# Restricted Maximum Likelihood Estimates of Multilevel Regression Model

A two-level regression model :

$$Y_{ij} = \gamma_{00} + \gamma_{10}X_{ij} + \gamma_{01}Z_j + \gamma_{11}X_{ij}Z_j + u_{0j} + u_{1j}X_{ij} + e_{ij}$$

where $e_{ij}\sim N(0,\sigma^2_e)$ and , $$\begin{bmatrix} u_{0j} \\ u_{1j} \\ \end{bmatrix}\sim N \begin{pmatrix} \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix}, \begin{bmatrix} \sigma^2_{u0}&0 \\ 0&\sigma^2_{u1}\\ \end{bmatrix} \end{pmatrix}$$

Can anyone please give reference or explain in detail how is to derive the restricted maximum likelihood estimates of fixed effect parameters $(\gamma_{00},\gamma_{10},\gamma_{01},\gamma_{11})$ and random effect parameters $(\sigma^2_{u0},\sigma^2_{u1})$ ?

In this book, section 2.4.2 described restricted maximum likelihood estimation procedure for simpler model than the two-level regression model.

• G.M. Fitzmaurice, N.M. Laird, J.H. Waird, "Applied longitudinal analysis" has a section on REML. – user83346 Aug 15 '15 at 7:30
• Another reference is this one (page 75 and next): verde.esalq.usp.br/~jorge/cursos/modelos_longitudinais/… – user83346 Aug 15 '15 at 7:48