Multilevel Modeling

Random Slopes

Spring 2026 | CLAS | PSYC 894
Jeffrey M. Girard | Lecture 07a

Roadmap

1. Conceptual Foundations
   • Defining Models and Assumptions
2. Application and Interpretation
   • Fitting and Visualizing Models

Conceptual Foundations

Overview

• Just as we estimated a distribution of intercepts across clusters, we can do the same for lower-level slopes
  • The fixed slope is the mean slope across clusters
  • The random slopes are deviations from this mean
  • The SD of the deviations measures effect heterogeneity

• Random slopes increase the accuracy of the model
  • They enable partial pooling and generalization
  • When \(SD\gg0\), it can be misleading to omit them

Visualizing Models

Which Predictors Can Have Random Slopes?

In a two-level model, we can only estimate random slopes for Level 1 predictors. We cannot estimate random slopes for Level 2 predictors. Why?

The Prerequisite for a Slope

• A slope represents a relationship: how much \(y\) changes when \(x\) changes.
• To calculate a slope for a specific cluster, the predictor \(x\) must vary within that cluster.
• Because Level 1 predictors (like student sex or student study time) vary within schools, we can draw a unique regression line for each school.

The “Constant” Problem for L2 Predictors

Level 2 predictors (like school funding or school type) are measured at the cluster level.

• These variables are constants within a given cluster.
• Every student in School A has the exact same value for “School Funding.”
• If \(x\) never changes within a school, we cannot calculate how a change in \(x\) affects \(y\) within that school.

• Therefore, in a two-level model, L2 predictors can only have fixed effects (the overall effect of the L2 predictor on the outcome across all clusters).
• They cannot have a \(u\) deviation because there is no cluster-specific slope to deviate from!

Review: Fixed Slopes for L1

Let’s recall our model with a fixed slope of an L1 predictor.

Level One (Observation)

\[ y_{ij} = \beta_{0j} + \beta_{1j} x_{ij} + \color{#4daf4a}{e_{ij}} \]

Level Two (Cluster)

\[ \begin{align} \beta_{0j} &= \color{#377eb8}{\gamma_{00}} + \color{#e41a1c}{u_{0j}}\\ \beta_{1j} &= \color{#377eb8}{\gamma_{10}} \end{align} \]

We assumed the slope for \(x\) was identical across all clusters.

Adding a Random Slope

Now, we let that slope vary across clusters by adding a Level 2 deviation.

Level One (Observation)

\[ y_{ij} = \beta_{0j} + \beta_{1j} x_{ij} + \color{#4daf4a}{e_{ij}} \]

Level Two (Cluster)

\[ \begin{align} \beta_{0j} &= \color{#377eb8}{\gamma_{00}} + \color{#e41a1c}{u_{0j}} \\ \beta_{1j} &= \color{#377eb8}{\gamma_{10}} + \underbrace{\color{#e41a1c}{u_{1j}}}_{\text{New}} \end{align} \]

• \(\color{#377eb8}{\gamma_{10}}\) is the Fixed L1 Slope (The overall mean slope across all clusters)
• \(\color{#e41a1c}{u_{1j}}\) is the Random Slope Deviation (How cluster \(j\)’s slope differs from \(\color{#377eb8}{\gamma_{10}}\))

Decoding the Gammas

1. The first number says which Level 1 parameter is being predicted.
   • 0 = The Intercept \((\beta_{0j})\)
   • 1 = The L1 slope \((\beta_{1j})\)
2. The second number says which Level 2 predictor is doing the predicting.
   • 0 = The base parameter (no L2 predictors are predicting this yet)

Putting it together:

• \(\color{#377eb8}{\gamma_{00}}\) is the baseline effect for the intercept (the fixed intercept).
• \(\color{#377eb8}{\gamma_{10}}\) is the baseline effect for the Level 1 slope (the fixed slope).

Reading the Equations

\[ y_{ij} = \beta_{0j} + \beta_{1j} x_{ij} + \color{#4daf4a}{e_{ij}} \]

\(y_{ij}\) is the outcome for observation \(i\) in cluster \(j\) and equals…
…the Cluster Intercept \((\beta_{0j})\) plus the Cluster Effect of Predictor \(x\) \((\beta_{1j} x_{ij})\) plus the Observation Deviation \((\color{#4daf4a}{e_{ij}})\).

\[ \beta_{1j} = \color{#377eb8}{\gamma_{10}} + \color{#e41a1c}{u_{1j}} \]

\(\beta_{1j}\) is the Cluster Slope for cluster \(j\) and equals…
…the Fixed Population Slope \((\color{#377eb8}{\gamma_{10}})\) plus the Cluster Slope Deviation \((\color{#e41a1c}{u_{1j}})\). We now allow the effect of \(x\) to change across clusters.

Random Effect Assumptions

• With the random intercepts model, we assumed the random intercept deviations were normally distributed.
• Once we add in random slopes, we further assume…
  • The random slope deviations are normally distributed.
  • Their linear combinations are normally distributed.
  • The deviations may be correlated with each other.

• This is captured by the Multivariate Normal Distribution.

Normal and Multivariate Normal

\[x \sim \text{Normal}(\mu, \sigma)\]

where \(\mu\) is the mean and \(\sigma\) is the SD (nonnegative)

\[\begin{bmatrix}x_1\\x_2\end{bmatrix} \sim \text{MVN} \left(\begin{bmatrix}\mu_1\\\mu_2\end{bmatrix}, \begin{bmatrix}\sigma_1 & \\ \rho_{12} & \sigma_2\end{bmatrix}\right)\]

where each variable gets its own \(\mu\) and \(\sigma\) (as above)
and also each pair of variables gets a correlation \(\rho\)

Multilevel Equation Part 2

As we add random effects (e.g., \(u_{1j}\)), we assume they form a multivariate normal distribution with 0 means and estimated SDs (\(\tau_{00}\) and \(\tau_{11}\)) and correlations (\(\tau_{10}\)).

Level One (Observation)

\[ \color{#4daf4a}{e_{ij}} \sim \text{Normal}(0, \color{#4daf4a}{\sigma}) \]

Level Two (Cluster)

\[ \begin{bmatrix}\color{#e41a1c}{u_{0j}}\\ \color{#e41a1c}{u_{1j}} \end{bmatrix} \sim \text{MVN}\left(\begin{bmatrix}0\\0\end{bmatrix}, \begin{bmatrix}\color{#e41a1c}{\tau_{00}} & \\ \color{#e41a1c}{\tau_{10}} & \color{#e41a1c}{\tau_{11}}\end{bmatrix}\right) \]

Random Effects Example

\[ \begin{align} e_{ij} &\sim \text{Normal}(0, \sigma) \\ \begin{bmatrix}u_{0j}\\u_{1j}\end{bmatrix} &\sim \text{MVN}\left(\begin{bmatrix}0 \\ 0\end{bmatrix}, \begin{bmatrix}\tau_{00} & \\ \tau_{10} & \tau_{11}\end{bmatrix}\right) \end{align} \]

• \(\sigma\) = SD of the L1 residuals \((e_{ij})\)
• \(\tau_{00}\) = SD of the distribution of random intercept deviations \((u_{0j})\)
• \(\tau_{11}\) = SD of the distribution of random slope deviations \((u_{1j})\)
• \(\tau_{10}\) = Correlation between the deviations (\(u_{0j}\) and \(u_{1j}\))

For compactness, we sometimes write \(\mathbf{u}_j \sim \text{MVN}(\mathbf{0}, \mathbf{T})\)

This is useful because it represents any number of \(u\) and \(\tau\) terms

Path Diagram

Mixed (or Reduced) Equation

\[ \begin{alignat}{2} &\text{L1:} \quad & y_{ij} &= \beta_{0j} + \beta_{1j} x_{ij} + \color{#4daf4a}{e_{ij}} \\ &\text{L2:} \quad & \beta_{0j} &= \color{#377eb8}{\gamma_{00}} + \color{#e41a1c}{u_{0j}} \\ & & \beta_{1j} &= \color{#377eb8}{\gamma_{10}} + \color{#e41a1c}{u_{1j}} \end{alignat} \]

Substitute the L2 equations directly into the L1 equation:

\[ y_{ij} = (\color{#377eb8}{\gamma_{00}} + \color{#e41a1c}{u_{0j}}) + (\color{#377eb8}{\gamma_{10}} + \color{#e41a1c}{u_{1j}}) x_{ij} + \color{#4daf4a}{e_{ij}} \]

Rearrange to group the fixed and random parts:

\[ y_{ij} = \underbrace{\color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{10}} x_{ij}}_{\text{Fixed}} + \underbrace{\color{#e41a1c}{u_{0j}} + \color{#e41a1c}{u_{1j}} x_{ij}}_{\text{Random}} + \color{#4daf4a}{e_{ij}} \]

Formula

\[ y_{ij} = \underbrace{\color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{10}} x_{ij}}_{\text{Fixed}} + \underbrace{\color{#e41a1c}{u_{0j}} + \color{#e41a1c}{u_{1j}} x_{ij}}_{\text{(Random)}} + \color{#4daf4a}{e_{ij}} \]

Generic

y ~ 1 + x + (1 + x | cluster)

where the 1 + x part is fixed
and the (1 + x | cluster) part is random

Example

math ~ 1 + female + (1 + female | school)

Application and Interpretation

Setup

library(tidyverse)
library(easystats)
library(lme4)

dat <- read_csv("heck2011.csv") |>
  mutate(school = factor(school), female = factor(female))
glimpse(dat)

Rows: 6,871
Columns: 7
$ school  <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2,…
$ student <dbl> 6701, 6702, 6703, 6704, 6705, 6706, 6707, 6708, 6709, 6710, 67…
$ female  <fct> 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
$ ses     <dbl> 0.586, 0.304, -0.544, -0.848, 0.001, -0.106, -0.330, -0.891, 0…
$ math    <dbl> 47.1400, 63.6100, 57.7100, 53.9000, 58.0100, 59.8700, 62.5556,…
$ puniv   <dbl> 0.08333333, 0.08333333, 0.08333333, 0.08333333, 0.08333333, 0.…
$ public  <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…

Fitting the Model

fit <- lmer(
  formula = math ~ 1 + female + (1 + female | school), 
  data = dat,
  REML = TRUE
)
model_parameters(fit)

# Fixed Effects

Parameter   | Coefficient |   SE |         95% CI | t(6865) |      p
--------------------------------------------------------------------
(Intercept) |       58.27 | 0.22 | [57.85, 58.70] |  268.38 | < .001
female [1]  |       -1.20 | 0.22 | [-1.64, -0.77] |   -5.48 | < .001

# Random Effects

Parameter                       | Coefficient
---------------------------------------------
SD (Intercept: school)          |        3.34
SD (female1: school)            |        1.70
Cor (Intercept~female1: school) |       -0.25
SD (Residual)                   |        8.10

Interpreting the Fixed Effects

• The fixed intercept \((\gamma_{00})\) is…
  • The average male student’s math score at the average school.
  • This value is significantly different from zero.

• The fixed slope of female \((\gamma_{10})\) is…
  • The average sex difference at the average school.
  • This value is significantly different from zero.
  • i.e., the female average is lower than the male average.

Interpreting the Random Effects 1

• The random intercept SD \((\tau_{00})\) is…
  • The spread of the random intercept deviations.
  • How much do male averages vary between schools?

• The random slope SD \((\tau_{11})\) is…
  • The spread of the random slope deviations.
  • How much do sex differences vary between schools?

Interpreting the Random Effects 2

• The correlation between deviations \((\tau_{10})\) is…
  • The relationship between \(u_{0j}\) and \(u_{1j}\).
  • Schools with higher average male math scores tend to have more negative slopes (i.e., stronger sex differences).

• The L1 residual SD \((\sigma)\) is…
  • The spread of the residuals at the lowest model level.
  • The spread of math scores within schools (after accounting for sex).

Information Sharing Across Parameters

• Partial pooling allows the model to share information between clusters
  • Small clusters’ estimates are shrunk toward the grand mean

• RE Correlations (\(\tau_{10}\)) share information between parameters.
  • A cluster’s intercept \((u_{0j})\) helps predict its slope \((u_{1j})\).
  • Example: If \(\tau_{10}\) is strongly negative, schools with high male math scores tend to have larger sex differences.
  • If we observe a new school with a very high intercept but only a few female students, the model “knows” to estimate a more negative slope for that school.

Do We Care About This Correlation?

How do applied researchers actually use \(\tau_{10}\)? It depends on the context.

1. Model Accuracy (The Rule)

• Often, it is just a necessary parameter to make the model more accurate.
• Assuming the correlation is exactly zero is a strong, restrictive assumption.
• Freeing (i.e., estimating) this parameter ensures that our standard errors and fixed effects are estimated correctly.

2. Substantive Interest (The Exception)

• Sometimes, the correlation is a research question!
• e.g., in longitudinal models, the correlation might answer: Do students who start with higher scores (intercept) learn at a faster rate (slope)?

