glmerMod.md
In equatiomatic: Transform Models into 'LaTeX' Equations

colorizing works

$$
\begin{aligned}
  {\color{blue}{\operatorname{Stops}}}_{i}  &\sim \operatorname{Poisson}(\lambda_i) \\
    \log(\lambda_i) &=\alpha_{j[i]} + \beta_{1}({\color{red}{\operatorname{Ethnicity}}}{\color{purple}{_{\operatorname{hispanic}}}}) + \beta_{2}({\color{red}{\operatorname{Ethnicity}}}{\color{purple}{_{\operatorname{white}}}}) \\
    \alpha_{j}  &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{Total\ Arrests}), \sigma^2_{\alpha_{j}} \right)
    \text{, for precinct j = 1,} \dots \text{,J}
\end{aligned}
$$

Renaming Variables works

$$
\begin{aligned}
  \operatorname{stops}_{i}  &\sim \operatorname{Poisson}(\lambda_i) \\
    \log(\lambda_i) &=\alpha_{j[i]} + \beta_{1}(\operatorname{Ethnicity}_{\operatorname{Hispanic/Latino}}) + \beta_{2}(\operatorname{Ethnicity}_{\operatorname{White}}) \\
    \alpha_{j}  &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{Total\ Arrests}), \sigma^2_{\alpha_{j}} \right)
    \text{, for precinct j = 1,} \dots \text{,J}
\end{aligned}
$$

Standard Poisson regression models work

$$
\begin{aligned}
  \operatorname{stops}_{i}  &\sim \operatorname{Poisson}(\lambda_i) \\
    \log(\lambda_i) &=\alpha_{j[i]} + \beta_{1}(\operatorname{eth}_{\operatorname{hispanic}}) + \beta_{2}(\operatorname{eth}_{\operatorname{white}}) \\
    \alpha_{j}  &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
    \text{, for precinct j = 1,} \dots \text{,J}
\end{aligned}
$$

$$
\begin{aligned}
  \operatorname{stops}_{i}  &\sim \operatorname{Poisson}(\lambda_i) \\
    \log(\lambda_i) &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{eth}_{\operatorname{hispanic}}) + \beta_{2j[i]}(\operatorname{eth}_{\operatorname{white}}) \\    
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\alpha_{j} \\
      &\beta_{1j} \\
      &\beta_{2j}
    \end{aligned}
  \end{array}
\right)
  &\sim N \left(
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{total\_arrests}) \\
      &\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{total\_arrests}) \\
      &\gamma^{\beta_{2}}_{0} + \gamma^{\beta_{2}}_{1}(\operatorname{total\_arrests})
    \end{aligned}
  \end{array}
\right)
, 
\left(
  \begin{array}{ccc}
     \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} \\ 
     \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} \\ 
     \rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}}
  \end{array}
\right)
 \right)
    \text{, for precinct j = 1,} \dots \text{,J}
\end{aligned}
$$

Poisson regression models with an offset work

$$
\begin{aligned}
  \operatorname{stops}_{i}  &\sim \operatorname{Poisson}(\lambda_i) \\
    \log(\lambda_i) &=\log(\operatorname{arrests}) + \alpha_{j[i]} + \beta_{1}(\operatorname{eth}_{\operatorname{hispanic}}) + \beta_{2}(\operatorname{eth}_{\operatorname{white}}) \\
    \alpha_{j}  &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
    \text{, for precinct j = 1,} \dots \text{,J}
\end{aligned}
$$

$$
\begin{aligned}
  \operatorname{stops}_{i}  &\sim \operatorname{Poisson}(\lambda_i) \\
    \log(\lambda_i) &=\log(\operatorname{arrests}) + \alpha_{j[i]} + \beta_{1j[i]}(\operatorname{eth}_{\operatorname{hispanic}}) + \beta_{2j[i]}(\operatorname{eth}_{\operatorname{white}}) \\    
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\alpha_{j} \\
      &\beta_{1j} \\
      &\beta_{2j}
    \end{aligned}
  \end{array}
\right)
  &\sim N \left(
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{total\_arrests}) \\
      &\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{total\_arrests}) \\
      &\gamma^{\beta_{2}}_{0} + \gamma^{\beta_{2}}_{1}(\operatorname{total\_arrests})
    \end{aligned}
  \end{array}
\right)
, 
\left(
  \begin{array}{ccc}
     \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} \\ 
     \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} \\ 
     \rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}}
  \end{array}
\right)
 \right)
    \text{, for precinct j = 1,} \dots \text{,J}
\end{aligned}
$$

Binomial Logistic Regression models work

$$
\begin{aligned}
  \operatorname{bush}_{i}  &\sim \operatorname{Binomial}(n = 1, \operatorname{prob}_{\operatorname{bush} = 1} = \widehat{P}) \\
    \log\left[\frac{\hat{P}}{1 - \hat{P}} \right] &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{black}) + \beta_{2}(\operatorname{female}) + \beta_{3}(\operatorname{edu}) \\    
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\alpha_{j} \\
      &\beta_{1j}
    \end{aligned}
  \end{array}
\right)
  &\sim N \left(
\left(
  \begin{array}{c} 
    \begin{aligned}
      &\mu_{\alpha_{j}} \\
      &\mu_{\beta_{1j}}
    \end{aligned}
  \end{array}
\right)
, 
\left(
  \begin{array}{cc}
     \sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\ 
     \rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
  \end{array}
\right)
 \right)
    \text{, for state j = 1,} \dots \text{,J}
\end{aligned}
$$