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Express the following model in matrix form, ie: specify $\beta$ and $X$ so that the model can be written as $Y = X \beta + \epsilon$.

The model $Y_{ij} = \mu_i + \epsilon_i$ where $Y_{ij}$ represents the $j$-th observation observed at level $i$ of a qualitative explanatory variable $T$, for $i=0,...,4$ and $j=1,2$.

I know how to write the vector $\beta$ (just a column of $\mu_0$ to $\mu_4$), but how do you write the matrix X?

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Write $\tilde{Y}$ as a vector of length $10,$ by stacking $Y_{01},Y_{02},Y_{11},Y_{12},...,Y_{42}.$ Similarly define $\tilde{\varepsilon}.$ You are now looking for a $10\times 5$ matrix $X$.

$\beta^T=(\mu_0,\mu_1,\mu_2,\mu_3,\mu_4)$

$$X=\begin{bmatrix}1&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&0&0&0&1\end{bmatrix}$$

Then $\tilde{Y}=X\beta+\tilde{\varepsilon}$ is an expression of $Y_{ij}=\mu_i+\varepsilon_{ij}.$

If instead you had $Y_{ij}=\mu_0+\mu_i+\varepsilon_{ij},$ $\tilde{Y}$ and $\tilde{\varepsilon}$ will have length $8.$ Then

$$X=\begin{bmatrix}1&1&0&0\\ 1&1&0&0\\ 1&1&0&0\\ 1&1&0&0\\ 1&0&1&0\\ 1&0&1&0\\ 1&0&0&1\\ 1&0&0&1\end{bmatrix}$$

will be your design matrix so that $\tilde{Y}=X\beta+\tilde{\varepsilon}.$

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  • $\begingroup$ I understand that, my problem was with how to construct the design matrix X. I know it is made up of 1's and 0's, and that the first column is just 1's, but how do you determine the other entries? $\endgroup$ – Student_1996 Apr 20 at 21:03
  • $\begingroup$ The first column will not be all $1$'s in the form you wrote. To get $Y_{ij}=\mu_i+\epsilon_{ij},$ take $X_{ij}=1$ and $0$ otherwise. See if this helps. $\endgroup$ – Arnab Auddy Apr 20 at 22:00
  • $\begingroup$ I still don't understand for which cases the ij-th entry is 1? $\endgroup$ – Student_1996 Apr 21 at 8:18
  • $\begingroup$ Sorry I was not clear enough. I have edited the answer. $\endgroup$ – Arnab Auddy Apr 21 at 19:48

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