# normal equations

• May 8th 2010, 09:18 PM
Dgphru
normal equations
Hi, for this model,

$y_{ijkl} = \mu + T_i + B_j + S_k + TB_{ij} + e_{ijkl},$ $i = 1,2,3; j = 1..3; k = 1..2; l=1..5$

am i supposed to have 1 + 3 + 3 + 2 + 5 = 14 normal equations?
or am i just meant to have 1 + 3 + 3 + 2 = 9 normal equations?
• May 8th 2010, 10:04 PM
matheagle
I use matrices

$\hat\beta=\left(X^t X\right)^{-1}X^tY$
• May 8th 2010, 11:10 PM
Dgphru
Quote:

Originally Posted by matheagle
I use matrices

$\hat\beta=\left(X^t X\right)^{-1}X^tY$

Thanks but im wondering actually how many normal equations am i supposed to find?

Because the way we do it, is we find the error sum of squares of the model and then differentiate it wrt each of the terms i.e. $\mu, T_1, T_2, T_3, B_1$ etc.

so so far i have 9 normal equations, but I'm wondering if for the interaction term $TB_{ij}$

which now i realise i didnt put into my question, (and i will after this post) do i differentiate it with respect to $TB_{11}, TB_{12}, TB_{13}, TB_{21}$ etc. ??
• May 8th 2010, 11:21 PM
matheagle
You should have one equation per parameter that you're differentiating with respect to.
• May 8th 2010, 11:30 PM
Dgphru
Quote:

Originally Posted by matheagle
You should have one equation per parameter that you're differentiating with respect to.

Thanks but like I'm really stupid as in when i just read your post, i dont get it.. so like when you say parameter are you refering to the i, j, k and l's? or are you referring to T, B, S, TBs?
• May 8th 2010, 11:52 PM
matheagle
Quote:

Originally Posted by Dgphru
Thanks but like I'm really stupid as in when i just read your post, i dont get it.. so like when you say parameter are you refering to the i, j, k and l's? or are you referring to T, B, S, TBs?

The mu's tau's beta's... are the unknown parameters
and you need to differentiate (yuk, the projection matrix is so much easier) wrt each of these and solve all those equations.
• May 9th 2010, 12:32 AM
Dgphru
Quote:

Originally Posted by matheagle
The mu's tau's beta's... are the unknown parameters
and you need to differentiate (yuk, the projection matrix is so much easier) wrt each of these and solve all those equations.

Thank you again. So just confirming, this means that I should have 5 normal equations?