Random Effect Deviations

These are called Empirical Bayes (EB) Estimates.

estimate_grouplevel(fit, type = "random") # u_0j and u_1j

Group  | Level | Parameter   | Coefficient |   SE |          95% CI
-------------------------------------------------------------------
school | 1     | (Intercept) |        1.00 | 2.05 | [ -3.03,  5.02]
school | 1     | female [1]  |       -0.25 | 1.60 | [ -3.39,  2.88]
school | 2     | (Intercept) |        3.70 | 1.86 | [  0.04,  7.35]
school | 2     | female1     |       -0.47 | 1.66 | [ -3.73,  2.79]
school | 3     | (Intercept) |       -7.30 | 1.88 | [-10.98, -3.63]
school | 3     | female1     |       -0.16 | 1.56 | [ -3.21,  2.90]
school | 4     | (Intercept) |        5.56 | 2.21 | [  1.24,  9.88]
school | 4     | female1     |        0.84 | 1.66 | [ -2.41,  4.09]
school | 5     | (Intercept) |       -6.65 | 1.84 | [-10.25, -3.05]
school | 5     | female1     |       -0.77 | 1.56 | [ -3.83,  2.30]
school | 6     | (Intercept) |        1.74 | 1.89 | [ -1.95,  5.44]
school | 6     | female1     |        1.35 | 1.57 | [ -1.72,  4.43]
school | 7     | (Intercept) |        0.19 | 1.89 | [ -3.50,  3.89]
school | 7     | female1     |       -0.81 | 1.57 | [ -3.89,  2.26]
school | 8     | (Intercept) |        2.73 | 1.61 | [ -0.42,  5.89]
school | 8     | female1     |        0.67 | 1.50 | [ -2.27,  3.61]
school | 9     | (Intercept) |       -1.43 | 1.93 | [ -5.21,  2.35]
school | 9     | female1     |       -0.87 | 1.57 | [ -3.95,  2.20]
school | 10    | (Intercept) |        2.82 | 1.78 | [ -0.67,  6.32]
school | 10    | female1     |        0.37 | 1.54 | [ -2.65,  3.39]
school | 11    | (Intercept) |       -1.17 | 1.73 | [ -4.55,  2.21]
school | 11    | female1     |        0.11 | 1.57 | [ -2.98,  3.19]
school | 12    | (Intercept) |       -1.93 | 2.09 | [ -6.03,  2.17]
school | 12    | female1     |        0.44 | 1.60 | [ -2.70,  3.57]
school | 13    | (Intercept) |        0.80 | 1.99 | [ -3.10,  4.69]
school | 13    | female1     |       -0.02 | 1.58 | [ -3.12,  3.08]
school | 14    | (Intercept) |       -0.43 | 2.03 | [ -4.41,  3.56]
school | 14    | female1     |        1.18 | 1.59 | [ -1.93,  4.30]
school | 15    | (Intercept) |        4.40 | 1.73 | [  1.01,  7.79]
school | 15    | female1     |        0.03 | 1.54 | [ -2.99,  3.05]
school | 16    | (Intercept) |       -1.20 | 2.06 | [ -5.24,  2.83]
school | 16    | female1     |       -0.50 | 1.59 | [ -3.62,  2.63]
school | 17    | (Intercept) |       -2.24 | 1.80 | [ -5.77,  1.28]
school | 17    | female1     |       -0.38 | 1.55 | [ -3.41,  2.65]
school | 18    | (Intercept) |        3.59 | 1.97 | [ -0.28,  7.45]
school | 18    | female1     |       -0.61 | 1.58 | [ -3.70,  2.49]
school | 19    | (Intercept) |       -0.07 | 1.91 | [ -3.81,  3.67]
school | 19    | female1     |        0.22 | 1.58 | [ -2.87,  3.31]
school | 20    | (Intercept) |        1.43 | 2.10 | [ -2.69,  5.55]
school | 20    | female1     |       -0.26 | 1.60 | [ -3.40,  2.87]
school | 21    | (Intercept) |        1.21 | 1.81 | [ -2.34,  4.76]
school | 21    | female1     |        0.89 | 1.54 | [ -2.13,  3.92]
school | 22    | (Intercept) |       -1.39 | 2.05 | [ -5.42,  2.63]
school | 22    | female1     |       -0.02 | 1.60 | [ -3.16,  3.11]
school | 23    | (Intercept) |        1.30 | 1.93 | [ -2.49,  5.09]
school | 23    | female1     |        0.63 | 1.59 | [ -2.48,  3.74]
school | 24    | (Intercept) |        0.52 | 2.01 | [ -3.41,  4.46]
school | 24    | female1     |   -2.66e-03 | 1.58 | [ -3.11,  3.10]
school | 25    | (Intercept) |       -1.07 | 1.77 | [ -4.54,  2.41]
school | 25    | female1     |       -1.41 | 1.55 | [ -4.44,  1.63]
school | 26    | (Intercept) |        2.09 | 1.77 | [ -1.37,  5.56]
school | 26    | female1     |        0.88 | 1.56 | [ -2.18,  3.93]
school | 27    | (Intercept) |       -2.97 | 1.80 | [ -6.50,  0.56]
school | 27    | female1     |       -0.15 | 1.55 | [ -3.18,  2.89]
school | 28    | (Intercept) |        5.71 | 1.72 | [  2.35,  9.08]
school | 28    | female1     |       -1.13 | 1.53 | [ -4.13,  1.88]
school | 29    | (Intercept) |        1.70 | 1.94 | [ -2.10,  5.50]
school | 29    | female1     |       -0.08 | 1.58 | [ -3.17,  3.01]
school | 30    | (Intercept) |       -0.98 | 1.81 | [ -4.53,  2.57]
school | 30    | female1     |       -1.05 | 1.54 | [ -4.08,  1.98]
school | 31    | (Intercept) |       -0.02 | 1.88 | [ -3.70,  3.65]
school | 31    | female1     |       -1.77 | 1.56 | [ -4.82,  1.28]
school | 32    | (Intercept) |        3.60 | 1.42 | [  0.81,  6.38]
school | 32    | female1     |        0.33 | 1.44 | [ -2.49,  3.15]
school | 33    | (Intercept) |       -4.40 | 1.67 | [ -7.68, -1.13]
school | 33    | female1     |       -0.10 | 1.54 | [ -3.12,  2.92]
school | 34    | (Intercept) |       -0.28 | 1.79 | [ -3.79,  3.23]
school | 34    | female1     |       -0.48 | 1.56 | [ -3.53,  2.57]
school | 35    | (Intercept) |        2.41 | 1.96 | [ -1.44,  6.25]
school | 35    | female1     |       -0.37 | 1.60 | [ -3.50,  2.76]
school | 36    | (Intercept) |       -3.46 | 2.13 | [ -7.64,  0.71]
school | 36    | female1     |       -0.50 | 1.61 | [ -3.64,  2.65]
school | 37    | (Intercept) |        2.39 | 1.91 | [ -1.35,  6.13]
school | 37    | female1     |        0.63 | 1.58 | [ -2.46,  3.72]
school | 38    | (Intercept) |       -1.15 | 2.09 | [ -5.25,  2.94]
school | 38    | female1     |       -0.85 | 1.60 | [ -3.99,  2.29]
school | 39    | (Intercept) |        0.40 | 1.78 | [ -3.10,  3.89]
school | 39    | female1     |       -0.94 | 1.54 | [ -3.96,  2.08]
school | 40    | (Intercept) |       -3.53 | 1.99 | [ -7.42,  0.37]
school | 40    | female1     |       -0.23 | 1.58 | [ -3.33,  2.87]
school | 41    | (Intercept) |        1.46 | 1.69 | [ -1.85,  4.77]
school | 41    | female1     |        0.69 | 1.52 | [ -2.29,  3.67]
school | 42    | (Intercept) |        0.20 | 2.10 | [ -3.91,  4.32]
school | 42    | female1     |        0.05 | 1.60 | [ -3.09,  3.18]
school | 43    | (Intercept) |        0.51 | 1.97 | [ -3.36,  4.37]
school | 43    | female1     |        0.83 | 1.58 | [ -2.26,  3.93]
school | 44    | (Intercept) |        3.06 | 1.89 | [ -0.65,  6.77]
school | 44    | female1     |        0.36 | 1.56 | [ -2.70,  3.42]
school | 45    | (Intercept) |        1.90 | 1.73 | [ -1.49,  5.28]
school | 45    | female1     |        0.11 | 1.55 | [ -2.94,  3.15]
school | 46    | (Intercept) |       -0.82 | 1.85 | [ -4.44,  2.79]
school | 46    | female1     |        0.64 | 1.56 | [ -2.41,  3.69]
school | 47    | (Intercept) |       -4.51 | 1.97 | [ -8.37, -0.64]
school | 47    | female1     |        0.68 | 1.58 | [ -2.42,  3.77]
school | 48    | (Intercept) |       -0.66 | 1.99 | [ -4.57,  3.25]
school | 48    | female1     |        0.37 | 1.61 | [ -2.78,  3.52]
school | 49    | (Intercept) |        0.15 | 1.82 | [ -3.41,  3.71]
school | 49    | female1     |       -0.59 | 1.55 | [ -3.64,  2.45]
school | 50    | (Intercept) |       -3.96 | 1.99 | [ -7.87, -0.05]
school | 50    | female1     |       -0.54 | 1.61 | [ -3.69,  2.61]
school | 51    | (Intercept) |        1.95 | 2.22 | [ -2.41,  6.31]
school | 51    | female1     |        0.34 | 1.63 | [ -2.85,  3.52]
school | 52    | (Intercept) |        3.06 | 1.99 | [ -0.84,  6.97]
school | 52    | female1     |       -0.12 | 1.59 | [ -3.24,  3.00]
school | 53    | (Intercept) |        2.37 | 1.79 | [ -1.14,  5.88]
school | 53    | female1     |       -1.09 | 1.56 | [ -4.14,  1.96]
school | 54    | (Intercept) |        4.94 | 1.88 | [  1.25,  8.63]
school | 54    | female1     |       -0.53 | 1.60 | [ -3.66,  2.60]
school | 55    | (Intercept) |       -6.35 | 1.93 | [-10.13, -2.57]
school | 55    | female1     |       -1.54 | 1.57 | [ -4.62,  1.53]
school | 56    | (Intercept) |       -0.89 | 2.07 | [ -4.95,  3.17]
school | 56    | female1     |        0.04 | 1.60 | [ -3.09,  3.16]
school | 57    | (Intercept) |        0.56 | 1.88 | [ -3.13,  4.24]
school | 57    | female1     |       -0.29 | 1.58 | [ -3.39,  2.81]
school | 58    | (Intercept) |        0.30 | 1.94 | [ -3.50,  4.10]
school | 58    | female1     |        0.42 | 1.58 | [ -2.67,  3.51]
school | 59    | (Intercept) |        1.00 | 1.79 | [ -2.51,  4.51]
school | 59    | female1     |       -0.05 | 1.56 | [ -3.10,  3.00]
school | 60    | (Intercept) |       -3.80 | 2.00 | [ -7.71,  0.11]
school | 60    | female1     |        0.59 | 1.59 | [ -2.52,  3.69]
school | 61    | (Intercept) |        2.05 | 1.73 | [ -1.34,  5.44]
school | 61    | female1     |       -1.15 | 1.54 | [ -4.18,  1.87]
school | 62    | (Intercept) |       -8.74 | 2.05 | [-12.75, -4.72]
school | 62    | female1     |        0.59 | 1.59 | [ -2.53,  3.71]
school | 63    | (Intercept) |        1.13 | 1.79 | [ -2.38,  4.64]
school | 63    | female1     |        0.14 | 1.56 | [ -2.91,  3.18]
school | 64    | (Intercept) |        0.93 | 1.89 | [ -2.77,  4.62]
school | 64    | female1     |       -0.16 | 1.57 | [ -3.24,  2.91]
school | 65    | (Intercept) |        3.86 | 1.93 | [  0.07,  7.65]
school | 65    | female1     |       -0.25 | 1.59 | [ -3.35,  2.86]
school | 66    | (Intercept) |       -1.38 | 1.81 | [ -4.93,  2.17]
school | 66    | female1     |       -0.25 | 1.56 | [ -3.32,  2.81]
school | 67    | (Intercept) |        1.95 | 1.99 | [ -1.95,  5.86]
school | 67    | female1     |       -0.25 | 1.59 | [ -3.37,  2.87]
school | 68    | (Intercept) |       -0.63 | 2.03 | [ -4.62,  3.35]
school | 68    | female1     |        0.78 | 1.59 | [ -2.33,  3.90]
school | 69    | (Intercept) |       -3.58 | 2.01 | [ -7.51,  0.36]
school | 69    | female1     |       -0.23 | 1.58 | [ -3.34,  2.88]
school | 70    | (Intercept) |        4.67 | 2.01 | [  0.73,  8.60]
school | 70    | female1     |        0.19 | 1.58 | [ -2.91,  3.30]
school | 71    | (Intercept) |       -1.69 | 1.94 | [ -5.49,  2.10]
school | 71    | female1     |       -0.56 | 1.58 | [ -3.65,  2.53]
school | 72    | (Intercept) |       -5.46 | 1.83 | [ -9.04, -1.87]
school | 72    | female1     |       -0.04 | 1.55 | [ -3.08,  2.99]
school | 73    | (Intercept) |       -0.71 | 1.74 | [ -4.13,  2.71]
school | 73    | female1     |        0.54 | 1.57 | [ -2.52,  3.61]
school | 74    | (Intercept) |        5.24 | 1.89 | [  1.53,  8.95]
school | 74    | female1     |        0.02 | 1.56 | [ -3.05,  3.08]
school | 75    | (Intercept) |       -0.39 | 1.81 | [ -3.94,  3.16]
school | 75    | female1     |        0.91 | 1.58 | [ -2.19,  4.01]
school | 76    | (Intercept) |       -0.47 | 1.91 | [ -4.21,  3.27]
school | 76    | female1     |        1.24 | 1.58 | [ -1.85,  4.33]
school | 77    | (Intercept) |        4.46 | 1.57 | [  1.38,  7.53]
school | 77    | female1     |        0.06 | 1.50 | [ -2.89,  3.00]
school | 78    | (Intercept) |       -1.11 | 2.09 | [ -5.21,  2.99]
school | 78    | female1     |       -0.08 | 1.60 | [ -3.23,  3.06]
school | 79    | (Intercept) |       -2.56 | 2.05 | [ -6.59,  1.46]
school | 79    | female1     |       -0.26 | 1.60 | [ -3.39,  2.88]
school | 80    | (Intercept) |        4.11 | 1.74 | [  0.69,  7.53]
school | 80    | female1     |        0.22 | 1.57 | [ -2.84,  3.29]
school | 81    | (Intercept) |       -2.47 | 2.06 | [ -6.51,  1.57]
school | 81    | female1     |       -0.15 | 1.59 | [ -3.28,  2.97]
school | 82    | (Intercept) |        1.76 | 1.76 | [ -1.68,  5.21]
school | 82    | female1     |       -0.83 | 1.54 | [ -3.85,  2.19]
school | 83    | (Intercept) |        1.87 | 1.55 | [ -1.17,  4.92]
school | 83    | female1     |        1.36 | 1.48 | [ -1.53,  4.25]
school | 84    | (Intercept) |        3.16 | 2.05 | [ -0.86,  7.19]
school | 84    | female1     |        0.09 | 1.60 | [ -3.04,  3.22]
school | 85    | (Intercept) |        0.19 | 1.97 | [ -3.68,  4.05]
school | 85    | female1     |        0.87 | 1.58 | [ -2.23,  3.96]
school | 86    | (Intercept) |       -1.63 | 1.79 | [ -5.13,  1.87]
school | 86    | female1     |        0.21 | 1.57 | [ -2.86,  3.29]
school | 87    | (Intercept) |        2.00 | 1.96 | [ -1.85,  5.85]
school | 87    | female1     |       -0.16 | 1.58 | [ -3.27,  2.94]
school | 88    | (Intercept) |        1.75 | 1.73 | [ -1.64,  5.14]
school | 88    | female1     |       -0.47 | 1.54 | [ -3.49,  2.55]
school | 89    | (Intercept) |        0.44 | 2.17 | [ -3.82,  4.70]
school | 89    | female1     |        0.02 | 1.61 | [ -3.14,  3.19]
school | 90    | (Intercept) |        3.64 | 1.78 | [  0.14,  7.13]
school | 90    | female1     |       -0.46 | 1.53 | [ -3.46,  2.55]
school | 91    | (Intercept) |        0.26 | 1.84 | [ -3.34,  3.86]
school | 91    | female1     |       -0.25 | 1.56 | [ -3.31,  2.81]
school | 92    | (Intercept) |        2.34 | 1.94 | [ -1.46,  6.14]
school | 92    | female1     |        0.13 | 1.58 | [ -2.96,  3.22]
school | 93    | (Intercept) |       -2.39 | 1.57 | [ -5.47,  0.68]
school | 93    | female1     |        0.67 | 1.49 | [ -2.25,  3.59]
school | 94    | (Intercept) |        1.97 | 2.06 | [ -2.06,  6.01]
school | 94    | female1     |        0.13 | 1.59 | [ -2.99,  3.26]
school | 95    | (Intercept) |       -0.25 | 1.76 | [ -3.69,  3.19]
school | 95    | female1     |        0.19 | 1.54 | [ -2.83,  3.21]
school | 96    | (Intercept) |        1.77 | 1.99 | [ -2.14,  5.68]
school | 96    | female1     |       -0.02 | 1.61 | [ -3.17,  3.13]
school | 97    | (Intercept) |        3.36 | 1.76 | [ -0.08,  6.80]
school | 97    | female1     |        0.02 | 1.54 | [ -3.00,  3.04]
school | 98    | (Intercept) |        0.29 | 1.88 | [ -3.40,  3.99]
school | 98    | female1     |        0.15 | 1.60 | [ -2.98,  3.28]
school | 99    | (Intercept) |       -0.92 | 2.13 | [ -5.09,  3.26]
school | 99    | female1     |        0.37 | 1.61 | [ -2.78,  3.51]
school | 100   | (Intercept) |        1.07 | 1.73 | [ -2.31,  4.46]
school | 100   | female1     |       -0.89 | 1.55 | [ -3.93,  2.16]
school | 101   | (Intercept) |       -0.20 | 1.83 | [ -3.78,  3.38]
school | 101   | female1     |        0.01 | 1.55 | [ -3.02,  3.05]
school | 102   | (Intercept) |       -0.23 | 1.99 | [ -4.12,  3.66]
school | 102   | female1     |        0.43 | 1.58 | [ -2.67,  3.53]
school | 103   | (Intercept) |        6.78 | 1.94 | [  2.98, 10.57]
school | 103   | female1     |       -0.33 | 1.60 | [ -3.47,  2.81]
school | 104   | (Intercept) |        0.05 | 2.05 | [ -3.98,  4.08]
school | 104   | female1     |        0.40 | 1.60 | [ -2.73,  3.54]
school | 105   | (Intercept) |        1.14 | 1.86 | [ -2.50,  4.78]
school | 105   | female1     |        0.23 | 1.57 | [ -2.85,  3.31]
school | 106   | (Intercept) |       -4.93 | 1.74 | [ -8.34, -1.52]
school | 106   | female1     |        0.84 | 1.53 | [ -2.16,  3.85]
school | 107   | (Intercept) |        5.17 | 1.86 | [  1.53,  8.81]
school | 107   | female1     |        0.74 | 1.59 | [ -2.37,  3.85]
school | 108   | (Intercept) |        1.08 | 1.81 | [ -2.47,  4.63]
school | 108   | female1     |       -1.16 | 1.56 | [ -4.23,  1.90]
school | 109   | (Intercept) |        4.14 | 1.91 | [  0.40,  7.88]
school | 109   | female1     |        0.03 | 1.58 | [ -3.06,  3.12]
school | 110   | (Intercept) |       -1.52 | 1.88 | [ -5.21,  2.17]
school | 110   | female1     |       -0.32 | 1.58 | [ -3.42,  2.77]
school | 111   | (Intercept) |       -1.42 | 1.86 | [ -5.07,  2.23]
school | 111   | female1     |        0.40 | 1.56 | [ -2.66,  3.46]
school | 112   | (Intercept) |        0.55 | 1.74 | [ -2.86,  3.96]
school | 112   | female1     |        1.54 | 1.53 | [ -1.46,  4.55]
school | 113   | (Intercept) |        1.11 | 1.67 | [ -2.16,  4.37]
school | 113   | female1     |        0.21 | 1.51 | [ -2.74,  3.17]
school | 114   | (Intercept) |       -1.52 | 1.86 | [ -5.17,  2.13]
school | 114   | female1     |       -0.55 | 1.61 | [ -3.71,  2.60]
school | 115   | (Intercept) |        3.22 | 2.09 | [ -0.87,  7.32]
school | 115   | female1     |       -0.15 | 1.60 | [ -3.29,  2.99]
school | 116   | (Intercept) |       -8.56 | 1.83 | [-12.14, -4.98]
school | 116   | female1     |        0.01 | 1.55 | [ -3.02,  3.05]
school | 117   | (Intercept) |       -2.64 | 2.05 | [ -6.66,  1.39]
school | 117   | female1     |        0.62 | 1.60 | [ -2.52,  3.75]
school | 118   | (Intercept) |       -0.82 | 1.99 | [ -4.73,  3.08]
school | 118   | female1     |        0.82 | 1.59 | [ -2.30,  3.94]
school | 119   | (Intercept) |        1.93 | 2.05 | [ -2.10,  5.95]
school | 119   | female1     |        0.30 | 1.60 | [ -2.83,  3.44]
school | 120   | (Intercept) |        3.66 | 1.84 | [  0.06,  7.26]
school | 120   | female1     |        0.36 | 1.56 | [ -2.70,  3.42]
school | 121   | (Intercept) |        0.74 | 1.92 | [ -3.02,  4.49]
school | 121   | female1     |       -0.50 | 1.57 | [ -3.58,  2.58]
school | 122   | (Intercept) |       -3.23 | 2.06 | [ -7.27,  0.80]
school | 122   | female1     |       -0.27 | 1.59 | [ -3.40,  2.86]
school | 123   | (Intercept) |        0.13 | 1.85 | [ -3.48,  3.75]
school | 123   | female1     |        0.68 | 1.56 | [ -2.37,  3.72]
school | 124   | (Intercept) |        1.45 | 2.06 | [ -2.59,  5.49]
school | 124   | female1     |        0.70 | 1.59 | [ -2.43,  3.82]
school | 125   | (Intercept) |       -1.57 | 1.86 | [ -5.22,  2.09]
school | 125   | female1     |        0.80 | 1.56 | [ -2.26,  3.86]
school | 126   | (Intercept) |       -0.19 | 1.65 | [ -3.43,  3.05]
school | 126   | female1     |       -0.46 | 1.51 | [ -3.42,  2.51]
school | 127   | (Intercept) |        2.96 | 2.02 | [ -1.00,  6.92]
school | 127   | female1     |       -0.02 | 1.60 | [ -3.16,  3.11]
school | 128   | (Intercept) |        0.46 | 1.77 | [ -3.01,  3.93]
school | 128   | female1     |        0.07 | 1.55 | [ -2.96,  3.10]
school | 129   | (Intercept) |       -2.75 | 1.79 | [ -6.25,  0.76]
school | 129   | female1     |        0.50 | 1.56 | [ -2.55,  3.55]
school | 130   | (Intercept) |       -1.25 | 2.03 | [ -5.22,  2.72]
school | 130   | female1     |       -0.07 | 1.59 | [ -3.18,  3.05]
school | 131   | (Intercept) |       -1.40 | 1.79 | [ -4.90,  2.11]
school | 131   | female1     |        0.41 | 1.59 | [ -2.70,  3.53]
school | 132   | (Intercept) |       -3.02 | 2.03 | [ -7.00,  0.97]
school | 132   | female1     |       -0.32 | 1.59 | [ -3.44,  2.79]
school | 133   | (Intercept) |        4.28 | 1.73 | [  0.88,  7.67]
school | 133   | female1     |       -0.39 | 1.60 | [ -3.53,  2.75]
school | 134   | (Intercept) |        0.81 | 1.97 | [ -3.04,  4.66]
school | 134   | female1     |       -0.54 | 1.58 | [ -3.63,  2.55]
school | 135   | (Intercept) |       -5.23 | 1.69 | [ -8.54, -1.92]
school | 135   | female1     |        0.99 | 1.55 | [ -2.05,  4.02]
school | 136   | (Intercept) |        0.02 | 2.00 | [ -3.89,  3.94]
school | 136   | female1     |       -0.98 | 1.59 | [ -4.08,  2.13]
school | 137   | (Intercept) |        0.36 | 1.82 | [ -3.20,  3.92]
school | 137   | female1     |       -0.42 | 1.55 | [ -3.46,  2.63]
school | 138   | (Intercept) |       -0.09 | 2.17 | [ -4.35,  4.17]
school | 138   | female1     |       -0.18 | 1.61 | [ -3.34,  2.98]
school | 139   | (Intercept) |       -2.01 | 2.01 | [ -5.94,  1.93]
school | 139   | female1     |        1.46 | 1.58 | [ -1.65,  4.57]
school | 140   | (Intercept) |        5.96 | 1.80 | [  2.43,  9.48]
school | 140   | female1     |       -0.58 | 1.54 | [ -3.60,  2.44]
school | 141   | (Intercept) |        1.56 | 2.06 | [ -2.47,  5.60]
school | 141   | female1     |       -0.13 | 1.59 | [ -3.26,  2.99]
school | 142   | (Intercept) |        2.77 | 1.77 | [ -0.70,  6.24]
school | 142   | female1     |       -0.21 | 1.55 | [ -3.24,  2.82]
school | 143   | (Intercept) |        1.87 | 2.03 | [ -2.11,  5.86]
school | 143   | female1     |        0.84 | 1.59 | [ -2.28,  3.95]
school | 144   | (Intercept) |       -1.46 | 1.88 | [ -5.15,  2.22]
school | 144   | female1     |        0.35 | 1.58 | [ -2.74,  3.45]
school | 145   | (Intercept) |        0.09 | 1.83 | [ -3.49,  3.67]
school | 145   | female1     |    7.80e-03 | 1.55 | [ -3.03,  3.04]
school | 146   | (Intercept) |        0.30 | 1.69 | [ -3.02,  3.62]
school | 146   | female1     |        0.82 | 1.54 | [ -2.19,  3.83]
school | 147   | (Intercept) |        0.29 | 1.83 | [ -3.30,  3.87]
school | 147   | female1     |        0.01 | 1.55 | [ -3.02,  3.05]
school | 148   | (Intercept) |       -3.58 | 1.86 | [ -7.22,  0.06]
school | 148   | female1     |       -0.54 | 1.59 | [ -3.65,  2.57]
school | 149   | (Intercept) |       -0.95 | 1.85 | [ -4.56,  2.67]
school | 149   | female1     |        1.28 | 1.56 | [ -1.77,  4.33]
school | 150   | (Intercept) |        3.30 | 1.62 | [  0.12,  6.48]
school | 150   | female1     |       -0.42 | 1.66 | [ -3.68,  2.84]
school | 151   | (Intercept) |       -2.88 | 1.89 | [ -6.59,  0.83]
school | 151   | female1     |        0.32 | 1.56 | [ -2.74,  3.38]
school | 152   | (Intercept) |        3.10 | 1.99 | [ -0.79,  7.00]
school | 152   | female1     |        0.65 | 1.58 | [ -2.45,  3.75]
school | 153   | (Intercept) |        6.79 | 1.91 | [  3.05, 10.53]
school | 153   | female1     |       -0.13 | 1.58 | [ -3.22,  2.96]
school | 154   | (Intercept) |       -1.58 | 1.97 | [ -5.44,  2.28]
school | 154   | female1     |        0.38 | 1.62 | [ -2.78,  3.55]
school | 155   | (Intercept) |       -3.53 | 1.85 | [ -7.15,  0.08]
school | 155   | female1     |        1.37 | 1.56 | [ -1.68,  4.42]
school | 156   | (Intercept) |        3.79 | 1.88 | [  0.10,  7.48]
school | 156   | female1     |       -0.29 | 1.60 | [ -3.42,  2.85]
school | 157   | (Intercept) |        1.49 | 1.86 | [ -2.15,  5.13]
school | 157   | female1     |        0.22 | 1.59 | [ -2.89,  3.33]
school | 158   | (Intercept) |        1.89 | 1.82 | [ -1.67,  5.45]
school | 158   | female1     |       -0.73 | 1.55 | [ -3.77,  2.32]
school | 159   | (Intercept) |       -0.04 | 1.83 | [ -3.63,  3.55]
school | 159   | female1     |        0.43 | 1.57 | [ -2.66,  3.51]
school | 160   | (Intercept) |        1.58 | 2.17 | [ -2.68,  5.85]
school | 160   | female1     |        0.48 | 1.61 | [ -2.69,  3.64]
school | 161   | (Intercept) |       -2.59 | 1.93 | [ -6.38,  1.20]
school | 161   | female1     |        0.67 | 1.59 | [ -2.44,  3.78]
school | 162   | (Intercept) |   -2.64e-03 | 2.02 | [ -3.97,  3.96]
school | 162   | female1     |       -0.80 | 1.59 | [ -3.92,  2.32]
school | 163   | (Intercept) |       -2.43 | 1.93 | [ -6.22,  1.36]
school | 163   | female1     |        0.12 | 1.59 | [ -2.98,  3.23]
school | 164   | (Intercept) |        0.08 | 1.79 | [ -3.43,  3.59]
school | 164   | female1     |    4.37e-03 | 1.56 | [ -3.04,  3.05]
school | 165   | (Intercept) |       -0.97 | 2.13 | [ -5.14,  3.21]
school | 165   | female1     |       -0.68 | 1.61 | [ -3.83,  2.47]
school | 166   | (Intercept) |       -3.51 | 1.97 | [ -7.37,  0.35]
school | 166   | female1     |        0.02 | 1.62 | [ -3.14,  3.19]
school | 167   | (Intercept) |        0.68 | 2.14 | [ -3.52,  4.87]
