Nothing
tests/testthat/_snaps/lmerMod.md
In equatiomatic: Transform Models into 'LaTeX' Equations
colorizing works
$$
\begin{aligned}
{\color{#FF00CC}{\operatorname{score}}}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}({\color{blue}{\operatorname{wave}}}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}}) + \gamma_{2}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}}) + \gamma_{3l[i]}^{\alpha}({\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma_{4}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma_{5}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}}) + \gamma^{\beta_{1}}_{2}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}}) + \gamma^{\beta_{1}}_{3}({\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{4}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{5}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}})
\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low}) + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}})
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Math extraction works
Code
extract_eq(m1)
Output
$$
\begin{aligned}
\operatorname{Reaction}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{\log(Days\ +\ 1)}) + \beta_{2}(\operatorname{\exp(Days)}) + \beta_{3}(\operatorname{Days}) + \beta_{4}(\operatorname{Days^2}) + \beta_{5}(\operatorname{Days^3}) + \beta_{6}(\operatorname{Days^4}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for Subject j = 1,} \dots \text{,J}
\end{aligned}
$$
Code
extract_eq(m2)
Output
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i]} + \beta_{1}(\operatorname{\log(wave\ +\ 1)}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{\exp(prop\_low)}) \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J}
\end{aligned}
$$
Code
extract_eq(m3)
Output
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low^2}) + \gamma_{3}^{\alpha}(\operatorname{prop\_low^3}) + \gamma_{4}^{\alpha}(\operatorname{prop\_low^4}) \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Implicit ID variables are handled
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]} + \beta_{1j[i],k[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for school j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
Renaming Variables works
$$
\begin{aligned}
\operatorname{Student\ Scores}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{Wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1k[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2k[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i],l[i]}^{\alpha}(\operatorname{treatment}) \\
&\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{1k} \\
&\gamma_{2k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1l[i]}^{\alpha}(\operatorname{P(low\ income)}) \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{1k}} \\
&\mu_{\gamma_{2k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{1k}} & \rho_{\alpha_{k}\gamma_{2k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{1k}} & \rho_{\beta_{1k}\gamma_{2k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \rho_{\gamma_{1k}\beta_{1k}} & \sigma^2_{\gamma_{1k}} & \rho_{\gamma_{1k}\gamma_{2k}} & \rho_{\gamma_{1k}\gamma_{3k}} \\
\rho_{\gamma_{2k}\alpha_{k}} & \rho_{\gamma_{2k}\beta_{1k}} & \rho_{\gamma_{2k}\gamma_{1k}} & \sigma^2_{\gamma_{2k}} & \rho_{\gamma_{2k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \rho_{\gamma_{3k}\gamma_{1k}} & \rho_{\gamma_{3k}\gamma_{2k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l} \\
&\gamma_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}} \\
&\mu_{\gamma_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} & \rho_{\alpha_{l}\gamma_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} & \rho_{\beta_{1l}\gamma_{1l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}} & \rho_{\gamma_{3l}\gamma_{1l}} \\
\rho_{\gamma_{1l}\alpha_{l}} & \rho_{\gamma_{1l}\beta_{1l}} & \rho_{\gamma_{1l}\gamma_{3l}} & \sigma^2_{\gamma_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Really big models work
Code
extract_eq(big_mod)
Output
$$
\begin{aligned}
\operatorname{rt}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\beta_{1j[i]}(\operatorname{n1\_intercept}) + \beta_{2j[i]}(\operatorname{n1\_warning1}) + \beta_{3j[i]}(\operatorname{n1\_cuing1}) + \beta_{4j[i]}(\operatorname{x1\_intercept}) + \beta_{5j[i]}(\operatorname{x1\_warning1}) + \beta_{6j[i]}(\operatorname{x1\_cuing1}) + \beta_{7j[i]}(\operatorname{n2\_intercept}) + \beta_{8j[i]}(\operatorname{n2\_warning1}) + \beta_{9j[i]}(\operatorname{n2\_cuing1}) + \beta_{10j[i]}(\operatorname{x2\_intercept}) + \beta_{11j[i]}(\operatorname{x2\_warning1}) + \beta_{12j[i]}(\operatorname{x2\_cuing1}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j} \\
&\beta_{4j} \\
&\beta_{5j} \\
&\beta_{6j} \\
&\beta_{7j} \\
&\beta_{8j} \\
&\beta_{9j} \\
&\beta_{10j} \\
&\beta_{11j} \\
&\beta_{12j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}} \\
&\mu_{\beta_{4j}} \\
&\mu_{\beta_{5j}} \\
&\mu_{\beta_{6j}} \\
&\mu_{\beta_{7j}} \\
&\mu_{\beta_{8j}} \\
&\mu_{\beta_{9j}} \\
&\mu_{\beta_{10j}} \\
&\mu_{\beta_{11j}} \\
&\mu_{\beta_{12j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccccccccccc}
\sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} & \rho_{\beta_{1j}\beta_{3j}} & \rho_{\beta_{1j}\beta_{4j}} & \rho_{\beta_{1j}\beta_{5j}} & \rho_{\beta_{1j}\beta_{6j}} & \rho_{\beta_{1j}\beta_{7j}} & \rho_{\beta_{1j}\beta_{8j}} & \rho_{\beta_{1j}\beta_{9j}} & \rho_{\beta_{1j}\beta_{10j}} & \rho_{\beta_{1j}\beta_{11j}} & \rho_{\beta_{1j}\beta_{12j}} \\
\rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}} & \rho_{\beta_{2j}\beta_{3j}} & \rho_{\beta_{2j}\beta_{4j}} & \rho_{\beta_{2j}\beta_{5j}} & \rho_{\beta_{2j}\beta_{6j}} & \rho_{\beta_{2j}\beta_{7j}} & \rho_{\beta_{2j}\beta_{8j}} & \rho_{\beta_{2j}\beta_{9j}} & \rho_{\beta_{2j}\beta_{10j}} & \rho_{\beta_{2j}\beta_{11j}} & \rho_{\beta_{2j}\beta_{12j}} \\
\rho_{\beta_{3j}\beta_{1j}} & \rho_{\beta_{3j}\beta_{2j}} & \sigma^2_{\beta_{3j}} & \rho_{\beta_{3j}\beta_{4j}} & \rho_{\beta_{3j}\beta_{5j}} & \rho_{\beta_{3j}\beta_{6j}} & \rho_{\beta_{3j}\beta_{7j}} & \rho_{\beta_{3j}\beta_{8j}} & \rho_{\beta_{3j}\beta_{9j}} & \rho_{\beta_{3j}\beta_{10j}} & \rho_{\beta_{3j}\beta_{11j}} & \rho_{\beta_{3j}\beta_{12j}} \\
\rho_{\beta_{4j}\beta_{1j}} & \rho_{\beta_{4j}\beta_{2j}} & \rho_{\beta_{4j}\beta_{3j}} & \sigma^2_{\beta_{4j}} & \rho_{\beta_{4j}\beta_{5j}} & \rho_{\beta_{4j}\beta_{6j}} & \rho_{\beta_{4j}\beta_{7j}} & \rho_{\beta_{4j}\beta_{8j}} & \rho_{\beta_{4j}\beta_{9j}} & \rho_{\beta_{4j}\beta_{10j}} & \rho_{\beta_{4j}\beta_{11j}} & \rho_{\beta_{4j}\beta_{12j}} \\
\rho_{\beta_{5j}\beta_{1j}} & \rho_{\beta_{5j}\beta_{2j}} & \rho_{\beta_{5j}\beta_{3j}} & \rho_{\beta_{5j}\beta_{4j}} & \sigma^2_{\beta_{5j}} & \rho_{\beta_{5j}\beta_{6j}} & \rho_{\beta_{5j}\beta_{7j}} & \rho_{\beta_{5j}\beta_{8j}} & \rho_{\beta_{5j}\beta_{9j}} & \rho_{\beta_{5j}\beta_{10j}} & \rho_{\beta_{5j}\beta_{11j}} & \rho_{\beta_{5j}\beta_{12j}} \\
\rho_{\beta_{6j}\beta_{1j}} & \rho_{\beta_{6j}\beta_{2j}} & \rho_{\beta_{6j}\beta_{3j}} & \rho_{\beta_{6j}\beta_{4j}} & \rho_{\beta_{6j}\beta_{5j}} & \sigma^2_{\beta_{6j}} & \rho_{\beta_{6j}\beta_{7j}} & \rho_{\beta_{6j}\beta_{8j}} & \rho_{\beta_{6j}\beta_{9j}} & \rho_{\beta_{6j}\beta_{10j}} & \rho_{\beta_{6j}\beta_{11j}} & \rho_{\beta_{6j}\beta_{12j}} \\
\rho_{\beta_{7j}\beta_{1j}} & \rho_{\beta_{7j}\beta_{2j}} & \rho_{\beta_{7j}\beta_{3j}} & \rho_{\beta_{7j}\beta_{4j}} & \rho_{\beta_{7j}\beta_{5j}} & \rho_{\beta_{7j}\beta_{6j}} & \sigma^2_{\beta_{7j}} & \rho_{\beta_{7j}\beta_{8j}} & \rho_{\beta_{7j}\beta_{9j}} & \rho_{\beta_{7j}\beta_{10j}} & \rho_{\beta_{7j}\beta_{11j}} & \rho_{\beta_{7j}\beta_{12j}} \\
\rho_{\beta_{8j}\beta_{1j}} & \rho_{\beta_{8j}\beta_{2j}} & \rho_{\beta_{8j}\beta_{3j}} & \rho_{\beta_{8j}\beta_{4j}} & \rho_{\beta_{8j}\beta_{5j}} & \rho_{\beta_{8j}\beta_{6j}} & \rho_{\beta_{8j}\beta_{7j}} & \sigma^2_{\beta_{8j}} & \rho_{\beta_{8j}\beta_{9j}} & \rho_{\beta_{8j}\beta_{10j}} & \rho_{\beta_{8j}\beta_{11j}} & \rho_{\beta_{8j}\beta_{12j}} \\
\rho_{\beta_{9j}\beta_{1j}} & \rho_{\beta_{9j}\beta_{2j}} & \rho_{\beta_{9j}\beta_{3j}} & \rho_{\beta_{9j}\beta_{4j}} & \rho_{\beta_{9j}\beta_{5j}} & \rho_{\beta_{9j}\beta_{6j}} & \rho_{\beta_{9j}\beta_{7j}} & \rho_{\beta_{9j}\beta_{8j}} & \sigma^2_{\beta_{9j}} & \rho_{\beta_{9j}\beta_{10j}} & \rho_{\beta_{9j}\beta_{11j}} & \rho_{\beta_{9j}\beta_{12j}} \\
\rho_{\beta_{10j}\beta_{1j}} & \rho_{\beta_{10j}\beta_{2j}} & \rho_{\beta_{10j}\beta_{3j}} & \rho_{\beta_{10j}\beta_{4j}} & \rho_{\beta_{10j}\beta_{5j}} & \rho_{\beta_{10j}\beta_{6j}} & \rho_{\beta_{10j}\beta_{7j}} & \rho_{\beta_{10j}\beta_{8j}} & \rho_{\beta_{10j}\beta_{9j}} & \sigma^2_{\beta_{10j}} & \rho_{\beta_{10j}\beta_{11j}} & \rho_{\beta_{10j}\beta_{12j}} \\
\rho_{\beta_{11j}\beta_{1j}} & \rho_{\beta_{11j}\beta_{2j}} & \rho_{\beta_{11j}\beta_{3j}} & \rho_{\beta_{11j}\beta_{4j}} & \rho_{\beta_{11j}\beta_{5j}} & \rho_{\beta_{11j}\beta_{6j}} & \rho_{\beta_{11j}\beta_{7j}} & \rho_{\beta_{11j}\beta_{8j}} & \rho_{\beta_{11j}\beta_{9j}} & \rho_{\beta_{11j}\beta_{10j}} & \sigma^2_{\beta_{11j}} & \rho_{\beta_{11j}\beta_{12j}} \\
\rho_{\beta_{12j}\beta_{1j}} & \rho_{\beta_{12j}\beta_{2j}} & \rho_{\beta_{12j}\beta_{3j}} & \rho_{\beta_{12j}\beta_{4j}} & \rho_{\beta_{12j}\beta_{5j}} & \rho_{\beta_{12j}\beta_{6j}} & \rho_{\beta_{12j}\beta_{7j}} & \rho_{\beta_{12j}\beta_{8j}} & \rho_{\beta_{12j}\beta_{9j}} & \rho_{\beta_{12j}\beta_{10j}} & \rho_{\beta_{12j}\beta_{11j}} & \sigma^2_{\beta_{12j}}
\end{array}
\right)
\right)
\text{, for id j = 1,} \dots \text{,J}
\end{aligned}
$$
Categorical variable level parsing works (from issue #140)
$$
\begin{aligned}
\operatorname{error}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{brochure}_{\operatorname{standard}}) + \beta_{2}(\operatorname{disease}_{\operatorname{DS}}) + \beta_{3}(\operatorname{brochure}_{\operatorname{standard}} \times \operatorname{disease}_{\operatorname{DS}}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for ID j = 1,} \dots \text{,J}
\end{aligned}
$$
Unconditional lmer models work
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Level 1 predictors work
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{female}) + \beta_{2}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}), \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Mean separate works as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\alpha_{j[i]} + \beta_{1}(\operatorname{female}) + \beta_{2}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}), \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Wrapping works as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{female})\ + \\
&\quad \beta_{2}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
Unstructured variance-covariances work as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{female}) + \beta_{2}(\operatorname{ses}) + \beta_{3j[i]}(\operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{3j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{3j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{3j}} \\
\rho_{\beta_{3j}\alpha_{j}} & \sigma^2_{\beta_{3j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{female}) + \beta_{2j[i]}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} \\
\rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{female}) + \beta_{2j[i]}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) + \beta_{4j[i]}(\operatorname{female} \times \operatorname{ses}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{4j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{4j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} & \rho_{\alpha_{j}\beta_{4j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} & \rho_{\beta_{1j}\beta_{4j}} \\
\rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}} & \rho_{\beta_{2j}\beta_{4j}} \\
\rho_{\beta_{4j}\alpha_{j}} & \rho_{\beta_{4j}\beta_{1j}} & \rho_{\beta_{4j}\beta_{2j}} & \sigma^2_{\beta_{4j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{female}) + \beta_{2j[i]}(\operatorname{ses}) + \beta_{3j[i]}(\operatorname{minority}) + \beta_{4j[i]}(\operatorname{female} \times \operatorname{ses}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j} \\
&\beta_{4j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}} \\
&\mu_{\beta_{4j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} & \rho_{\alpha_{j}\beta_{3j}} & \rho_{\alpha_{j}\beta_{4j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} & \rho_{\beta_{1j}\beta_{3j}} & \rho_{\beta_{1j}\beta_{4j}} \\
\rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}} & \rho_{\beta_{2j}\beta_{3j}} & \rho_{\beta_{2j}\beta_{4j}} \\
\rho_{\beta_{3j}\alpha_{j}} & \rho_{\beta_{3j}\beta_{1j}} & \rho_{\beta_{3j}\beta_{2j}} & \sigma^2_{\beta_{3j}} & \rho_{\beta_{3j}\beta_{4j}} \\
\rho_{\beta_{4j}\alpha_{j}} & \rho_{\beta_{4j}\beta_{1j}} & \rho_{\beta_{4j}\beta_{2j}} & \rho_{\beta_{4j}\beta_{3j}} & \sigma^2_{\beta_{4j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Group-level predictors work as expected
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1k[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2k[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i],l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{1k} \\
&\gamma_{2k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{1k}} \\
&\mu_{\gamma_{2k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{1k}} & \rho_{\alpha_{k}\gamma_{2k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{1k}} & \rho_{\beta_{1k}\gamma_{2k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \rho_{\gamma_{1k}\beta_{1k}} & \sigma^2_{\gamma_{1k}} & \rho_{\gamma_{1k}\gamma_{2k}} & \rho_{\gamma_{1k}\gamma_{3k}} \\
\rho_{\gamma_{2k}\alpha_{k}} & \rho_{\gamma_{2k}\beta_{1k}} & \rho_{\gamma_{2k}\gamma_{1k}} & \sigma^2_{\gamma_{2k}} & \rho_{\gamma_{2k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \rho_{\gamma_{3k}\gamma_{1k}} & \rho_{\gamma_{3k}\gamma_{2k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1k[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2k[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i],l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{1k} \\
&\gamma_{2k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1l[i]}^{\alpha}(\operatorname{prop\_low}) \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{1k}} \\
&\mu_{\gamma_{2k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{1k}} & \rho_{\alpha_{k}\gamma_{2k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{1k}} & \rho_{\beta_{1k}\gamma_{2k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \rho_{\gamma_{1k}\beta_{1k}} & \sigma^2_{\gamma_{1k}} & \rho_{\gamma_{1k}\gamma_{2k}} & \rho_{\gamma_{1k}\gamma_{3k}} \\
\rho_{\gamma_{2k}\alpha_{k}} & \rho_{\gamma_{2k}\beta_{1k}} & \rho_{\gamma_{2k}\gamma_{1k}} & \sigma^2_{\gamma_{2k}} & \rho_{\gamma_{2k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \rho_{\gamma_{3k}\gamma_{1k}} & \rho_{\gamma_{3k}\gamma_{2k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l} \\
&\gamma_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}} \\
&\mu_{\gamma_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} & \rho_{\alpha_{l}\gamma_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} & \rho_{\beta_{1l}\gamma_{1l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}} & \rho_{\gamma_{3l}\gamma_{1l}} \\
\rho_{\gamma_{1l}\alpha_{l}} & \rho_{\gamma_{1l}\beta_{1l}} & \rho_{\gamma_{1l}\gamma_{3l}} & \sigma^2_{\gamma_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{dist\_mean}) \\
&\mu_{\beta_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Interactions work as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{minority}) + \beta_{2}(\operatorname{female}) + \beta_{3}(\operatorname{female} \times \operatorname{minority}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]}, \sigma^2 \right) \\ \alpha_{j} &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1k[i],l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{2l[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{3l[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{4l[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma_{5l[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}}), \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\gamma_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\gamma_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\gamma_{1k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \sigma^2_{\gamma_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\gamma_{1l} \\
&\gamma_{2l} \\
&\gamma_{3l} \\
&\gamma_{4l} \\
&\gamma_{5l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\gamma_{1l}} \\
&\mu_{\gamma_{2l}} \\
&\mu_{\gamma_{3l}} \\
&\mu_{\gamma_{4l}} \\
