# Two-Level Model in Matrix Notation

#### Cynderella

##### New Member
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.

Last edited: