# normal equations

#### Dgphru

Hi, for this model,

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

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?

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#### matheagle

MHF Hall of Honor
I use matrices

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

• Dgphru

#### Dgphru

I use matrices

$$\displaystyle \hat\beta=\left(X^t X\right)^{-1}X^tY$$
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. $$\displaystyle \mu, T_1, T_2, T_3, B_1$$ etc.

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

which now i realise i didnt put into my question, (and i will after this post) do i differentiate it with respect to $$\displaystyle TB_{11}, TB_{12}, TB_{13}, TB_{21}$$ etc. ??

#### matheagle

MHF Hall of Honor
You should have one equation per parameter that you're differentiating with respect to.

• Dgphru

#### Dgphru

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?

#### matheagle

MHF Hall of Honor
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.

• Dgphru

#### Dgphru

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?