&\mu_{\gamma_{5l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\gamma_{1l}} & \rho_{\alpha_{l}\gamma_{2l}} & \rho_{\alpha_{l}\gamma_{3l}} & \rho_{\alpha_{l}\gamma_{4l}} & \rho_{\alpha_{l}\gamma_{5l}} \\
\rho_{\gamma_{1l}\alpha_{l}} & \sigma^2_{\gamma_{1l}} & \rho_{\gamma_{1l}\gamma_{2l}} & \rho_{\gamma_{1l}\gamma_{3l}} & \rho_{\gamma_{1l}\gamma_{4l}} & \rho_{\gamma_{1l}\gamma_{5l}} \\
\rho_{\gamma_{2l}\alpha_{l}} & \rho_{\gamma_{2l}\gamma_{1l}} & \sigma^2_{\gamma_{2l}} & \rho_{\gamma_{2l}\gamma_{3l}} & \rho_{\gamma_{2l}\gamma_{4l}} & \rho_{\gamma_{2l}\gamma_{5l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\gamma_{1l}} & \rho_{\gamma_{3l}\gamma_{2l}} & \sigma^2_{\gamma_{3l}} & \rho_{\gamma_{3l}\gamma_{4l}} & \rho_{\gamma_{3l}\gamma_{5l}} \\
\rho_{\gamma_{4l}\alpha_{l}} & \rho_{\gamma_{4l}\gamma_{1l}} & \rho_{\gamma_{4l}\gamma_{2l}} & \rho_{\gamma_{4l}\gamma_{3l}} & \sigma^2_{\gamma_{4l}} & \rho_{\gamma_{4l}\gamma_{5l}} \\
\rho_{\gamma_{5l}\alpha_{l}} & \rho_{\gamma_{5l}\gamma_{1l}} & \rho_{\gamma_{5l}\gamma_{2l}} & \rho_{\gamma_{5l}\gamma_{3l}} & \rho_{\gamma_{5l}\gamma_{4l}} & \sigma^2_{\gamma_{5l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{treatment}_{\operatorname{1}})
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\ \alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}), \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{2}^{\alpha}(\operatorname{treatment}_{\operatorname{1}} \times \operatorname{wave}), \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{4}^{\alpha}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma_{5}^{\alpha}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{group}_{\operatorname{low}}) + \gamma^{\beta_{1}}_{2}(\operatorname{group}_{\operatorname{medium}}) + \gamma^{\beta_{1}}_{3}(\operatorname{treatment}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{4}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{5}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}})
\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low} \times \operatorname{treatment}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{2}(\operatorname{prop\_low}) + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low} \times \operatorname{treatment}_{\operatorname{1}})
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Alternate random effect VCV structures work
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{minority}) + \beta_{2j[i]}(\operatorname{female}) + \beta_{3j[i]}(\operatorname{female} \times \operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{j}} & 0 & 0 & 0 \\
0 & \sigma^2_{\beta_{1j}} & 0 & 0 \\
0 & 0 & \sigma^2_{\beta_{2j}} & 0 \\
0 & 0 & 0 & \sigma^2_{\beta_{3j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{minority}) + \beta_{2j[i]}(\operatorname{female}) + \beta_{3j[i]}(\operatorname{female} \times \operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{j}} & 0 & 0 & 0 \\
0 & \sigma^2_{\beta_{1j}} & 0 & 0 \\
0 & 0 & \sigma^2_{\beta_{2j}} & 0 \\
0 & 0 & 0 & \sigma^2_{\beta_{3j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i],m[i]} + \beta_{1j[i],k[i],m[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & 0 \\
0 & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for school.1 l = 1,} \dots \text{,L} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{m} \\
&\beta_{1m}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{m}} \\
&\mu_{\beta_{1m}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{m}} & 0 \\
0 & \sigma^2_{\beta_{1m}}
\end{array}
\right)
\right)
\text{, for district m = 1,} \dots \text{,M}
\end{aligned}
$$
Nested model syntax works
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for school:sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for school:sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for school:sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
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colorizing works
$$
\begin{aligned}
{\color{#FF00CC}{\operatorname{score}}}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}({\color{blue}{\operatorname{wave}}}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}}) + \gamma_{2}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}}) + \gamma_{3l[i]}^{\alpha}({\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma_{4}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma_{5}^{\alpha}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}}) + \gamma^{\beta_{1}}_{2}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}}) + \gamma^{\beta_{1}}_{3}({\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{4}(\operatorname{group}{\color{orange}{_{\operatorname{low}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{5}(\operatorname{group}{\color{orange}{_{\operatorname{medium}}}} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}})
\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low}) + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low} \times {\color{red}{\operatorname{treatment}}}_{\operatorname{1}})
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Math extraction works
Code
extract_eq(m1)
Output
$$
\begin{aligned}
\operatorname{Reaction}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{\log(Days\ +\ 1)}) + \beta_{2}(\operatorname{\exp(Days)}) + \beta_{3}(\operatorname{Days}) + \beta_{4}(\operatorname{Days^2}) + \beta_{5}(\operatorname{Days^3}) + \beta_{6}(\operatorname{Days^4}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for Subject j = 1,} \dots \text{,J}
\end{aligned}
$$
Code
extract_eq(m2)
Output
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i]} + \beta_{1}(\operatorname{\log(wave\ +\ 1)}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{\exp(prop\_low)}) \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J}
\end{aligned}
$$
Code
extract_eq(m3)
Output
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low^2}) + \gamma_{3}^{\alpha}(\operatorname{prop\_low^3}) + \gamma_{4}^{\alpha}(\operatorname{prop\_low^4}) \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Implicit ID variables are handled
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]} + \beta_{1j[i],k[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for school j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
Renaming Variables works
$$
\begin{aligned}
\operatorname{Student\ Scores}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{Wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1k[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2k[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i],l[i]}^{\alpha}(\operatorname{treatment}) \\
&\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{1k} \\
&\gamma_{2k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1l[i]}^{\alpha}(\operatorname{P(low\ income)}) \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{1k}} \\
&\mu_{\gamma_{2k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{1k}} & \rho_{\alpha_{k}\gamma_{2k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{1k}} & \rho_{\beta_{1k}\gamma_{2k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \rho_{\gamma_{1k}\beta_{1k}} & \sigma^2_{\gamma_{1k}} & \rho_{\gamma_{1k}\gamma_{2k}} & \rho_{\gamma_{1k}\gamma_{3k}} \\
\rho_{\gamma_{2k}\alpha_{k}} & \rho_{\gamma_{2k}\beta_{1k}} & \rho_{\gamma_{2k}\gamma_{1k}} & \sigma^2_{\gamma_{2k}} & \rho_{\gamma_{2k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \rho_{\gamma_{3k}\gamma_{1k}} & \rho_{\gamma_{3k}\gamma_{2k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l} \\
&\gamma_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}} \\
&\mu_{\gamma_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} & \rho_{\alpha_{l}\gamma_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} & \rho_{\beta_{1l}\gamma_{1l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}} & \rho_{\gamma_{3l}\gamma_{1l}} \\
\rho_{\gamma_{1l}\alpha_{l}} & \rho_{\gamma_{1l}\beta_{1l}} & \rho_{\gamma_{1l}\gamma_{3l}} & \sigma^2_{\gamma_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Really big models work
Code
extract_eq(big_mod)
Output
$$
\begin{aligned}
\operatorname{rt}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\beta_{1j[i]}(\operatorname{n1\_intercept}) + \beta_{2j[i]}(\operatorname{n1\_warning1}) + \beta_{3j[i]}(\operatorname{n1\_cuing1}) + \beta_{4j[i]}(\operatorname{x1\_intercept}) + \beta_{5j[i]}(\operatorname{x1\_warning1}) + \beta_{6j[i]}(\operatorname{x1\_cuing1}) + \beta_{7j[i]}(\operatorname{n2\_intercept}) + \beta_{8j[i]}(\operatorname{n2\_warning1}) + \beta_{9j[i]}(\operatorname{n2\_cuing1}) + \beta_{10j[i]}(\operatorname{x2\_intercept}) + \beta_{11j[i]}(\operatorname{x2\_warning1}) + \beta_{12j[i]}(\operatorname{x2\_cuing1}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j} \\
&\beta_{4j} \\
&\beta_{5j} \\
&\beta_{6j} \\
&\beta_{7j} \\
&\beta_{8j} \\
&\beta_{9j} \\
&\beta_{10j} \\
&\beta_{11j} \\
&\beta_{12j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}} \\
&\mu_{\beta_{4j}} \\
&\mu_{\beta_{5j}} \\
&\mu_{\beta_{6j}} \\
&\mu_{\beta_{7j}} \\
&\mu_{\beta_{8j}} \\
&\mu_{\beta_{9j}} \\
&\mu_{\beta_{10j}} \\
&\mu_{\beta_{11j}} \\
&\mu_{\beta_{12j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccccccccccc}
\sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} & \rho_{\beta_{1j}\beta_{3j}} & \rho_{\beta_{1j}\beta_{4j}} & \rho_{\beta_{1j}\beta_{5j}} & \rho_{\beta_{1j}\beta_{6j}} & \rho_{\beta_{1j}\beta_{7j}} & \rho_{\beta_{1j}\beta_{8j}} & \rho_{\beta_{1j}\beta_{9j}} & \rho_{\beta_{1j}\beta_{10j}} & \rho_{\beta_{1j}\beta_{11j}} & \rho_{\beta_{1j}\beta_{12j}} \\
\rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}} & \rho_{\beta_{2j}\beta_{3j}} & \rho_{\beta_{2j}\beta_{4j}} & \rho_{\beta_{2j}\beta_{5j}} & \rho_{\beta_{2j}\beta_{6j}} & \rho_{\beta_{2j}\beta_{7j}} & \rho_{\beta_{2j}\beta_{8j}} & \rho_{\beta_{2j}\beta_{9j}} & \rho_{\beta_{2j}\beta_{10j}} & \rho_{\beta_{2j}\beta_{11j}} & \rho_{\beta_{2j}\beta_{12j}} \\
\rho_{\beta_{3j}\beta_{1j}} & \rho_{\beta_{3j}\beta_{2j}} & \sigma^2_{\beta_{3j}} & \rho_{\beta_{3j}\beta_{4j}} & \rho_{\beta_{3j}\beta_{5j}} & \rho_{\beta_{3j}\beta_{6j}} & \rho_{\beta_{3j}\beta_{7j}} & \rho_{\beta_{3j}\beta_{8j}} & \rho_{\beta_{3j}\beta_{9j}} & \rho_{\beta_{3j}\beta_{10j}} & \rho_{\beta_{3j}\beta_{11j}} & \rho_{\beta_{3j}\beta_{12j}} \\
\rho_{\beta_{4j}\beta_{1j}} & \rho_{\beta_{4j}\beta_{2j}} & \rho_{\beta_{4j}\beta_{3j}} & \sigma^2_{\beta_{4j}} & \rho_{\beta_{4j}\beta_{5j}} & \rho_{\beta_{4j}\beta_{6j}} & \rho_{\beta_{4j}\beta_{7j}} & \rho_{\beta_{4j}\beta_{8j}} & \rho_{\beta_{4j}\beta_{9j}} & \rho_{\beta_{4j}\beta_{10j}} & \rho_{\beta_{4j}\beta_{11j}} & \rho_{\beta_{4j}\beta_{12j}} \\
\rho_{\beta_{5j}\beta_{1j}} & \rho_{\beta_{5j}\beta_{2j}} & \rho_{\beta_{5j}\beta_{3j}} & \rho_{\beta_{5j}\beta_{4j}} & \sigma^2_{\beta_{5j}} & \rho_{\beta_{5j}\beta_{6j}} & \rho_{\beta_{5j}\beta_{7j}} & \rho_{\beta_{5j}\beta_{8j}} & \rho_{\beta_{5j}\beta_{9j}} & \rho_{\beta_{5j}\beta_{10j}} & \rho_{\beta_{5j}\beta_{11j}} & \rho_{\beta_{5j}\beta_{12j}} \\
\rho_{\beta_{6j}\beta_{1j}} & \rho_{\beta_{6j}\beta_{2j}} & \rho_{\beta_{6j}\beta_{3j}} & \rho_{\beta_{6j}\beta_{4j}} & \rho_{\beta_{6j}\beta_{5j}} & \sigma^2_{\beta_{6j}} & \rho_{\beta_{6j}\beta_{7j}} & \rho_{\beta_{6j}\beta_{8j}} & \rho_{\beta_{6j}\beta_{9j}} & \rho_{\beta_{6j}\beta_{10j}} & \rho_{\beta_{6j}\beta_{11j}} & \rho_{\beta_{6j}\beta_{12j}} \\
\rho_{\beta_{7j}\beta_{1j}} & \rho_{\beta_{7j}\beta_{2j}} & \rho_{\beta_{7j}\beta_{3j}} & \rho_{\beta_{7j}\beta_{4j}} & \rho_{\beta_{7j}\beta_{5j}} & \rho_{\beta_{7j}\beta_{6j}} & \sigma^2_{\beta_{7j}} & \rho_{\beta_{7j}\beta_{8j}} & \rho_{\beta_{7j}\beta_{9j}} & \rho_{\beta_{7j}\beta_{10j}} & \rho_{\beta_{7j}\beta_{11j}} & \rho_{\beta_{7j}\beta_{12j}} \\
\rho_{\beta_{8j}\beta_{1j}} & \rho_{\beta_{8j}\beta_{2j}} & \rho_{\beta_{8j}\beta_{3j}} & \rho_{\beta_{8j}\beta_{4j}} & \rho_{\beta_{8j}\beta_{5j}} & \rho_{\beta_{8j}\beta_{6j}} & \rho_{\beta_{8j}\beta_{7j}} & \sigma^2_{\beta_{8j}} & \rho_{\beta_{8j}\beta_{9j}} & \rho_{\beta_{8j}\beta_{10j}} & \rho_{\beta_{8j}\beta_{11j}} & \rho_{\beta_{8j}\beta_{12j}} \\
\rho_{\beta_{9j}\beta_{1j}} & \rho_{\beta_{9j}\beta_{2j}} & \rho_{\beta_{9j}\beta_{3j}} & \rho_{\beta_{9j}\beta_{4j}} & \rho_{\beta_{9j}\beta_{5j}} & \rho_{\beta_{9j}\beta_{6j}} & \rho_{\beta_{9j}\beta_{7j}} & \rho_{\beta_{9j}\beta_{8j}} & \sigma^2_{\beta_{9j}} & \rho_{\beta_{9j}\beta_{10j}} & \rho_{\beta_{9j}\beta_{11j}} & \rho_{\beta_{9j}\beta_{12j}} \\
\rho_{\beta_{10j}\beta_{1j}} & \rho_{\beta_{10j}\beta_{2j}} & \rho_{\beta_{10j}\beta_{3j}} & \rho_{\beta_{10j}\beta_{4j}} & \rho_{\beta_{10j}\beta_{5j}} & \rho_{\beta_{10j}\beta_{6j}} & \rho_{\beta_{10j}\beta_{7j}} & \rho_{\beta_{10j}\beta_{8j}} & \rho_{\beta_{10j}\beta_{9j}} & \sigma^2_{\beta_{10j}} & \rho_{\beta_{10j}\beta_{11j}} & \rho_{\beta_{10j}\beta_{12j}} \\
\rho_{\beta_{11j}\beta_{1j}} & \rho_{\beta_{11j}\beta_{2j}} & \rho_{\beta_{11j}\beta_{3j}} & \rho_{\beta_{11j}\beta_{4j}} & \rho_{\beta_{11j}\beta_{5j}} & \rho_{\beta_{11j}\beta_{6j}} & \rho_{\beta_{11j}\beta_{7j}} & \rho_{\beta_{11j}\beta_{8j}} & \rho_{\beta_{11j}\beta_{9j}} & \rho_{\beta_{11j}\beta_{10j}} & \sigma^2_{\beta_{11j}} & \rho_{\beta_{11j}\beta_{12j}} \\
\rho_{\beta_{12j}\beta_{1j}} & \rho_{\beta_{12j}\beta_{2j}} & \rho_{\beta_{12j}\beta_{3j}} & \rho_{\beta_{12j}\beta_{4j}} & \rho_{\beta_{12j}\beta_{5j}} & \rho_{\beta_{12j}\beta_{6j}} & \rho_{\beta_{12j}\beta_{7j}} & \rho_{\beta_{12j}\beta_{8j}} & \rho_{\beta_{12j}\beta_{9j}} & \rho_{\beta_{12j}\beta_{10j}} & \rho_{\beta_{12j}\beta_{11j}} & \sigma^2_{\beta_{12j}}
\end{array}
\right)
\right)
\text{, for id j = 1,} \dots \text{,J}
\end{aligned}
$$
Categorical variable level parsing works (from issue #140)
$$
\begin{aligned}
\operatorname{error}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{brochure}_{\operatorname{standard}}) + \beta_{2}(\operatorname{disease}_{\operatorname{DS}}) + \beta_{3}(\operatorname{brochure}_{\operatorname{standard}} \times \operatorname{disease}_{\operatorname{DS}}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for ID j = 1,} \dots \text{,J}
\end{aligned}
$$
Unconditional lmer models work
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Level 1 predictors work
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{female}) + \beta_{2}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}), \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Mean separate works as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\alpha_{j[i]} + \beta_{1}(\operatorname{female}) + \beta_{2}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}), \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Wrapping works as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{female})\ + \\
&\quad \beta_{2}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
Unstructured variance-covariances work as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{female}) + \beta_{2}(\operatorname{ses}) + \beta_{3j[i]}(\operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{3j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{3j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{3j}} \\
\rho_{\beta_{3j}\alpha_{j}} & \sigma^2_{\beta_{3j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{female}) + \beta_{2j[i]}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} \\
\rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{female}) + \beta_{2j[i]}(\operatorname{ses}) + \beta_{3}(\operatorname{minority}) + \beta_{4j[i]}(\operatorname{female} \times \operatorname{ses}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{4j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{4j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} & \rho_{\alpha_{j}\beta_{4j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} & \rho_{\beta_{1j}\beta_{4j}} \\
\rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}} & \rho_{\beta_{2j}\beta_{4j}} \\
\rho_{\beta_{4j}\alpha_{j}} & \rho_{\beta_{4j}\beta_{1j}} & \rho_{\beta_{4j}\beta_{2j}} & \sigma^2_{\beta_{4j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{female}) + \beta_{2j[i]}(\operatorname{ses}) + \beta_{3j[i]}(\operatorname{minority}) + \beta_{4j[i]}(\operatorname{female} \times \operatorname{ses}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j} \\
&\beta_{4j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}} \\
&\mu_{\beta_{4j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} & \rho_{\alpha_{j}\beta_{2j}} & \rho_{\alpha_{j}\beta_{3j}} & \rho_{\alpha_{j}\beta_{4j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}} & \rho_{\beta_{1j}\beta_{2j}} & \rho_{\beta_{1j}\beta_{3j}} & \rho_{\beta_{1j}\beta_{4j}} \\
\rho_{\beta_{2j}\alpha_{j}} & \rho_{\beta_{2j}\beta_{1j}} & \sigma^2_{\beta_{2j}} & \rho_{\beta_{2j}\beta_{3j}} & \rho_{\beta_{2j}\beta_{4j}} \\
\rho_{\beta_{3j}\alpha_{j}} & \rho_{\beta_{3j}\beta_{1j}} & \rho_{\beta_{3j}\beta_{2j}} & \sigma^2_{\beta_{3j}} & \rho_{\beta_{3j}\beta_{4j}} \\