school | 167   | female1     |        0.10 | 1.62 | [ -3.07,  3.28]
school | 168   | (Intercept) |        1.62 | 1.81 | [ -1.93,  5.17]
school | 168   | female1     |        0.07 | 1.56 | [ -2.99,  3.14]
school | 169   | (Intercept) |       -0.37 | 2.05 | [ -4.39,  3.66]
school | 169   | female1     |       -0.62 | 1.60 | [ -3.76,  2.51]
school | 170   | (Intercept) |        0.36 | 1.94 | [ -3.44,  4.15]
school | 170   | female1     |       -0.52 | 1.60 | [ -3.66,  2.62]
school | 171   | (Intercept) |       -0.63 | 1.51 | [ -3.58,  2.33]
school | 171   | female1     |       -0.03 | 1.44 | [ -2.86,  2.80]
school | 172   | (Intercept) |        0.39 | 2.02 | [ -3.58,  4.35]
school | 172   | female1     |       -0.33 | 1.59 | [ -3.45,  2.79]
school | 173   | (Intercept) |        4.31 | 1.83 | [  0.72,  7.91]
school | 173   | female1     |       -1.20 | 1.59 | [ -4.32,  1.93]
school | 174   | (Intercept) |       -1.79 | 2.02 | [ -5.76,  2.17]
school | 174   | female1     |        0.26 | 1.60 | [ -2.88,  3.40]
school | 175   | (Intercept) |        1.22 | 2.02 | [ -2.74,  5.19]
school | 175   | female1     |       -0.21 | 1.59 | [ -3.33,  2.90]
school | 176   | (Intercept) |        1.54 | 1.92 | [ -2.21,  5.30]
school | 176   | female1     |       -0.38 | 1.57 | [ -3.46,  2.70]
school | 177   | (Intercept) |       -0.32 | 2.01 | [ -4.25,  3.62]
school | 177   | female1     |        0.11 | 1.58 | [ -2.99,  3.22]
school | 178   | (Intercept) |        5.09 | 1.97 | [  1.23,  8.95]
school | 178   | female1     |       -0.65 | 1.67 | [ -3.91,  2.62]
school | 179   | (Intercept) |       -1.82 | 1.79 | [ -5.32,  1.69]
school | 179   | female1     |       -0.87 | 1.57 | [ -3.95,  2.21]
school | 180   | (Intercept) |        1.07 | 1.88 | [ -2.60,  4.75]
school | 180   | female1     |       -0.10 | 1.56 | [ -3.16,  2.96]
school | 181   | (Intercept) |        0.41 | 1.69 | [ -2.90,  3.72]
school | 181   | female1     |        0.23 | 1.55 | [ -2.81,  3.27]
school | 182   | (Intercept) |        3.08 | 1.86 | [ -0.56,  6.72]
school | 182   | female1     |       -0.12 | 1.57 | [ -3.19,  2.96]
school | 183   | (Intercept) |        1.28 | 2.13 | [ -2.90,  5.45]
school | 183   | female1     |        0.09 | 1.61 | [ -3.06,  3.24]
school | 184   | (Intercept) |       -0.80 | 1.81 | [ -4.35,  2.75]
school | 184   | female1     |        0.83 | 1.56 | [ -2.24,  3.89]
school | 185   | (Intercept) |       -5.55 | 1.99 | [ -9.45, -1.65]
school | 185   | female1     |        0.19 | 1.59 | [ -2.93,  3.31]
school | 186   | (Intercept) |       -5.24 | 1.93 | [ -9.03, -1.45]
school | 186   | female1     |       -0.17 | 1.59 | [ -3.27,  2.94]
school | 187   | (Intercept) |       -0.93 | 1.99 | [ -4.82,  2.96]
school | 187   | female1     |       -0.56 | 1.58 | [ -3.65,  2.54]
school | 188   | (Intercept) |        2.78 | 2.02 | [ -1.18,  6.75]
school | 188   | female1     |        0.84 | 1.59 | [ -2.28,  3.96]
school | 189   | (Intercept) |       -3.09 | 1.69 | [ -6.40,  0.23]
school | 189   | female1     |       -1.41 | 1.51 | [ -4.37,  1.55]
school | 190   | (Intercept) |       -1.70 | 2.03 | [ -5.69,  2.28]
school | 190   | female1     |        1.51 | 1.59 | [ -1.60,  4.63]
school | 191   | (Intercept) |       -0.76 | 2.03 | [ -4.75,  3.22]
school | 191   | female1     |       -0.18 | 1.59 | [ -3.29,  2.94]
school | 192   | (Intercept) |        3.11 | 1.59 | [ -0.01,  6.23]
school | 192   | female1     |       -0.40 | 1.66 | [ -3.65,  2.86]
school | 193   | (Intercept) |       -2.76 | 1.99 | [ -6.67,  1.14]
school | 193   | female1     |       -0.42 | 1.59 | [ -3.54,  2.70]
school | 194   | (Intercept) |        2.68 | 2.09 | [ -1.41,  6.78]
school | 194   | female1     |        0.27 | 1.60 | [ -2.87,  3.41]
school | 195   | (Intercept) |        0.12 | 1.77 | [ -3.34,  3.59]
school | 195   | female1     |        0.84 | 1.54 | [ -2.17,  3.85]
school | 196   | (Intercept) |       -1.64 | 2.00 | [ -5.56,  2.27]
school | 196   | female1     |        0.93 | 1.59 | [ -2.17,  4.04]
school | 197   | (Intercept) |       -8.24 | 1.91 | [-11.98, -4.49]
school | 197   | female1     |        0.40 | 1.56 | [ -2.67,  3.46]
school | 198   | (Intercept) |       -1.77 | 1.77 | [ -5.23,  1.70]
school | 198   | female1     |        0.03 | 1.54 | [ -2.98,  3.04]
school | 199   | (Intercept) |        0.10 | 1.85 | [ -3.51,  3.72]
school | 199   | female1     |       -0.84 | 1.56 | [ -3.89,  2.21]
school | 200   | (Intercept) |       -3.72 | 1.77 | [ -7.18, -0.26]
school | 200   | female1     |        2.05 | 1.58 | [ -1.04,  5.14]
school | 201   | (Intercept) |        0.10 | 1.93 | [ -3.69,  3.89]
school | 201   | female1     |        1.42 | 1.59 | [ -1.69,  4.53]
school | 202   | (Intercept) |        2.69 | 1.59 | [ -0.42,  5.80]
school | 202   | female1     |        0.71 | 1.48 | [ -2.19,  3.60]
school | 203   | (Intercept) |       -0.05 | 1.99 | [ -3.96,  3.85]
school | 203   | female1     |       -0.24 | 1.59 | [ -3.36,  2.88]
school | 204   | (Intercept) |        0.81 | 2.07 | [ -3.25,  4.87]
school | 204   | female1     |        0.04 | 1.60 | [ -3.10,  3.18]
school | 205   | (Intercept) |       -2.04 | 1.62 | [ -5.21,  1.14]
school | 205   | female1     |       -1.05 | 1.51 | [ -4.01,  1.90]
school | 206   | (Intercept) |       -1.67 | 2.01 | [ -5.60,  2.27]
school | 206   | female1     |        0.10 | 1.58 | [ -3.01,  3.21]
school | 207   | (Intercept) |       -0.59 | 1.83 | [ -4.18,  3.01]
school | 207   | female1     |        0.25 | 1.59 | [ -2.87,  3.38]
school | 208   | (Intercept) |       -2.98 | 1.69 | [ -6.30,  0.34]
school | 208   | female1     |       -0.10 | 1.54 | [ -3.11,  2.91]
school | 209   | (Intercept) |        2.64 | 1.83 | [ -0.95,  6.22]
school | 209   | female1     |        0.59 | 1.55 | [ -2.45,  3.62]
school | 210   | (Intercept) |       -4.49 | 1.67 | [ -7.77, -1.22]
school | 210   | female1     |        1.30 | 1.54 | [ -1.72,  4.31]
school | 211   | (Intercept) |       -2.37 | 1.94 | [ -6.17,  1.43]
school | 211   | female1     |        1.70 | 1.58 | [ -1.39,  4.79]
school | 212   | (Intercept) |       -3.57 | 1.81 | [ -7.12, -0.02]
school | 212   | female1     |        0.03 | 1.56 | [ -3.04,  3.09]
school | 213   | (Intercept) |        2.86 | 1.88 | [ -0.81,  6.54]
school | 213   | female1     |       -0.30 | 1.56 | [ -3.35,  2.76]
school | 214   | (Intercept) |       -3.94 | 1.96 | [ -7.79, -0.09]
school | 214   | female1     |        0.65 | 1.58 | [ -2.45,  3.76]
school | 215   | (Intercept) |        0.79 | 1.81 | [ -2.75,  4.34]
school | 215   | female1     |       -0.51 | 1.56 | [ -3.58,  2.56]
school | 216   | (Intercept) |        1.33 | 1.99 | [ -2.57,  5.24]
school | 216   | female1     |        0.46 | 1.59 | [ -2.66,  3.58]
school | 217   | (Intercept) |        2.91 | 2.00 | [ -1.00,  6.82]
school | 217   | female1     |        0.31 | 1.59 | [ -2.79,  3.42]
school | 218   | (Intercept) |       -0.33 | 1.93 | [ -4.11,  3.45]
school | 218   | female1     |       -0.88 | 1.57 | [ -3.95,  2.20]
school | 219   | (Intercept) |        0.29 | 2.09 | [ -3.81,  4.38]
school | 219   | female1     |       -1.11 | 1.60 | [ -4.25,  2.03]
school | 220   | (Intercept) |       -5.50 | 2.09 | [ -9.60, -1.40]
school | 220   | female1     |        0.18 | 1.60 | [ -2.96,  3.31]
school | 221   | (Intercept) |        3.31 | 2.03 | [ -0.67,  7.30]
school | 221   | female1     |        0.01 | 1.59 | [ -3.10,  3.13]
school | 222   | (Intercept) |       -2.75 | 1.84 | [ -6.37,  0.86]
school | 222   | female1     |       -0.70 | 1.55 | [ -3.73,  2.34]
school | 223   | (Intercept) |       -5.20 | 1.97 | [ -9.07, -1.34]
school | 223   | female1     |       -0.83 | 1.58 | [ -3.93,  2.26]
school | 224   | (Intercept) |        1.92 | 2.01 | [ -2.01,  5.85]
school | 224   | female1     |        0.12 | 1.59 | [ -2.99,  3.23]
school | 225   | (Intercept) |       -3.43 | 1.79 | [ -6.94,  0.08]
school | 225   | female1     |       -0.51 | 1.56 | [ -3.56,  2.54]
school | 226   | (Intercept) |        3.36 | 2.02 | [ -0.60,  7.33]
school | 226   | female1     |        0.32 | 1.59 | [ -2.80,  3.44]
school | 227   | (Intercept) |       -0.98 | 2.05 | [ -4.99,  3.04]
school | 227   | female1     |       -0.54 | 1.59 | [ -3.66,  2.59]
school | 228   | (Intercept) |        0.39 | 2.00 | [ -3.53,  4.30]
school | 228   | female1     |       -0.53 | 1.59 | [ -3.64,  2.58]
school | 229   | (Intercept) |        2.29 | 1.74 | [ -1.13,  5.71]
school | 229   | female1     |       -0.06 | 1.57 | [ -3.13,  3.00]
school | 230   | (Intercept) |        1.54 | 1.99 | [ -2.37,  5.45]
school | 230   | female1     |       -0.51 | 1.61 | [ -3.66,  2.64]
school | 231   | (Intercept) |       -2.28 | 2.00 | [ -6.20,  1.63]
school | 231   | female1     |       -0.11 | 1.59 | [ -3.22,  2.99]
school | 232   | (Intercept) |       -0.45 | 1.86 | [ -4.09,  3.19]
school | 232   | female1     |        0.76 | 1.57 | [ -2.32,  3.84]
school | 233   | (Intercept) |        0.49 | 1.99 | [ -3.42,  4.40]
school | 233   | female1     |        0.20 | 1.61 | [ -2.95,  3.35]
school | 234   | (Intercept) |        0.85 | 1.94 | [ -2.95,  4.65]
school | 234   | female1     |        0.35 | 1.58 | [ -2.74,  3.44]
school | 235   | (Intercept) |        3.04 | 1.78 | [ -0.46,  6.53]
school | 235   | female1     |        0.04 | 1.54 | [ -2.98,  3.06]
school | 236   | (Intercept) |       -0.92 | 1.84 | [ -4.51,  2.68]
school | 236   | female1     |        0.76 | 1.56 | [ -2.30,  3.82]
school | 237   | (Intercept) |       -1.71 | 1.83 | [ -5.29,  1.87]
school | 237   | female1     |       -0.71 | 1.55 | [ -3.74,  2.32]
school | 238   | (Intercept) |        0.37 | 2.09 | [ -3.73,  4.47]
school | 238   | female1     |        0.44 | 1.60 | [ -2.69,  3.58]
school | 239   | (Intercept) |       -2.77 | 1.65 | [ -6.01,  0.47]
school | 239   | female1     |       -0.13 | 1.51 | [ -3.09,  2.84]
school | 240   | (Intercept) |        3.35 | 2.02 | [ -0.62,  7.32]
school | 240   | female1     |       -0.34 | 1.59 | [ -3.46,  2.78]
school | 241   | (Intercept) |        4.59 | 2.10 | [  0.47,  8.71]
school | 241   | female1     |       -0.23 | 1.60 | [ -3.37,  2.90]
school | 242   | (Intercept) |       -0.55 | 2.01 | [ -4.49,  3.38]
school | 242   | female1     |       -0.15 | 1.58 | [ -3.25,  2.96]
school | 243   | (Intercept) |       -4.95 | 2.09 | [ -9.05, -0.86]
school | 243   | female1     |       -0.19 | 1.60 | [ -3.32,  2.95]
school | 244   | (Intercept) |        0.92 | 2.02 | [ -3.05,  4.89]
school | 244   | female1     |        1.09 | 1.59 | [ -2.02,  4.21]
school | 245   | (Intercept) |        2.17 | 1.77 | [ -1.30,  5.63]
school | 245   | female1     |        0.06 | 1.56 | [ -2.99,  3.12]
school | 246   | (Intercept) |       -6.42 | 2.02 | [-10.39, -2.46]
school | 246   | female1     |       -0.23 | 1.60 | [ -3.36,  2.91]
school | 247   | (Intercept) |       -1.69 | 1.68 | [ -4.98,  1.61]
school | 247   | female1     |        0.55 | 1.53 | [ -2.45,  3.54]
school | 248   | (Intercept) |        1.39 | 1.99 | [ -2.51,  5.29]
school | 248   | female1     |        0.14 | 1.59 | [ -2.98,  3.26]
school | 249   | (Intercept) |       -4.23 | 1.99 | [ -8.14, -0.33]
school | 249   | female1     |        0.72 | 1.59 | [ -2.40,  3.84]
school | 250   | (Intercept) |       -3.17 | 1.42 | [ -5.95, -0.40]
school | 250   | female1     |       -1.33 | 1.46 | [ -4.18,  1.53]
school | 251   | (Intercept) |        2.88 | 1.81 | [ -0.67,  6.43]
school | 251   | female1     |       -1.06 | 1.58 | [ -4.16,  2.03]
school | 252   | (Intercept) |        4.34 | 1.89 | [  0.63,  8.05]
school | 252   | female1     |        0.23 | 1.56 | [ -2.83,  3.30]
school | 253   | (Intercept) |       -2.44 | 1.86 | [ -6.10,  1.21]
school | 253   | female1     |        1.41 | 1.56 | [ -1.66,  4.47]
school | 254   | (Intercept) |       -2.36 | 1.75 | [ -5.78,  1.06]
school | 254   | female1     |        0.18 | 1.59 | [ -2.93,  3.30]
school | 255   | (Intercept) |        0.51 | 2.06 | [ -3.53,  4.54]
school | 255   | female1     |        0.18 | 1.59 | [ -2.95,  3.30]
school | 256   | (Intercept) |        0.57 | 1.95 | [ -3.25,  4.39]
school | 256   | female1     |        0.24 | 1.63 | [ -2.95,  3.43]
school | 257   | (Intercept) |        5.57 | 2.21 | [  1.24,  9.89]
school | 257   | female1     |        0.84 | 1.66 | [ -2.41,  4.09]
school | 258   | (Intercept) |        0.57 | 2.00 | [ -3.34,  4.49]
school | 258   | female1     |        0.61 | 1.59 | [ -2.49,  3.72]
school | 259   | (Intercept) |       -3.63 | 1.91 | [ -7.37,  0.11]
school | 259   | female1     |       -1.18 | 1.59 | [ -4.29,  1.94]
school | 260   | (Intercept) |       -1.14 | 1.99 | [ -5.05,  2.76]
school | 260   | female1     |       -0.72 | 1.59 | [ -3.84,  2.40]
school | 261   | (Intercept) |       -1.05 | 2.00 | [ -4.96,  2.86]
school | 261   | female1     |       -0.31 | 1.59 | [ -3.41,  2.80]
school | 262   | (Intercept) |        3.81 | 1.80 | [  0.28,  7.34]
school | 262   | female1     |       -0.68 | 1.55 | [ -3.71,  2.35]
school | 263   | (Intercept) |       -0.47 | 1.86 | [ -4.13,  3.18]
school | 263   | female1     |       -0.61 | 1.56 | [ -3.68,  2.45]
school | 264   | (Intercept) |       -3.56 | 1.93 | [ -7.35,  0.23]
school | 264   | female1     |       -1.02 | 1.59 | [ -4.13,  2.09]
school | 265   | (Intercept) |       -1.90 | 1.92 | [ -5.65,  1.86]
school | 265   | female1     |       -0.77 | 1.57 | [ -3.85,  2.30]
school | 266   | (Intercept) |       -2.60 | 1.83 | [ -6.20,  0.99]
school | 266   | female1     |       -0.89 | 1.57 | [ -3.98,  2.19]
school | 267   | (Intercept) |        0.70 | 2.05 | [ -3.33,  4.72]
school | 267   | female1     |       -0.04 | 1.60 | [ -3.17,  3.10]
school | 268   | (Intercept) |        2.33 | 2.05 | [ -1.70,  6.35]
school | 268   | female1     |       -0.28 | 1.60 | [ -3.42,  2.85]
school | 269   | (Intercept) |       -2.42 | 2.00 | [ -6.33,  1.49]
school | 269   | female1     |        0.80 | 1.59 | [ -2.31,  3.91]
school | 270   | (Intercept) |        2.12 | 1.61 | [ -1.03,  5.27]
school | 270   | female1     |        0.08 | 1.50 | [ -2.86,  3.02]
school | 271   | (Intercept) |        2.57 | 1.84 | [ -1.03,  6.17]
school | 271   | female1     |       -0.78 | 1.56 | [ -3.84,  2.28]
school | 272   | (Intercept) |       -1.05 | 2.17 | [ -5.31,  3.21]
school | 272   | female1     |        0.51 | 1.61 | [ -2.65,  3.68]
school | 273   | (Intercept) |       -0.30 | 1.99 | [ -4.20,  3.61]
school | 273   | female1     |       -0.05 | 1.61 | [ -3.20,  3.09]
school | 274   | (Intercept) |        3.89 | 1.83 | [  0.31,  7.47]
school | 274   | female1     |       -1.01 | 1.55 | [ -4.05,  2.03]
school | 275   | (Intercept) |        4.64 | 1.79 | [  1.14,  8.15]
school | 275   | female1     |       -0.09 | 1.56 | [ -3.14,  2.96]
school | 276   | (Intercept) |       -1.23 | 1.97 | [ -5.09,  2.63]
school | 276   | female1     |       -0.02 | 1.62 | [ -3.18,  3.15]
school | 277   | (Intercept) |       -0.19 | 1.91 | [ -3.93,  3.55]
school | 277   | female1     |        0.46 | 1.56 | [ -2.60,  3.53]
school | 278   | (Intercept) |        1.69 | 1.91 | [ -2.06,  5.43]
school | 278   | female1     |        0.02 | 1.56 | [ -3.04,  3.09]
school | 279   | (Intercept) |        0.71 | 1.80 | [ -2.82,  4.24]
school | 279   | female1     |       -0.16 | 1.55 | [ -3.20,  2.87]
school | 280   | (Intercept) |        0.19 | 1.59 | [ -2.92,  3.31]
school | 280   | female1     |       -0.52 | 1.50 | [ -3.46,  2.43]
school | 281   | (Intercept) |        5.16 | 1.77 | [  1.70,  8.62]
school | 281   | female1     |       -0.14 | 1.56 | [ -3.20,  2.92]
school | 282   | (Intercept) |        5.56 | 1.27 | [  3.08,  8.05]
school | 282   | female1     |       -0.71 | 1.66 | [ -3.95,  2.54]
school | 283   | (Intercept) |       -2.04 | 1.82 | [ -5.60,  1.52]
school | 283   | female1     |        0.49 | 1.55 | [ -2.56,  3.54]
school | 284   | (Intercept) |        2.28 | 1.50 | [ -0.65,  5.21]
school | 284   | female1     |       -1.66 | 1.48 | [ -4.55,  1.24]
school | 285   | (Intercept) |       -1.41 | 2.06 | [ -5.45,  2.63]
school | 285   | female1     |       -0.46 | 1.59 | [ -3.59,  2.66]
school | 286   | (Intercept) |       -1.36 | 1.78 | [ -4.86,  2.13]
school | 286   | female1     |       -0.03 | 1.54 | [ -3.05,  2.99]
school | 287   | (Intercept) |       -3.42 | 1.86 | [ -7.07,  0.22]
school | 287   | female1     |       -0.71 | 1.55 | [ -3.75,  2.34]
school | 288   | (Intercept) |        3.58 | 1.81 | [  0.04,  7.13]
school | 288   | female1     |        0.33 | 1.56 | [ -2.74,  3.40]
school | 289   | (Intercept) |       -0.79 | 2.02 | [ -4.75,  3.18]
school | 289   | female1     |       -0.23 | 1.59 | [ -3.35,  2.89]
school | 290   | (Intercept) |       -2.26 | 1.75 | [ -5.69,  1.17]
school | 290   | female1     |       -0.79 | 1.55 | [ -3.83,  2.25]
school | 291   | (Intercept) |       -0.57 | 1.80 | [ -4.10,  2.96]
school | 291   | female1     |       -1.05 | 1.55 | [ -4.08,  1.98]
school | 292   | (Intercept) |       -1.36 | 1.97 | [ -5.22,  2.51]
school | 292   | female1     |        0.12 | 1.58 | [ -2.97,  3.22]
school | 293   | (Intercept) |       -2.05 | 1.81 | [ -5.60,  1.50]
school | 293   | female1     |        0.47 | 1.56 | [ -2.60,  3.53]
school | 294   | (Intercept) |        0.44 | 2.13 | [ -3.73,  4.62]
school | 294   | female1     |        1.24 | 1.61 | [ -1.90,  4.39]
school | 295   | (Intercept) |       -5.30 | 1.99 | [ -9.21, -1.40]
school | 295   | female1     |        0.20 | 1.59 | [ -2.92,  3.32]
school | 296   | (Intercept) |        1.64 | 1.88 | [ -2.05,  5.33]
school | 296   | female1     |       -0.18 | 1.58 | [ -3.27,  2.92]
school | 297   | (Intercept) |       -6.68 | 1.94 | [-10.48, -2.88]
school | 297   | female1     |       -0.15 | 1.60 | [ -3.29,  2.99]
school | 298   | (Intercept) |        0.59 | 2.05 | [ -3.44,  4.61]
school | 298   | female1     |       -0.47 | 1.60 | [ -3.60,  2.67]
school | 299   | (Intercept) |        0.02 | 1.96 | [ -3.83,  3.87]
school | 299   | female1     |        1.64 | 1.58 | [ -1.47,  4.74]
school | 300   | (Intercept) |       -0.89 | 1.52 | [ -3.86,  2.09]
school | 300   | female1     |        0.54 | 1.52 | [ -2.44,  3.52]
school | 301   | (Intercept) |       -2.54 | 1.99 | [ -6.44,  1.36]
school | 301   | female1     |       -0.33 | 1.59 | [ -3.45,  2.79]
school | 302   | (Intercept) |       -2.29 | 2.10 | [ -6.41,  1.83]
school | 302   | female1     |       -1.09 | 1.60 | [ -4.23,  2.04]
school | 303   | (Intercept) |        4.57 | 1.97 | [  0.71,  8.44]
school | 303   | female1     |        0.16 | 1.58 | [ -2.94,  3.25]
school | 304   | (Intercept) |        2.40 | 1.94 | [ -1.39,  6.20]
school | 304   | female1     |       -0.74 | 1.60 | [ -3.88,  2.40]
school | 305   | (Intercept) |       -2.12 | 1.96 | [ -5.96,  1.73]
school | 305   | female1     |        0.68 | 1.60 | [ -2.45,  3.80]
school | 306   | (Intercept) |       -2.61 | 1.77 | [ -6.09,  0.86]
school | 306   | female1     |       -0.09 | 1.55 | [ -3.12,  2.94]
school | 307   | (Intercept) |        2.02 | 1.81 | [ -1.52,  5.57]
school | 307   | female1     |       -0.09 | 1.58 | [ -3.19,  3.01]
school | 308   | (Intercept) |        1.48 | 1.91 | [ -2.27,  5.23]
school | 308   | female1     |       -0.64 | 1.61 | [ -3.80,  2.52]
school | 309   | (Intercept) |        0.70 | 1.89 | [ -3.01,  4.41]
school | 309   | female1     |        0.68 | 1.56 | [ -2.38,  3.75]
school | 310   | (Intercept) |       -1.87 | 1.99 | [ -5.77,  2.03]
school | 310   | female1     |        0.64 | 1.59 | [ -2.48,  3.76]
school | 311   | (Intercept) |        4.23 | 1.88 | [  0.55,  7.91]
school | 311   | female1     |        0.15 | 1.56 | [ -2.90,  3.21]
school | 312   | (Intercept) |       -1.45 | 2.09 | [ -5.55,  2.64]
school | 312   | female1     |        1.40 | 1.60 | [ -1.74,  4.54]
school | 313   | (Intercept) |       -3.58 | 1.96 | [ -7.43,  0.27]
school | 313   | female1     |        0.33 | 1.58 | [ -2.78,  3.43]
school | 314   | (Intercept) |        0.38 | 1.85 | [ -3.23,  4.00]
school | 314   | female1     |       -0.46 | 1.56 | [ -3.51,  2.58]
school | 315   | (Intercept) |        1.32 | 2.10 | [ -2.80,  5.44]
school | 315   | female1     |       -0.90 | 1.60 | [ -4.03,  2.24]
school | 316   | (Intercept) |       -4.36 | 1.83 | [ -7.95, -0.77]
school | 316   | female1     |        0.86 | 1.57 | [ -2.23,  3.94]
school | 317   | (Intercept) |        1.27 | 2.00 | [ -2.64,  5.19]
school | 317   | female1     |       -1.00 | 1.59 | [ -4.11,  2.11]
school | 318   | (Intercept) |       -0.14 | 2.05 | [ -4.17,  3.88]
school | 318   | female1     |       -0.21 | 1.60 | [ -3.34,  2.92]
school | 319   | (Intercept) |       -0.63 | 2.14 | [ -4.83,  3.57]
school | 319   | female1     |        0.39 | 1.61 | [ -2.77,  3.55]
school | 320   | (Intercept) |        2.11 | 2.09 | [ -1.99,  6.21]
school | 320   | female1     |       -0.53 | 1.60 | [ -3.67,  2.61]
school | 321   | (Intercept) |        1.47 | 1.56 | [ -1.59,  4.53]
school | 321   | female1     |       -0.93 | 1.48 | [ -3.84,  1.97]
school | 322   | (Intercept) |       -0.23 | 1.91 | [ -3.98,  3.52]
school | 322   | female1     |       -0.10 | 1.57 | [ -3.17,  2.97]
school | 323   | (Intercept) |        1.02 | 1.96 | [ -2.83,  4.86]
school | 323   | female1     |        0.08 | 1.60 | [ -3.05,  3.21]
school | 324   | (Intercept) |        0.63 | 1.81 | [ -2.92,  4.18]
school | 324   | female1     |        1.52 | 1.54 | [ -1.51,  4.55]
school | 325   | (Intercept) |        3.01 | 1.80 | [ -0.52,  6.54]
school | 325   | female1     |        0.22 | 1.55 | [ -2.81,  3.25]
school | 326   | (Intercept) |        0.70 | 1.97 | [ -3.15,  4.55]
school | 326   | female1     |       -1.60 | 1.58 | [ -4.69,  1.49]
school | 327   | (Intercept) |        2.83 | 2.06 | [ -1.21,  6.86]
school | 327   | female1     |        0.68 | 1.59 | [ -2.45,  3.80]
school | 328   | (Intercept) |        4.03 | 1.83 | [  0.45,  7.62]
school | 328   | female1     |       -0.01 | 1.55 | [ -3.05,  3.02]
school | 329   | (Intercept) |        3.59 | 1.93 | [ -0.20,  7.38]
school | 329   | female1     |        0.22 | 1.59 | [ -2.89,  3.33]
school | 330   | (Intercept) |       -3.67 | 1.84 | [ -7.27, -0.07]
school | 330   | female1     |        0.16 | 1.56 | [ -2.90,  3.22]
school | 331   | (Intercept) |        0.41 | 2.07 | [ -3.65,  4.47]
school | 331   | female1     |        0.12 | 1.60 | [ -3.01,  3.25]
school | 332   | (Intercept) |        0.28 | 1.92 | [ -3.48,  4.03]
school | 332   | female1     |       -0.25 | 1.57 | [ -3.32,  2.83]
school | 333   | (Intercept) |        2.08 | 1.53 | [ -0.92,  5.08]
school | 333   | female1     |        0.73 | 1.53 | [ -2.28,  3.73]
school | 334   | (Intercept) |        2.51 | 2.09 | [ -1.59,  6.61]
school | 334   | female1     |       -0.08 | 1.60 | [ -3.21,  3.06]
school | 335   | (Intercept) |        4.30 | 1.86 | [  0.66,  7.94]
school | 335   | female1     |    3.93e-03 | 1.57 | [ -3.07,  3.08]
school | 336   | (Intercept) |       -0.85 | 1.74 | [ -4.27,  2.56]
school | 336   | female1     |        0.79 | 1.53 | [ -2.22,  3.80]
school | 337   | (Intercept) |        0.08 | 1.81 | [ -3.47,  3.63]
school | 337   | female1     |       -0.35 | 1.54 | [ -3.38,  2.68]
school | 338   | (Intercept) |       -1.16 | 1.93 | [ -4.95,  2.63]
school | 338   | female1     |       -0.46 | 1.59 | [ -3.57,  2.65]
school | 339   | (Intercept) |       -4.66 | 1.81 | [ -8.21, -1.12]
school | 339   | female1     |       -0.11 | 1.56 | [ -3.18,  2.95]
school | 340   | (Intercept) |       -4.56 | 1.91 | [ -8.31, -0.81]
school | 340   | female1     |        0.44 | 1.61 | [ -2.72,  3.60]
school | 341   | (Intercept) |        2.47 | 1.83 | [ -1.11,  6.05]
school | 341   | female1     |       -0.25 | 1.55 | [ -3.29,  2.79]
school | 342   | (Intercept) |       -2.86 | 1.93 | [ -6.65,  0.93]
school | 342   | female1     |        0.37 | 1.59 | [ -2.74,  3.47]
school | 343   | (Intercept) |        2.52 | 1.97 | [ -1.34,  6.39]
school | 343   | female1     |       -0.07 | 1.58 | [ -3.17,  3.02]
school | 344   | (Intercept) |        0.26 | 1.86 | [ -3.39,  3.92]
school | 344   | female1     |       -1.46 | 1.56 | [ -4.52,  1.61]
school | 345   | (Intercept) |        0.32 | 1.86 | [ -3.34,  3.97]
school | 345   | female1     |        0.19 | 1.56 | [ -2.87,  3.25]
school | 346   | (Intercept) |        1.47 | 1.86 | [ -2.18,  5.12]
school | 346   | female1     |       -0.78 | 1.56 | [ -3.84,  2.28]
school | 347   | (Intercept) |        0.21 | 2.02 | [ -3.76,  4.18]
school | 347   | female1     |       -0.32 | 1.59 | [ -3.43,  2.80]
school | 348   | (Intercept) |        3.84 | 1.97 | [ -0.03,  7.70]
school | 348   | female1     |        1.13 | 1.58 | [ -1.96,  4.23]
school | 349   | (Intercept) |        0.53 | 2.05 | [ -3.50,  4.55]
school | 349   | female1     |       -0.62 | 1.60 | [ -3.76,  2.51]
school | 350   | (Intercept) |        0.43 | 1.69 | [ -2.89,  3.75]
school | 350   | female1     |       -0.05 | 1.66 | [ -3.31,  3.20]
school | 351   | (Intercept) |        1.15 | 1.86 | [ -2.49,  4.79]
school | 351   | female1     |        0.13 | 1.57 | [ -2.95,  3.21]
school | 352   | (Intercept) |        2.77 | 2.09 | [ -1.33,  6.87]
school | 352   | female1     |        1.11 | 1.60 | [ -2.03,  4.24]
school | 353   | (Intercept) |        1.69 | 1.91 | [ -2.05,  5.43]
school | 353   | female1     |       -0.90 | 1.59 | [ -4.02,  2.22]
school | 354   | (Intercept) |        3.13 | 1.56 | [  0.07,  6.18]
school | 354   | female1     |       -0.13 | 1.50 | [ -3.06,  2.80]
school | 355   | (Intercept) |       -0.08 | 1.61 | [ -3.23,  3.08]
school | 355   | female1     |       -0.43 | 1.52 | [ -3.40,  2.54]
school | 356   | (Intercept) |        1.52 | 1.91 | [ -2.22,  5.26]
school | 356   | female1     |       -0.07 | 1.59 | [ -3.19,  3.04]
school | 357   | (Intercept) |        0.61 | 1.73 | [ -2.78,  3.99]
school | 357   | female1     |       -1.01 | 1.55 | [ -4.06,  2.04]
school | 358   | (Intercept) |       -0.35 | 1.99 | [ -4.26,  3.56]
school | 358   | female1     |   -7.86e-03 | 1.61 | [ -3.16,  3.14]
school | 359   | (Intercept) |        0.24 | 1.96 | [ -3.60,  4.09]
school | 359   | female1     |       -0.91 | 1.60 | [ -4.04,  2.21]
school | 360   | (Intercept) |       -2.49 | 2.03 | [ -6.46,  1.48]
school | 360   | female1     |       -0.04 | 1.59 | [ -3.16,  3.07]
school | 361   | (Intercept) |       -1.75 | 1.77 | [ -5.21,  1.71]
school | 361   | female1     |        0.40 | 1.58 | [ -2.69,  3.49]
school | 362   | (Intercept) |       -5.86 | 1.82 | [ -9.42, -2.30]
school | 362   | female1     |       -0.63 | 1.55 | [ -3.67,  2.42]
school | 363   | (Intercept) |       -5.01 | 2.02 | [ -8.97, -1.04]
school | 363   | female1     |        1.51 | 1.60 | [ -1.63,  4.64]
school | 364   | (Intercept) |       -2.80 | 2.05 | [ -6.82,  1.23]
school | 364   | female1     |       -0.14 | 1.60 | [ -3.27,  3.00]
school | 365   | (Intercept) |        2.50 | 1.92 | [ -1.26,  6.25]
school | 365   | female1     |       -0.32 | 1.67 | [ -3.58,  2.95]
school | 366   | (Intercept) |        2.80 | 1.78 | [ -0.69,  6.30]
school | 366   | female1     |       -0.90 | 1.54 | [ -3.92,  2.12]
school | 367   | (Intercept) |        2.16 | 1.76 | [ -1.28,  5.60]
school | 367   | female1     |       -0.64 | 1.54 | [ -3.66,  2.38]
school | 368   | (Intercept) |        5.00 | 2.01 | [  1.06,  8.93]
school | 368   | female1     |       -0.40 | 1.58 | [ -3.50,  2.71]
school | 369   | (Intercept) |        4.68 | 2.06 | [  0.64,  8.72]
school | 369   | female1     |       -0.05 | 1.59 | [ -3.17,  3.08]
school | 370   | (Intercept) |       -2.43 | 1.84 | [ -6.03,  1.17]
school | 370   | female1     |        0.69 | 1.56 | [ -2.37,  3.75]
school | 371   | (Intercept) |       -4.02 | 1.91 | [ -7.76, -0.28]
school | 371   | female1     |        0.33 | 1.58 | [ -2.76,  3.42]
school | 372   | (Intercept) |       -0.90 | 1.96 | [ -4.75,  2.95]
school | 372   | female1     |       -0.03 | 1.58 | [ -3.13,  3.08]
school | 373   | (Intercept) |        0.64 | 1.86 | [ -3.00,  4.29]
school | 373   | female1     |        0.96 | 1.55 | [ -2.09,  4.00]
school | 374   | (Intercept) |        1.86 | 1.91 | [ -1.88,  5.61]
school | 374   | female1     |       -0.74 | 1.56 | [ -3.80,  2.33]
school | 375   | (Intercept) |       -3.03 | 2.03 | [ -7.01,  0.96]
school | 375   | female1     |        0.06 | 1.59 | [ -3.05,  3.18]
school | 376   | (Intercept) |        0.95 | 1.91 | [ -2.78,  4.69]
school | 376   | female1     |        0.26 | 1.59 | [ -2.86,  3.37]
school | 377   | (Intercept) |       -2.33 | 1.74 | [ -5.75,  1.08]
school | 377   | female1     |       -0.03 | 1.53 | [ -3.04,  2.97]
school | 378   | (Intercept) |       -1.77 | 2.09 | [ -5.87,  2.32]
school | 378   | female1     |       -0.16 | 1.60 | [ -3.29,  2.98]
school | 379   | (Intercept) |        0.53 | 2.05 | [ -3.48,  4.55]
school | 379   | female1     |        0.04 | 1.59 | [ -3.08,  3.16]
school | 380   | (Intercept) |        0.57 | 1.78 | [ -2.93,  4.06]
school | 380   | female1     |        0.30 | 1.54 | [ -2.72,  3.33]
school | 381   | (Intercept) |        2.91 | 1.73 | [ -0.47,  6.29]
school | 381   | female1     |       -0.73 | 1.55 | [ -3.78,  2.32]
school | 382   | (Intercept) |       -4.67 | 1.75 | [ -8.11, -1.24]
school | 382   | female1     |        1.67 | 1.53 | [ -1.33,  4.67]
school | 383   | (Intercept) |       -6.28 | 2.09 | [-10.38, -2.19]
school | 383   | female1     |        0.97 | 1.60 | [ -2.16,  4.11]
school | 384   | (Intercept) |        0.27 | 1.86 | [ -3.37,  3.91]
school | 384   | female1     |       -0.46 | 1.57 | [ -3.54,  2.62]
school | 385   | (Intercept) |       -3.83 | 1.81 | [ -7.38, -0.28]
school | 385   | female1     |        0.85 | 1.56 | [ -2.22,  3.91]
school | 386   | (Intercept) |        1.01 | 1.81 | [ -2.54,  4.56]
school | 386   | female1     |       -0.04 | 1.56 | [ -3.11,  3.03]
school | 387   | (Intercept) |       -3.60 | 1.99 | [ -7.49,  0.29]
school | 387   | female1     |       -0.30 | 1.58 | [ -3.40,  2.80]
school | 388   | (Intercept) |        0.24 | 1.89 | [ -3.47,  3.95]
school | 388   | female1     |        1.00 | 1.56 | [ -2.06,  4.06]
school | 389   | (Intercept) |       -0.44 | 2.00 | [ -4.35,  3.47]
school | 389   | female1     |        0.03 | 1.59 | [ -3.07,  3.14]
school | 390   | (Intercept) |        4.04 | 1.77 | [  0.57,  7.51]
school | 390   | female1     |       -0.51 | 1.66 | [ -3.77,  2.75]
school | 391   | (Intercept) |       -1.98 | 1.78 | [ -5.47,  1.51]
school | 391   | female1     |        0.17 | 1.54 | [ -2.85,  3.19]
school | 392   | (Intercept) |        2.07 | 1.73 | [ -1.33,  5.46]
school | 392   | female1     |       -0.26 | 1.66 | [ -3.52,  3.00]
school | 393   | (Intercept) |       -1.37 | 1.79 | [ -4.88,  2.13]
school | 393   | female1     |       -0.18 | 1.57 | [ -3.25,  2.90]
school | 394   | (Intercept) |       -3.12 | 1.86 | [ -6.76,  0.52]
school | 394   | female1     |       -0.55 | 1.59 | [ -3.65,  2.56]
school | 395   | (Intercept) |        3.38 | 1.96 | [ -0.47,  7.23]
school | 395   | female1     |       -0.41 | 1.58 | [ -3.52,  2.69]
school | 396   | (Intercept) |       -1.83 | 1.78 | [ -5.32,  1.67]
school | 396   | female1     |       -0.02 | 1.54 | [ -3.04,  3.00]
school | 397   | (Intercept) |        0.60 | 1.99 | [ -3.30,  4.51]
school | 397   | female1     |       -0.35 | 1.61 | [ -3.50,  2.80]
school | 398   | (Intercept) |        1.08 | 1.77 | [ -2.40,  4.55]
school | 398   | female1     |       -0.14 | 1.66 | [ -3.40,  3.12]
school | 399   | (Intercept) |       -3.97 | 2.22 | [ -8.33,  0.39]
school | 399   | female1     |        0.90 | 1.63 | [ -2.29,  4.09]
school | 400   | (Intercept) |       -0.29 | 1.76 | [ -3.74,  3.15]
school | 400   | female1     |       -0.54 | 1.54 | [ -3.56,  2.47]
school | 401   | (Intercept) |        0.72 | 1.86 | [ -2.92,  4.36]
school | 401   | female1     |        0.08 | 1.59 | [ -3.03,  3.19]
school | 402   | (Intercept) |       -2.52 | 2.14 | [ -6.72,  1.69]
school | 402   | female1     |       -0.65 | 1.61 | [ -3.81,  2.51]
school | 403   | (Intercept) |        2.15 | 1.88 | [ -1.54,  5.85]
school | 403   | female1     |        0.57 | 1.60 | [ -2.56,  3.70]
school | 404   | (Intercept) |        1.49 | 1.93 | [ -2.30,  5.28]
school | 404   | female1     |       -0.25 | 1.59 | [ -3.36,  2.85]
school | 405   | (Intercept) |       -0.20 | 1.96 | [ -4.05,  3.64]
school | 405   | female1     |        0.10 | 1.60 | [ -3.03,  3.23]
school | 406   | (Intercept) |        0.37 | 1.77 | [ -3.10,  3.85]
school | 406   | female1     |       -0.28 | 1.55 | [ -3.32,  2.75]
school | 407   | (Intercept) |       -0.21 | 1.99 | [ -4.11,  3.69]
school | 407   | female1     |       -1.19 | 1.59 | [ -4.31,  1.93]
school | 408   | (Intercept) |       -1.91 | 2.02 | [ -5.87,  2.06]
school | 408   | female1     |       -0.36 | 1.59 | [ -3.48,  2.76]
school | 409   | (Intercept) |       -6.43 | 1.94 | [-10.23, -2.63]
school | 409   | female1     |        1.17 | 1.58 | [ -1.92,  4.26]
school | 410   | (Intercept) |       -0.41 | 2.05 | [ -4.43,  3.60]
school | 410   | female1     |       -0.53 | 1.59 | [ -3.65,  2.60]
school | 411   | (Intercept) |        0.95 | 1.67 | [ -2.32,  4.23]
school | 411   | female1     |        1.36 | 1.54 | [ -1.65,  4.38]
school | 412   | (Intercept) |        0.72 | 1.82 | [ -2.84,  4.29]
school | 412   | female1     |       -0.51 | 1.55 | [ -3.55,  2.54]
school | 413   | (Intercept) |        1.75 | 1.83 | [ -1.83,  5.34]
school | 413   | female1     |       -0.30 | 1.55 | [ -3.34,  2.74]
school | 414   | (Intercept) |       -0.46 | 1.89 | [ -4.16,  3.23]
school | 414   | female1     |        0.05 | 1.57 | [ -3.02,  3.13]
school | 415   | (Intercept) |        3.66 | 2.03 | [ -0.32,  7.64]
school | 415   | female1     |        0.55 | 1.65 | [ -2.69,  3.80]
school | 416   | (Intercept) |        1.22 | 1.96 | [ -2.63,  5.07]
school | 416   | female1     |       -0.51 | 1.58 | [ -3.61,  2.60]
school | 417   | (Intercept) |       -5.74 | 2.07 | [ -9.80, -1.67]
school | 417   | female1     |       -0.57 | 1.60 | [ -3.70,  2.56]
school | 418   | (Intercept) |       -1.49 | 1.57 | [ -4.57,  1.60]
school | 418   | female1     |        1.09 | 1.53 | [ -1.90,  4.09]
school | 419   | (Intercept) |       -2.53 | 1.72 | [ -5.89,  0.83]
school | 419   | female1     |        0.19 | 1.53 | [ -2.81,  3.20]

