into the new \[\alpha_{i}\] as in the solution. If we were to include it then where will it go?

Thank you!

 

[John Lee]

Can you see they have different subscripts? We can't combine them - otherwise that will say the vehicle type is the same category as the gender.

 

[wzlee12]

Thank you sir for the clarification regarding why we cannot combine those together.

Just one more thing though (Hopefully I did not ask too much, as I struggle at this for quite long already ). Can we combine γ0βj with βj, since they are in a way similar? Note that in the solution that I posted there is no γ0βj. If yes, then I guess the βj in the solution is different with the one in the original equation. Correct? If not, hopefully you can clarify where I went wrong.

Thank you!

 

[John Lee]

Yes. You can combine them (as they are both constants with the same subscript). And yes - it will be a different \(\beta_j\) but they've just used the same name (which is confusing!)