\rho_{\beta_{4j}\alpha_{j}} & \rho_{\beta_{4j}\beta_{1j}} & \rho_{\beta_{4j}\beta_{2j}} & \rho_{\beta_{4j}\beta_{3j}} & \sigma^2_{\beta_{4j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Group-level predictors work as expected
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1k[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2k[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i],l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{1k} \\
&\gamma_{2k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{1k}} \\
&\mu_{\gamma_{2k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{1k}} & \rho_{\alpha_{k}\gamma_{2k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{1k}} & \rho_{\beta_{1k}\gamma_{2k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \rho_{\gamma_{1k}\beta_{1k}} & \sigma^2_{\gamma_{1k}} & \rho_{\gamma_{1k}\gamma_{2k}} & \rho_{\gamma_{1k}\gamma_{3k}} \\
\rho_{\gamma_{2k}\alpha_{k}} & \rho_{\gamma_{2k}\beta_{1k}} & \rho_{\gamma_{2k}\gamma_{1k}} & \sigma^2_{\gamma_{2k}} & \rho_{\gamma_{2k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \rho_{\gamma_{3k}\gamma_{1k}} & \rho_{\gamma_{3k}\gamma_{2k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1k[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2k[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i],l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{1k} \\
&\gamma_{2k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1l[i]}^{\alpha}(\operatorname{prop\_low}) \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{1k}} \\
&\mu_{\gamma_{2k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{1k}} & \rho_{\alpha_{k}\gamma_{2k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{1k}} & \rho_{\beta_{1k}\gamma_{2k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \rho_{\gamma_{1k}\beta_{1k}} & \sigma^2_{\gamma_{1k}} & \rho_{\gamma_{1k}\gamma_{2k}} & \rho_{\gamma_{1k}\gamma_{3k}} \\
\rho_{\gamma_{2k}\alpha_{k}} & \rho_{\gamma_{2k}\beta_{1k}} & \rho_{\gamma_{2k}\gamma_{1k}} & \sigma^2_{\gamma_{2k}} & \rho_{\gamma_{2k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \rho_{\gamma_{3k}\gamma_{1k}} & \rho_{\gamma_{3k}\gamma_{2k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l} \\
&\gamma_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}} \\
&\mu_{\gamma_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} & \rho_{\alpha_{l}\gamma_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} & \rho_{\beta_{1l}\gamma_{1l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}} & \rho_{\gamma_{3l}\gamma_{1l}} \\
\rho_{\gamma_{1l}\alpha_{l}} & \rho_{\gamma_{1l}\beta_{1l}} & \rho_{\gamma_{1l}\gamma_{3l}} & \sigma^2_{\gamma_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3k[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k} \\
&\gamma_{3k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) \\
&\mu_{\beta_{1k}} \\
&\mu_{\gamma_{3k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} & \rho_{\alpha_{k}\gamma_{3k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}} & \rho_{\beta_{1k}\gamma_{3k}} \\
\rho_{\gamma_{3k}\alpha_{k}} & \rho_{\gamma_{3k}\beta_{1k}} & \sigma^2_{\gamma_{3k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{dist\_mean}) \\
&\mu_{\beta_{1l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Interactions work as expected
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1}(\operatorname{minority}) + \beta_{2}(\operatorname{female}) + \beta_{3}(\operatorname{female} \times \operatorname{minority}) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]}, \sigma^2 \right) \\ \alpha_{j} &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1k[i],l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{2l[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{3l[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{4l[i]}^{\alpha}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma_{5l[i]}^{\alpha}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}}), \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\gamma_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\gamma_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\gamma_{1k}} \\
\rho_{\gamma_{1k}\alpha_{k}} & \sigma^2_{\gamma_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\gamma_{1l} \\
&\gamma_{2l} \\
&\gamma_{3l} \\
&\gamma_{4l} \\
&\gamma_{5l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\gamma_{1l}} \\
&\mu_{\gamma_{2l}} \\
&\mu_{\gamma_{3l}} \\
&\mu_{\gamma_{4l}} \\
&\mu_{\gamma_{5l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\gamma_{1l}} & \rho_{\alpha_{l}\gamma_{2l}} & \rho_{\alpha_{l}\gamma_{3l}} & \rho_{\alpha_{l}\gamma_{4l}} & \rho_{\alpha_{l}\gamma_{5l}} \\
\rho_{\gamma_{1l}\alpha_{l}} & \sigma^2_{\gamma_{1l}} & \rho_{\gamma_{1l}\gamma_{2l}} & \rho_{\gamma_{1l}\gamma_{3l}} & \rho_{\gamma_{1l}\gamma_{4l}} & \rho_{\gamma_{1l}\gamma_{5l}} \\
\rho_{\gamma_{2l}\alpha_{l}} & \rho_{\gamma_{2l}\gamma_{1l}} & \sigma^2_{\gamma_{2l}} & \rho_{\gamma_{2l}\gamma_{3l}} & \rho_{\gamma_{2l}\gamma_{4l}} & \rho_{\gamma_{2l}\gamma_{5l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\gamma_{1l}} & \rho_{\gamma_{3l}\gamma_{2l}} & \sigma^2_{\gamma_{3l}} & \rho_{\gamma_{3l}\gamma_{4l}} & \rho_{\gamma_{3l}\gamma_{5l}} \\
\rho_{\gamma_{4l}\alpha_{l}} & \rho_{\gamma_{4l}\gamma_{1l}} & \rho_{\gamma_{4l}\gamma_{2l}} & \rho_{\gamma_{4l}\gamma_{3l}} & \sigma^2_{\gamma_{4l}} & \rho_{\gamma_{4l}\gamma_{5l}} \\
\rho_{\gamma_{5l}\alpha_{l}} & \rho_{\gamma_{5l}\gamma_{1l}} & \rho_{\gamma_{5l}\gamma_{2l}} & \rho_{\gamma_{5l}\gamma_{3l}} & \rho_{\gamma_{5l}\gamma_{4l}} & \sigma^2_{\gamma_{5l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{treatment}_{\operatorname{1}})