Random Effect Predictions

These are called Best Linear Unbiased Predictions (BLUPs).

estimate_grouplevel(fit, type = "total") # beta_0j and beta_1j

Group  | Level | Parameter   | Coefficient
------------------------------------------
school | 1     | (Intercept) |       59.27
school | 1     | female1     |       -1.46
school | 2     | (Intercept) |       61.97
school | 2     | female1     |       -1.67
school | 3     | (Intercept) |       50.97
school | 3     | female1     |       -1.36
school | 4     | (Intercept) |       63.83
school | 4     | female1     |       -0.36
school | 5     | (Intercept) |       51.62
school | 5     | female1     |       -1.97
school | 6     | (Intercept) |       60.02
school | 6     | female1     |        0.15
school | 7     | (Intercept) |       58.46
school | 7     | female1     |       -2.02
school | 8     | (Intercept) |       61.00
school | 8     | female1     |       -0.54
school | 9     | (Intercept) |       56.85
school | 9     | female1     |       -2.08
school | 10    | (Intercept) |       61.10
school | 10    | female1     |       -0.83
school | 11    | (Intercept) |       57.10
school | 11    | female1     |       -1.10
school | 12    | (Intercept) |       56.34
school | 12    | female1     |       -0.77
school | 13    | (Intercept) |       59.07
school | 13    | female1     |       -1.22
school | 14    | (Intercept) |       57.84
school | 14    | female1     |       -0.02
school | 15    | (Intercept) |       62.67
school | 15    | female1     |       -1.17
school | 16    | (Intercept) |       57.07
school | 16    | female1     |       -1.70
school | 17    | (Intercept) |       56.03
school | 17    | female1     |       -1.59
school | 18    | (Intercept) |       61.86
school | 18    | female1     |       -1.81
school | 19    | (Intercept) |       58.20
school | 19    | female1     |       -0.99
school | 20    | (Intercept) |       59.70
school | 20    | female1     |       -1.47
school | 21    | (Intercept) |       59.48
school | 21    | female1     |       -0.31
school | 22    | (Intercept) |       56.88
school | 22    | female1     |       -1.23
school | 23    | (Intercept) |       59.57
school | 23    | female1     |       -0.57
school | 24    | (Intercept) |       58.80
school | 24    | female1     |       -1.21
school | 25    | (Intercept) |       57.21
school | 25    | female1     |       -2.61
school | 26    | (Intercept) |       60.37
school | 26    | female1     |       -0.33
school | 27    | (Intercept) |       55.30
school | 27    | female1     |       -1.35
school | 28    | (Intercept) |       63.98
school | 28    | female1     |       -2.33
school | 29    | (Intercept) |       59.97
school | 29    | female1     |       -1.29
school | 30    | (Intercept) |       57.29
school | 30    | female1     |       -2.26
school | 31    | (Intercept) |       58.25
school | 31    | female1     |       -2.98
school | 32    | (Intercept) |       61.87
school | 32    | female1     |       -0.88
school | 33    | (Intercept) |       53.87
school | 33    | female1     |       -1.30
school | 34    | (Intercept) |       58.00
school | 34    | female1     |       -1.69
school | 35    | (Intercept) |       60.68
school | 35    | female1     |       -1.58
school | 36    | (Intercept) |       54.81
school | 36    | female1     |       -1.70
school | 37    | (Intercept) |       60.66
school | 37    | female1     |       -0.57
school | 38    | (Intercept) |       57.12
school | 38    | female1     |       -2.06
school | 39    | (Intercept) |       58.67
school | 39    | female1     |       -2.15
school | 40    | (Intercept) |       54.75
school | 40    | female1     |       -1.44
school | 41    | (Intercept) |       59.73
school | 41    | female1     |       -0.52
school | 42    | (Intercept) |       58.48
school | 42    | female1     |       -1.16
school | 43    | (Intercept) |       58.78
school | 43    | female1     |       -0.37
school | 44    | (Intercept) |       61.33
school | 44    | female1     |       -0.84
school | 45    | (Intercept) |       60.17
school | 45    | female1     |       -1.10
school | 46    | (Intercept) |       57.45
school | 46    | female1     |       -0.57
school | 47    | (Intercept) |       53.77
school | 47    | female1     |       -0.53
school | 48    | (Intercept) |       57.61
school | 48    | female1     |       -0.83
school | 49    | (Intercept) |       58.42
school | 49    | female1     |       -1.80
school | 50    | (Intercept) |       54.31
school | 50    | female1     |       -1.75
school | 51    | (Intercept) |       60.23
school | 51    | female1     |       -0.87
school | 52    | (Intercept) |       61.34
school | 52    | female1     |       -1.33
school | 53    | (Intercept) |       60.64
school | 53    | female1     |       -2.29
school | 54    | (Intercept) |       63.21
school | 54    | female1     |       -1.74
school | 55    | (Intercept) |       51.92
school | 55    | female1     |       -2.75
school | 56    | (Intercept) |       57.38
school | 56    | female1     |       -1.17
school | 57    | (Intercept) |       58.83
school | 57    | female1     |       -1.50
school | 58    | (Intercept) |       58.57
school | 58    | female1     |       -0.78
school | 59    | (Intercept) |       59.27
school | 59    | female1     |       -1.26
school | 60    | (Intercept) |       54.47
school | 60    | female1     |       -0.62
school | 61    | (Intercept) |       60.32
school | 61    | female1     |       -2.36
school | 62    | (Intercept) |       49.54
school | 62    | female1     |       -0.62
school | 63    | (Intercept) |       59.40
school | 63    | female1     |       -1.07
school | 64    | (Intercept) |       59.20
school | 64    | female1     |       -1.37
school | 65    | (Intercept) |       62.13
school | 65    | female1     |       -1.45
school | 66    | (Intercept) |       56.89
school | 66    | female1     |       -1.46
school | 67    | (Intercept) |       60.23
school | 67    | female1     |       -1.45
school | 68    | (Intercept) |       57.64
school | 68    | female1     |       -0.42
school | 69    | (Intercept) |       54.70
school | 69    | female1     |       -1.43
school | 70    | (Intercept) |       62.94
school | 70    | female1     |       -1.01
school | 71    | (Intercept) |       56.58
school | 71    | female1     |       -1.76
school | 72    | (Intercept) |       52.82
school | 72    | female1     |       -1.25
school | 73    | (Intercept) |       57.57
school | 73    | female1     |       -0.66
school | 74    | (Intercept) |       63.51
school | 74    | female1     |       -1.19
school | 75    | (Intercept) |       57.88
school | 75    | female1     |       -0.29
school | 76    | (Intercept) |       57.80
school | 76    | female1     |        0.04
school | 77    | (Intercept) |       62.73
school | 77    | female1     |       -1.15
school | 78    | (Intercept) |       57.17
school | 78    | female1     |       -1.29
school | 79    | (Intercept) |       55.71
school | 79    | female1     |       -1.46
school | 80    | (Intercept) |       62.38
school | 80    | female1     |       -0.98
school | 81    | (Intercept) |       55.80
school | 81    | female1     |       -1.36
school | 82    | (Intercept) |       60.04
school | 82    | female1     |       -2.04
school | 83    | (Intercept) |       60.15
school | 83    | female1     |        0.15
school | 84    | (Intercept) |       61.44
school | 84    | female1     |       -1.11
school | 85    | (Intercept) |       58.46
school | 85    | female1     |       -0.34
school | 86    | (Intercept) |       56.64
school | 86    | female1     |       -0.99
school | 87    | (Intercept) |       60.27
school | 87    | female1     |       -1.37
school | 88    | (Intercept) |       60.02
school | 88    | female1     |       -1.68
school | 89    | (Intercept) |       58.71
school | 89    | female1     |       -1.18
school | 90    | (Intercept) |       61.91
school | 90    | female1     |       -1.66
school | 91    | (Intercept) |       58.53
school | 91    | female1     |       -1.45
school | 92    | (Intercept) |       60.62
school | 92    | female1     |       -1.07
school | 93    | (Intercept) |       55.88
school | 93    | female1     |       -0.53
school | 94    | (Intercept) |       60.25
school | 94    | female1     |       -1.07
school | 95    | (Intercept) |       58.02
school | 95    | female1     |       -1.01
school | 96    | (Intercept) |       60.04
school | 96    | female1     |       -1.23
school | 97    | (Intercept) |       61.63
school | 97    | female1     |       -1.18
school | 98    | (Intercept) |       58.57
school | 98    | female1     |       -1.06
school | 99    | (Intercept) |       57.35
school | 99    | female1     |       -0.84
school | 100   | (Intercept) |       59.35
school | 100   | female1     |       -2.09
school | 101   | (Intercept) |       58.07
school | 101   | female1     |       -1.19
school | 102   | (Intercept) |       58.04
school | 102   | female1     |       -0.78
school | 103   | (Intercept) |       65.05
school | 103   | female1     |       -1.53
school | 104   | (Intercept) |       58.32
school | 104   | female1     |       -0.80
school | 105   | (Intercept) |       59.41
school | 105   | female1     |       -0.97
school | 106   | (Intercept) |       53.34
school | 106   | female1     |       -0.36
school | 107   | (Intercept) |       63.44
school | 107   | female1     |       -0.47
school | 108   | (Intercept) |       59.35
school | 108   | female1     |       -2.37
school | 109   | (Intercept) |       62.41
school | 109   | female1     |       -1.17
school | 110   | (Intercept) |       56.75
school | 110   | female1     |       -1.53
school | 111   | (Intercept) |       56.85
school | 111   | female1     |       -0.81
school | 112   | (Intercept) |       58.83
school | 112   | female1     |        0.34
school | 113   | (Intercept) |       59.38
school | 113   | female1     |       -0.99
school | 114   | (Intercept) |       56.75
school | 114   | female1     |       -1.76
school | 115   | (Intercept) |       61.50
school | 115   | female1     |       -1.35
school | 116   | (Intercept) |       49.71
school | 116   | female1     |       -1.19
school | 117   | (Intercept) |       55.63
school | 117   | female1     |       -0.59
school | 118   | (Intercept) |       57.45
school | 118   | female1     |       -0.38
school | 119   | (Intercept) |       60.20
school | 119   | female1     |       -0.90
school | 120   | (Intercept) |       61.93
school | 120   | female1     |       -0.85
school | 121   | (Intercept) |       59.01
school | 121   | female1     |       -1.70
school | 122   | (Intercept) |       55.04
school | 122   | female1     |       -1.47
school | 123   | (Intercept) |       58.41
school | 123   | female1     |       -0.53
school | 124   | (Intercept) |       59.72
school | 124   | female1     |       -0.51
school | 125   | (Intercept) |       56.70
school | 125   | female1     |       -0.41
school | 126   | (Intercept) |       58.08
school | 126   | female1     |       -1.66
school | 127   | (Intercept) |       61.23
school | 127   | female1     |       -1.23
school | 128   | (Intercept) |       58.73
school | 128   | female1     |       -1.13
school | 129   | (Intercept) |       55.53
school | 129   | female1     |       -0.71
school | 130   | (Intercept) |       57.02
school | 130   | female1     |       -1.27
school | 131   | (Intercept) |       56.88
school | 131   | female1     |       -0.79
school | 132   | (Intercept) |       55.25
school | 132   | female1     |       -1.53
school | 133   | (Intercept) |       62.55
school | 133   | female1     |       -1.59
school | 134   | (Intercept) |       59.08
school | 134   | female1     |       -1.74
school | 135   | (Intercept) |       53.04
school | 135   | female1     |       -0.22
school | 136   | (Intercept) |       58.29
school | 136   | female1     |       -2.18
school | 137   | (Intercept) |       58.64
school | 137   | female1     |       -1.62
school | 138   | (Intercept) |       58.18
school | 138   | female1     |       -1.39
school | 139   | (Intercept) |       56.27
school | 139   | female1     |        0.26
school | 140   | (Intercept) |       64.23
school | 140   | female1     |       -1.78
school | 141   | (Intercept) |       59.84
school | 141   | female1     |       -1.34
school | 142   | (Intercept) |       61.04
school | 142   | female1     |       -1.41
school | 143   | (Intercept) |       60.15
school | 143   | female1     |       -0.37
school | 144   | (Intercept) |       56.81
school | 144   | female1     |       -0.85
school | 145   | (Intercept) |       58.36
school | 145   | female1     |       -1.20
school | 146   | (Intercept) |       58.57
school | 146   | female1     |       -0.39
school | 147   | (Intercept) |       58.56
school | 147   | female1     |       -1.19
school | 148   | (Intercept) |       54.69
school | 148   | female1     |       -1.75
school | 149   | (Intercept) |       57.33
school | 149   | female1     |        0.08
school | 150   | (Intercept) |       61.57
school | 150   | female1     |       -1.62
school | 151   | (Intercept) |       55.39
school | 151   | female1     |       -0.88
school | 152   | (Intercept) |       61.38
school | 152   | female1     |       -0.55
school | 153   | (Intercept) |       65.06
school | 153   | female1     |       -1.33
school | 154   | (Intercept) |       56.69
school | 154   | female1     |       -0.82
school | 155   | (Intercept) |       54.74
school | 155   | female1     |        0.16
school | 156   | (Intercept) |       62.06
school | 156   | female1     |       -1.49
school | 157   | (Intercept) |       59.76
school | 157   | female1     |       -0.98
school | 158   | (Intercept) |       60.16
school | 158   | female1     |       -1.93
school | 159   | (Intercept) |       58.23
school | 159   | female1     |       -0.78
school | 160   | (Intercept) |       59.86
school | 160   | female1     |       -0.73
school | 161   | (Intercept) |       55.68
school | 161   | female1     |       -0.53
school | 162   | (Intercept) |       58.27
school | 162   | female1     |       -2.01
