Single-level GLM

The two-level model written above can be re-written as a single-level model by substituting equation 2 into a block form of equation 1, giving

$$Y = X X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+ X \eta + \epsilon$$ (3)

where now all the first-level GLMs have been combined such that

$$Y\! =\! \left[\! \begin{array}{c} Y_1 \\ Y_2 \\ \vdots \\ Y_N \e... ...{c} \epsilon_1 \\ \epsilon_2 \\ \vdots \\ \epsilon_N \end{array}\! \right],\;$$

$$\textrm{E}(\eta) = 0 \; , \qquad \textrm{E}(\epsilon) = 0 \; , \q... ... 0 \\ \vdots& & \ddots & \vdots \\ 0 & 0 & \cdots & V_N \end{array} \right].$$

The two error terms in equation 3 can simply be combined such that

$$Y = X X_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}\beta_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}+ \gamma,$$ (4)

where

$$\textrm{E}(\gamma) = 0$$   and$$\quad \textrm{Cov}(\gamma) = W = X V_{\mbox{\tiny\textit{\sffamily {$\!$G}}}}^{\mbox{}}X^{\mbox{\scriptsize\textit{\sffamily {$\!$T}}}}+ V.$$

It is easy to see that this model is equivalent to the two-level version presented as equations 1 and 2.