I'm reading the following paper to learn about multilevel modeling:

Multilevel Modeling for Binary Data

Guang Guo and Hongxin Zhao

Annual Review of Sociology

Vol. 26, (2000), pp. 441-462

They start by discussing the multilevel linear model. On Page 445, they present Model 1, which seems to be a random coefficients model with a random intercept and a fixed slope.

$Y_{ij} = \beta_0 + \beta_1 x_{ij} + u_j + e_{ij}$

$E(u_j) = E(e_{ij}) = 0$

$Var(u_j) = \sigma_u^2$

$Var(e_{ij}) = \sigma_e^2$

$Cov(u_j, e_{ij}) = 0$

$Cov(u_j, u_{h}) = 0 \ \ for \ \ j \neq h$

They then extend this 2-level model to a 3-level model, and yet it contains a random slope. Thus, the extension seems inconsistent. Here is Model #2.

$Y_{ijk} = \beta_0 + \beta_1 x_{ijk} + u_{1jk}x_{ijk} + v_{0k} + u_{0jk} + e_{0ijk}$

$v_{0k}$ and $u_{0jk}$ are random intercepts for level 3 and level 2, respectively.

$x_{ijk}$ is the observed predictor at level 1.

$u_{1jk}$ is $x_{ijk}$'s random effect at level 2.

There wasn't a random slope in the 2-level model, so why is there a need for $u_{1jk}x_{ijk}$ in the 3-level model?


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