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I have some difficulties in understanding the notation of multilevel regression models. Let's consider, for example, a varying intercept and varying slope model with just one level-I predictor. We have:

Model of level I: $y_{ij}=\beta _{0j}+\beta _{1j}x_{ij}+\epsilon _{ij}$
Model of level II:
$\beta _{0j}=\gamma _{00}+u_{0j}$
$\beta _{1j}=\gamma _{10}+u_{1j}$
Combined model: $y_{ij}=\gamma _{00}+\gamma _{10}x_{ij}+u_{0j}+u_{1j}x_{ij}+\epsilon _{ij}$

where:
$\gamma _{00},\gamma _{10}$ are fixed

$u_{0j}\sim N(0,\sigma _{\beta_{0}}^{2})$
$u_{1j}\sim N(0,\sigma _{\beta_{1}}^{2})$
$\epsilon _{ij} \sim N(0,\sigma _{\epsilon}^{2})$

$\epsilon _{ij} \bot u_{0j}$
$\epsilon _{ij} \bot u_{1j}$

$\beta _{0j}\sim N(\gamma _{00},\sigma _{\beta_{0}}^{2})$
$\beta _{1j}\sim N(\gamma _{10},\sigma _{\beta_{1}}^{2})$
$Cov(\beta _{0j},\beta _{1j})= \rho \sigma _{\beta_{0}}\sigma _{\beta_{1}}$

$Var(y_{ij} |x_{ij})=\sigma _{\beta_{0}}^{2}+x_{ij}^{2}\sigma _{\beta_{1}}^{2}+2\rho \sigma _{\beta_{0}}\sigma _{\beta_{1}}x_{ij}+\sigma _{\epsilon}^{2}$
$Var(y_{ij} |x_{ij})=\sigma _{y_{i}}^{2}$

Now my question is the following: which of these notations are correct and why?
1) $y_{ij}\sim N(\beta _{0j}+\beta _{1j}x_{ij}, \, \sigma _{\epsilon}^{2})$
2) $y_{ij}\sim N(\beta _{0j}+\beta _{1j}x_{ij}, \, \sigma _{y_{i}}^{2})$
3) $y_{ij} | \beta _{0j},\beta _{1j}\sim N(\beta _{0j}+\beta _{1j}x_{ij}, \, \sigma _{\epsilon}^{2})$
4) $y_{ij}\sim N(\gamma _{00}+\gamma _{10}x_{ij},\,\sigma _{\epsilon}^{2})$
5) $y_{ij}\sim N(\gamma _{00}+\gamma _{10}x_{ij},\,\sigma _{y_{i}}^{2})$
The fact is that I don't now if at the individual level I should consider $\beta _{0j}$ and $\beta _{1j}$ fixed or not. So I don't know if I should add to the within variance the between variance or not.
Thank you.

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This is how I would denote this hierarchical model,

$$y_{ij} \,\vert\, \alpha_{j},\beta_{j} \sim \textsf{N}\left( \alpha_j+\beta_j x_{ij},\sigma^2 \right)$$ $$\alpha_j\sim \textsf{N}\left( \gamma,\tau^2 \right)$$ $$\beta_j \sim\textsf{N}\left( \kappa,\phi^2 \right)$$ where $\gamma,\tau,\kappa,\phi $ are given constants. I changed notation since I think this looks cleaner without all the subscripts and rids any idea of interdependence. If there is covariance, then this might work better, $$y_{ij}\,\vert\,\vec{\beta}\sim\textsf{N}\left( \vec{\beta}\mathbf{X},\sigma^2 \right)$$ $$\vec{\beta}\sim\textsf{MVN}\left( \vec{\gamma},\Sigma \right)$$

where $\vec{\beta},\vec{\gamma}$ are given $2\times1$ vectors, and $\Sigma$ is a given $2\times2$ matrix.

Unless you've fitted the model, you cannot hold the top level fixed since it is conditional on the lower levels.

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  • $\begingroup$ +1, but why $\textsf N$ and not $\mathcal N$? $\endgroup$ – amoeba Apr 7 '15 at 14:52
  • $\begingroup$ $y_{ij} | \alpha_j,\beta_j\sim N(\alpha_j+\beta_j x_ij,\sigma^2)$. * Is that the model : $y_{ij}=\alpha_j+\beta_j x_ij+\epsilon_{ij}$ , where $\epsilon_{ij}\sim N(0,\sigma^2)$ ? * Why is $y_{ij} | \alpha_j,\beta_j$ instead of $y_{ij} | x_{ij} $ as i usually seen in linear regression model ? $\endgroup$ – ABC Apr 8 '15 at 1:23
  • $\begingroup$ $\alpha_j\sim \textsf{N}\left( \gamma,\tau^2 \right)$. * Is that the model: $\alpha_j=\gamma+u_{\alpha_j}$ , where $\gamma$ is fixed and $u_{\alpha_j}\sim N(0,\tau^2)$ ? $\endgroup$ – ABC Apr 8 '15 at 1:28
  • $\begingroup$ $\beta_j\sim \textsf{N}\left( \kappa,\phi^2 \right)$. * Is that the model: $\beta_j=\kappa+u_{\beta_j}$ , where $\kappa$ is fixed and $u_{\beta_j}\sim N(0,\phi^2)$ ? $\endgroup$ – ABC Apr 8 '15 at 1:31
  • $\begingroup$ $y_{ij}\,\vert\,\vec{\beta}\sim\textsf{N}\left( \vec{\beta}\mathbf{X},\sigma^2 \right)$ , so that number(3) in the question is correct , which is 3) $y_{ij} | \beta _{0j},\beta _{1j}\sim N(\beta _{0j}+\beta _{1j}x_{ij}, \, \sigma _{\epsilon}^{2})$. Isn't it ? $\endgroup$ – ABC Apr 8 '15 at 1:36

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