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & \rho_{\alpha_{j}\beta_{1j}} \\
\rho_{\beta_{1j}\alpha_{j}} & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\ \alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1}(\operatorname{wave}), \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{2}^{\alpha}(\operatorname{treatment}_{\operatorname{1}} \times \operatorname{wave}), \sigma^2_{\alpha_{j}} \right)
\text{, for sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for school k = 1,} \dots \text{,K} \\
\alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i]} + \beta_{1j[i],k[i],l[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{group}_{\operatorname{low}}) + \gamma_{2}^{\alpha}(\operatorname{group}_{\operatorname{medium}}) + \gamma_{3l[i]}^{\alpha}(\operatorname{treatment}_{\operatorname{1}}) + \gamma_{4}^{\alpha}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma_{5}^{\alpha}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{1}(\operatorname{group}_{\operatorname{low}}) + \gamma^{\beta_{1}}_{2}(\operatorname{group}_{\operatorname{medium}}) + \gamma^{\beta_{1}}_{3}(\operatorname{treatment}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{4}(\operatorname{group}_{\operatorname{low}} \times \operatorname{treatment}_{\operatorname{1}}) + \gamma^{\beta_{1}}_{5}(\operatorname{group}_{\operatorname{medium}} \times \operatorname{treatment}_{\operatorname{1}})
\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 sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\gamma_{0}^{\alpha} + \gamma_{1}^{\alpha}(\operatorname{prop\_low}) + \gamma_{2}^{\alpha}(\operatorname{prop\_low} \times \operatorname{treatment}_{\operatorname{1}}) \\
&\gamma^{\beta_{1}}_{0} + \gamma^{\beta_{1}}_{2}(\operatorname{prop\_low}) + \gamma^{\beta_{1}}_{1}(\operatorname{prop\_low} \times \operatorname{treatment}_{\operatorname{1}})
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{l} \\
&\beta_{1l} \\
&\gamma_{3l}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{l}} \\
&\mu_{\beta_{1l}} \\
&\mu_{\gamma_{3l}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{ccc}
\sigma^2_{\alpha_{l}} & \rho_{\alpha_{l}\beta_{1l}} & \rho_{\alpha_{l}\gamma_{3l}} \\
\rho_{\beta_{1l}\alpha_{l}} & \sigma^2_{\beta_{1l}} & \rho_{\beta_{1l}\gamma_{3l}} \\
\rho_{\gamma_{3l}\alpha_{l}} & \rho_{\gamma_{3l}\beta_{1l}} & \sigma^2_{\gamma_{3l}}
\end{array}
\right)
\right)
\text{, for district l = 1,} \dots \text{,L}
\end{aligned}
$$
Alternate random effect VCV structures work
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{minority}) + \beta_{2j[i]}(\operatorname{female}) + \beta_{3j[i]}(\operatorname{female} \times \operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{j}} & 0 & 0 & 0 \\
0 & \sigma^2_{\beta_{1j}} & 0 & 0 \\
0 & 0 & \sigma^2_{\beta_{2j}} & 0 \\
0 & 0 & 0 & \sigma^2_{\beta_{3j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{math}_{i} &\sim N \left(\mu, \sigma^2 \right) \\
\mu &=\alpha_{j[i]} + \beta_{1j[i]}(\operatorname{minority}) + \beta_{2j[i]}(\operatorname{female}) + \beta_{3j[i]}(\operatorname{female} \times \operatorname{minority}) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j} \\
&\beta_{2j} \\
&\beta_{3j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}} \\
&\mu_{\beta_{2j}} \\
&\mu_{\beta_{3j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cccc}
\sigma^2_{\alpha_{j}} & 0 & 0 & 0 \\
0 & \sigma^2_{\beta_{1j}} & 0 & 0 \\
0 & 0 & \sigma^2_{\beta_{2j}} & 0 \\
0 & 0 & 0 & \sigma^2_{\beta_{3j}}
\end{array}
\right)
\right)
\text{, for sch.id j = 1,} \dots \text{,J}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i],l[i],m[i]} + \beta_{1j[i],k[i],m[i]}(\operatorname{wave}), \sigma^2 \right) \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{j} \\
&\beta_{1j}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{j}} \\
&\mu_{\beta_{1j}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{j}} & 0 \\
0 & \sigma^2_{\beta_{1j}}
\end{array}
\right)
\right)
\text{, for sid j = 1,} \dots \text{,J} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{k} \\
&\beta_{1k}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{k}} \\
&\mu_{\beta_{1k}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{k}} & \rho_{\alpha_{k}\beta_{1k}} \\
\rho_{\beta_{1k}\alpha_{k}} & \sigma^2_{\beta_{1k}}
\end{array}
\right)
\right)
\text{, for school k = 1,} \dots \text{,K} \\ \alpha_{l} &\sim N \left(\mu_{\alpha_{l}}, \sigma^2_{\alpha_{l}} \right)
\text{, for school.1 l = 1,} \dots \text{,L} \\
\left(
\begin{array}{c}
\begin{aligned}
&\alpha_{m} \\
&\beta_{1m}
\end{aligned}
\end{array}
\right)
&\sim N \left(
\left(
\begin{array}{c}
\begin{aligned}
&\mu_{\alpha_{m}} \\
&\mu_{\beta_{1m}}
\end{aligned}
\end{array}
\right)
,
\left(
\begin{array}{cc}
\sigma^2_{\alpha_{m}} & 0 \\
0 & \sigma^2_{\beta_{1m}}
\end{array}
\right)
\right)
\text{, for district m = 1,} \dots \text{,M}
\end{aligned}
$$
Nested model syntax works
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for school:sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for school:sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{score}_{i} &\sim N \left(\alpha_{j[i],k[i]}, \sigma^2 \right) \\
\alpha_{j} &\sim N \left(\mu_{\alpha_{j}}, \sigma^2_{\alpha_{j}} \right)
\text{, for school:sid j = 1,} \dots \text{,J} \\
\alpha_{k} &\sim N \left(\mu_{\alpha_{k}}, \sigma^2_{\alpha_{k}} \right)
\text{, for sid k = 1,} \dots \text{,K}
\end{aligned}
$$
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