[R-sig-ME] Multilevel equation
Brian Hudson
bhud@on@g@u @end|ng |rom gm@||@com
Mon Jul 19 17:42:39 CEST 2021
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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