A two-level model, with one explanatory variable at the individual level (X) and one explanatory variable at the group level (Z):

\(Y_{ij}=\gamma_{00}+\gamma_{10}X_{ij}+\gamma_{01}Z_{j}+\gamma_{11}X_{ij}Z_{j}+u_{0j}+u_{1j}X_{ij}+e_{ij}\ldots (1)\)

correlation between \(u_{0j}\) and \(u_{1j}\) is 0 .

The matrix form of a mixed model collects the fixed effects in a vector \(\beta\), and the random effects in a vector \(u\), and finally the random error term, which is also a random effect factor in the vector \(e\). A formal definition is

\(Y=X\beta+Zu+e\ldots (2)\)

with \(X\) the known design matrix for fixed effects and \(Z\) the known design matrix for random effects .

Now I want to write down equation (1) in matrix form. But I can't visualize what will be the dimension and elements in each vector/matrix in it.

Say, in equation (1), I have 3 groups (J=3) and 2 individuals (i=2) in each group so that the total sample size, N=6 .

Then equation (2) will be,

\(\boldsymbol Y=

\begin{bmatrix}

y_{11}\\

y_{21}\\

y_{12}\\

y_{22}\\

y_{13}\\

y_{23}\\

\end{bmatrix},\quad\quad \boldsymbol e=

\begin{bmatrix}

e_{11}\\

e_{21}\\

e_{12}\\

e_{22}\\

e_{13}\\

e_{23}\\

\end{bmatrix}

\)

and is \(\beta=

\begin{bmatrix}

\gamma_{00} \\

\gamma_{10} \\

\gamma_{01}\\

\gamma_{11}\\

\end{bmatrix} ?

\)

How will be \(X\) , \(Z\) and \(u\) in equation (2) look like ?

Any help is appreciated. Many thanks.

\(Y_{ij}=\gamma_{00}+\gamma_{10}X_{ij}+\gamma_{01}Z_{j}+\gamma_{11}X_{ij}Z_{j}+u_{0j}+u_{1j}X_{ij}+e_{ij}\ldots (1)\)

correlation between \(u_{0j}\) and \(u_{1j}\) is 0 .

The matrix form of a mixed model collects the fixed effects in a vector \(\beta\), and the random effects in a vector \(u\), and finally the random error term, which is also a random effect factor in the vector \(e\). A formal definition is

\(Y=X\beta+Zu+e\ldots (2)\)

with \(X\) the known design matrix for fixed effects and \(Z\) the known design matrix for random effects .

Now I want to write down equation (1) in matrix form. But I can't visualize what will be the dimension and elements in each vector/matrix in it.

Say, in equation (1), I have 3 groups (J=3) and 2 individuals (i=2) in each group so that the total sample size, N=6 .

Then equation (2) will be,

\(\boldsymbol Y=

\begin{bmatrix}

y_{11}\\

y_{21}\\

y_{12}\\

y_{22}\\

y_{13}\\

y_{23}\\

\end{bmatrix},\quad\quad \boldsymbol e=

\begin{bmatrix}

e_{11}\\

e_{21}\\

e_{12}\\

e_{22}\\

e_{13}\\

e_{23}\\

\end{bmatrix}

\)

and is \(\beta=

\begin{bmatrix}

\gamma_{00} \\

\gamma_{10} \\

\gamma_{01}\\

\gamma_{11}\\

\end{bmatrix} ?

\)

How will be \(X\) , \(Z\) and \(u\) in equation (2) look like ?

Any help is appreciated. Many thanks.

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