Multilevel equation

Dear Brian,

I'd write it as follows. In the case of a Gaussian model, you only have to
write $Y_{ijk} \sim \mathcal{N}(\eta_{ijk}, \sigma^2)$ and drop the link
function. (And you could replace \eta with \mu). Basically, Y depends on a
distribution defined by some parameters. And these parameters might need
some further definition.

$i$: state index
$j$: year index
$k$: observation index
$X_m$: state_mnthyr_pred
$X_y$: state_year_pred
$X_s$: state_pred
$$Y_{ijk} \sim Binom(\pi_{ijk})$$
$$\eta_{ijk} = \frac{\pi_{ijk}}{1- \pi_{ijk}}$$
$$\eta_{ijk} = \beta_0 + \beta_1X_m + \beta_2 X_y + \beta_3 X_s + b_i +
b_{ij}$$
$$b_i\sim \mathcal{N}(0, \sigma_s^2)$$
$$b_{ij}\sim \mathcal{N}(0, \sigma_{y}^2)$$

Best regards,


ir. Thierry Onkelinx
Statisticus / Statistician

Vlaamse Overheid / Government of Flanders
INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE AND
FOREST
Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
thierry.onkelinx at inbo.be
Havenlaan 88 bus 73, 1000 Brussel
www.inbo.be

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Op ma 19 jul. 2021 om 17:44 schreef Brian Hudson <bhudson.gsu at gmail.com>:

Hello,

I am fitting a multilevel model in `lme4` and am having trouble writing the
equation for it. I very much appreciate any help. The formula and code is
below, but I am not sure if the equation represents the error correctly -
do I need to include error terms or is that captured by the distributions?
I am also not sure if I am representing the logit function correctly with
the indexing or functional form.

The data are comprised of US-state months nested within US-state-years and
US-states. I include predictors at each level and a varying intercept for
both state-years and states.

The formula looks like this in R:

```
as.formula(outcome ~ state_mnthyr_pred + state_year_pred + state_pred +
                         (1 | state) + (1 | state_year))
```
Where the outcome is dichotomous. The state months (e.g. jan-2010, feb-2010
... jan-2013) are nested with state years and within states.

The formula I am using can be seen here:

https://quicklatex.com/cache3/e9/ql_038eeb4e4e1b0af94d3ef69fe4ff7be9_l3.png
And the LaTeX code:

$$
\begin{aligned}
    \mu &=\alpha_{j[i],k[i]} +
\beta_{0}(\operatorname{state\_mnthyr\_pred})\ \\
    \alpha_{j}  &\sim N \left(\gamma_{0}^{\alpha} +
\gamma_{1}^{\alpha}(\operatorname{\textrm{state\_year\_pred}}),
\sigma^2_{\alpha_{j}} \right)
    \text{, for \textrm{State-Year} j = 1,} \dots \text{, J} \\
    \alpha_{k}  &\sim N \left(\gamma_{0}^{\alpha} +
\gamma_{1}^{\alpha}(\operatorname{\textrm{state\_pred}}),
\sigma^2_{\alpha_{k}} \right)
    \text{, for State k = 1,} \dots \text{, K}\\
\pi_{i} &=\frac{e_{i}^{\mu}}{1+e_{i}^{\mu}}\\
y_{i j k} \sim & \operatorname{Binom}\left(1, \pi_{i}\right)\\
\end{aligned}
$$

I really appreciate any help. Thank you.

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