school | 163   | (Intercept) |       55.84
school | 163   | female1     |       -1.08
school | 164   | (Intercept) |       58.35
school | 164   | female1     |       -1.20
school | 165   | (Intercept) |       57.31
school | 165   | female1     |       -1.89
school | 166   | (Intercept) |       54.76
school | 166   | female1     |       -1.18
school | 167   | (Intercept) |       58.95
school | 167   | female1     |       -1.10
school | 168   | (Intercept) |       59.89
school | 168   | female1     |       -1.13
school | 169   | (Intercept) |       57.91
school | 169   | female1     |       -1.83
school | 170   | (Intercept) |       58.63
school | 170   | female1     |       -1.73
school | 171   | (Intercept) |       57.64
school | 171   | female1     |       -1.23
school | 172   | (Intercept) |       58.66
school | 172   | female1     |       -1.53
school | 173   | (Intercept) |       62.59
school | 173   | female1     |       -2.40
school | 174   | (Intercept) |       56.48
school | 174   | female1     |       -0.94
school | 175   | (Intercept) |       59.50
school | 175   | female1     |       -1.42
school | 176   | (Intercept) |       59.82
school | 176   | female1     |       -1.58
school | 177   | (Intercept) |       57.96
school | 177   | female1     |       -1.09
school | 178   | (Intercept) |       63.36
school | 178   | female1     |       -1.85
school | 179   | (Intercept) |       56.46
school | 179   | female1     |       -2.07
school | 180   | (Intercept) |       59.35
school | 180   | female1     |       -1.30
school | 181   | (Intercept) |       58.68
school | 181   | female1     |       -0.98
school | 182   | (Intercept) |       61.35
school | 182   | female1     |       -1.32
school | 183   | (Intercept) |       59.55
school | 183   | female1     |       -1.12
school | 184   | (Intercept) |       57.47
school | 184   | female1     |       -0.38
school | 185   | (Intercept) |       52.72
school | 185   | female1     |       -1.02
school | 186   | (Intercept) |       53.04
school | 186   | female1     |       -1.37
school | 187   | (Intercept) |       57.34
school | 187   | female1     |       -1.76
school | 188   | (Intercept) |       61.06
school | 188   | female1     |       -0.37
school | 189   | (Intercept) |       55.19
school | 189   | female1     |       -2.61
school | 190   | (Intercept) |       56.57
school | 190   | female1     |        0.31
school | 191   | (Intercept) |       57.51
school | 191   | female1     |       -1.38
school | 192   | (Intercept) |       61.38
school | 192   | female1     |       -1.60
school | 193   | (Intercept) |       55.51
school | 193   | female1     |       -1.63
school | 194   | (Intercept) |       60.96
school | 194   | female1     |       -0.93
school | 195   | (Intercept) |       58.40
school | 195   | female1     |       -0.36
school | 196   | (Intercept) |       56.63
school | 196   | female1     |       -0.27
school | 197   | (Intercept) |       50.04
school | 197   | female1     |       -0.81
school | 198   | (Intercept) |       56.51
school | 198   | female1     |       -1.17
school | 199   | (Intercept) |       58.38
school | 199   | female1     |       -2.04
school | 200   | (Intercept) |       54.55
school | 200   | female1     |        0.85
school | 201   | (Intercept) |       58.37
school | 201   | female1     |        0.21
school | 202   | (Intercept) |       60.96
school | 202   | female1     |       -0.50
school | 203   | (Intercept) |       58.22
school | 203   | female1     |       -1.44
school | 204   | (Intercept) |       59.08
school | 204   | female1     |       -1.16
school | 205   | (Intercept) |       56.23
school | 205   | female1     |       -2.26
school | 206   | (Intercept) |       56.61
school | 206   | female1     |       -1.10
school | 207   | (Intercept) |       57.68
school | 207   | female1     |       -0.95
school | 208   | (Intercept) |       55.29
school | 208   | female1     |       -1.30
school | 209   | (Intercept) |       60.91
school | 209   | female1     |       -0.62
school | 210   | (Intercept) |       53.78
school | 210   | female1     |        0.09
school | 211   | (Intercept) |       55.90
school | 211   | female1     |        0.49
school | 212   | (Intercept) |       54.70
school | 212   | female1     |       -1.18
school | 213   | (Intercept) |       61.14
school | 213   | female1     |       -1.50
school | 214   | (Intercept) |       54.33
school | 214   | female1     |       -0.55
school | 215   | (Intercept) |       59.07
school | 215   | female1     |       -1.71
school | 216   | (Intercept) |       59.61
school | 216   | female1     |       -0.74
school | 217   | (Intercept) |       61.18
school | 217   | female1     |       -0.89
school | 218   | (Intercept) |       57.94
school | 218   | female1     |       -2.08
school | 219   | (Intercept) |       58.56
school | 219   | female1     |       -2.31
school | 220   | (Intercept) |       52.77
school | 220   | female1     |       -1.03
school | 221   | (Intercept) |       61.59
school | 221   | female1     |       -1.19
school | 222   | (Intercept) |       55.52
school | 222   | female1     |       -1.90
school | 223   | (Intercept) |       53.07
school | 223   | female1     |       -2.04
school | 224   | (Intercept) |       60.19
school | 224   | female1     |       -1.08
school | 225   | (Intercept) |       54.84
school | 225   | female1     |       -1.72
school | 226   | (Intercept) |       61.64
school | 226   | female1     |       -0.88
school | 227   | (Intercept) |       57.30
school | 227   | female1     |       -1.74
school | 228   | (Intercept) |       58.66
school | 228   | female1     |       -1.74
school | 229   | (Intercept) |       60.57
school | 229   | female1     |       -1.27
school | 230   | (Intercept) |       59.81
school | 230   | female1     |       -1.71
school | 231   | (Intercept) |       55.99
school | 231   | female1     |       -1.32
school | 232   | (Intercept) |       57.82
school | 232   | female1     |       -0.44
school | 233   | (Intercept) |       58.76
school | 233   | female1     |       -1.00
school | 234   | (Intercept) |       59.12
school | 234   | female1     |       -0.85
school | 235   | (Intercept) |       61.31
school | 235   | female1     |       -1.17
school | 236   | (Intercept) |       57.36
school | 236   | female1     |       -0.45
school | 237   | (Intercept) |       56.56
school | 237   | female1     |       -1.91
school | 238   | (Intercept) |       58.64
school | 238   | female1     |       -0.76
school | 239   | (Intercept) |       55.50
school | 239   | female1     |       -1.33
school | 240   | (Intercept) |       61.62
school | 240   | female1     |       -1.55
school | 241   | (Intercept) |       62.86
school | 241   | female1     |       -1.44
school | 242   | (Intercept) |       57.72
school | 242   | female1     |       -1.35
school | 243   | (Intercept) |       53.32
school | 243   | female1     |       -1.39
school | 244   | (Intercept) |       59.19
school | 244   | female1     |       -0.11
school | 245   | (Intercept) |       60.44
school | 245   | female1     |       -1.14
school | 246   | (Intercept) |       51.85
school | 246   | female1     |       -1.43
school | 247   | (Intercept) |       56.59
school | 247   | female1     |       -0.66
school | 248   | (Intercept) |       59.66
school | 248   | female1     |       -1.07
school | 249   | (Intercept) |       54.04
school | 249   | female1     |       -0.48
school | 250   | (Intercept) |       55.10
school | 250   | female1     |       -2.53
school | 251   | (Intercept) |       61.15
school | 251   | female1     |       -2.27
school | 252   | (Intercept) |       62.61
school | 252   | female1     |       -0.97
school | 253   | (Intercept) |       55.83
school | 253   | female1     |        0.20
school | 254   | (Intercept) |       55.91
school | 254   | female1     |       -1.02
school | 255   | (Intercept) |       58.78
school | 255   | female1     |       -1.03
school | 256   | (Intercept) |       58.85
school | 256   | female1     |       -0.97
school | 257   | (Intercept) |       63.84
school | 257   | female1     |       -0.36
school | 258   | (Intercept) |       58.85
school | 258   | female1     |       -0.59
school | 259   | (Intercept) |       54.64
school | 259   | female1     |       -2.38
school | 260   | (Intercept) |       57.13
school | 260   | female1     |       -1.92
school | 261   | (Intercept) |       57.22
school | 261   | female1     |       -1.51
school | 262   | (Intercept) |       62.08
school | 262   | female1     |       -1.89
school | 263   | (Intercept) |       57.80
school | 263   | female1     |       -1.82
school | 264   | (Intercept) |       54.71
school | 264   | female1     |       -2.23
school | 265   | (Intercept) |       56.38
school | 265   | female1     |       -1.98
school | 266   | (Intercept) |       55.67
school | 266   | female1     |       -2.10
school | 267   | (Intercept) |       58.97
school | 267   | female1     |       -1.24
school | 268   | (Intercept) |       60.60
school | 268   | female1     |       -1.49
school | 269   | (Intercept) |       55.85
school | 269   | female1     |       -0.40
school | 270   | (Intercept) |       60.39
school | 270   | female1     |       -1.13
school | 271   | (Intercept) |       60.84
school | 271   | female1     |       -1.98
school | 272   | (Intercept) |       57.22
school | 272   | female1     |       -0.69
school | 273   | (Intercept) |       57.98
school | 273   | female1     |       -1.26
school | 274   | (Intercept) |       62.16
school | 274   | female1     |       -2.21
school | 275   | (Intercept) |       62.92
school | 275   | female1     |       -1.29
school | 276   | (Intercept) |       57.05
school | 276   | female1     |       -1.22
school | 277   | (Intercept) |       58.08
school | 277   | female1     |       -0.74
school | 278   | (Intercept) |       59.96
school | 278   | female1     |       -1.18
school | 279   | (Intercept) |       58.98
school | 279   | female1     |       -1.37
school | 280   | (Intercept) |       58.46
school | 280   | female1     |       -1.72
school | 281   | (Intercept) |       63.43
school | 281   | female1     |       -1.34
school | 282   | (Intercept) |       63.84
school | 282   | female1     |       -1.91
school | 283   | (Intercept) |       56.23
school | 283   | female1     |       -0.71
school | 284   | (Intercept) |       60.56
school | 284   | female1     |       -2.86
school | 285   | (Intercept) |       56.86
school | 285   | female1     |       -1.67
school | 286   | (Intercept) |       56.91
school | 286   | female1     |       -1.24
school | 287   | (Intercept) |       54.85
school | 287   | female1     |       -1.91
school | 288   | (Intercept) |       61.86
school | 288   | female1     |       -0.87
school | 289   | (Intercept) |       57.49
school | 289   | female1     |       -1.43
school | 290   | (Intercept) |       56.01
school | 290   | female1     |       -2.00
school | 291   | (Intercept) |       57.70
school | 291   | female1     |       -2.25
school | 292   | (Intercept) |       56.92
school | 292   | female1     |       -1.08
school | 293   | (Intercept) |       56.22
school | 293   | female1     |       -0.74
school | 294   | (Intercept) |       58.72
school | 294   | female1     |        0.04
school | 295   | (Intercept) |       52.97
school | 295   | female1     |       -1.01
school | 296   | (Intercept) |       59.91
school | 296   | female1     |       -1.38
school | 297   | (Intercept) |       51.59
school | 297   | female1     |       -1.35
school | 298   | (Intercept) |       58.86
school | 298   | female1     |       -1.67
school | 299   | (Intercept) |       58.29
school | 299   | female1     |        0.43
school | 300   | (Intercept) |       57.39
school | 300   | female1     |       -0.67
school | 301   | (Intercept) |       55.73
school | 301   | female1     |       -1.53
school | 302   | (Intercept) |       55.98
school | 302   | female1     |       -2.30
school | 303   | (Intercept) |       62.84
school | 303   | female1     |       -1.05
school | 304   | (Intercept) |       60.68
school | 304   | female1     |       -1.95
school | 305   | (Intercept) |       56.16
school | 305   | female1     |       -0.53
school | 306   | (Intercept) |       55.66
school | 306   | female1     |       -1.30
school | 307   | (Intercept) |       60.29
school | 307   | female1     |       -1.30
school | 308   | (Intercept) |       59.75
school | 308   | female1     |       -1.85
school | 309   | (Intercept) |       58.97
school | 309   | female1     |       -0.52
school | 310   | (Intercept) |       56.40
school | 310   | female1     |       -0.57
school | 311   | (Intercept) |       62.50
school | 311   | female1     |       -1.05
school | 312   | (Intercept) |       56.82
school | 312   | female1     |        0.20
school | 313   | (Intercept) |       54.69
school | 313   | female1     |       -0.88
school | 314   | (Intercept) |       58.66
school | 314   | female1     |       -1.67
school | 315   | (Intercept) |       59.59
school | 315   | female1     |       -2.10
school | 316   | (Intercept) |       53.91
school | 316   | female1     |       -0.35
school | 317   | (Intercept) |       59.55
school | 317   | female1     |       -2.20
school | 318   | (Intercept) |       58.13
school | 318   | female1     |       -1.41
school | 319   | (Intercept) |       57.64
school | 319   | female1     |       -0.82
school | 320   | (Intercept) |       60.38
school | 320   | female1     |       -1.73
school | 321   | (Intercept) |       59.75
school | 321   | female1     |       -2.14
school | 322   | (Intercept) |       58.04
school | 322   | female1     |       -1.31
school | 323   | (Intercept) |       59.29
school | 323   | female1     |       -1.12
school | 324   | (Intercept) |       58.90
school | 324   | female1     |        0.32
school | 325   | (Intercept) |       61.28
school | 325   | female1     |       -0.98
school | 326   | (Intercept) |       58.97
school | 326   | female1     |       -2.80
school | 327   | (Intercept) |       61.10
school | 327   | female1     |       -0.53
school | 328   | (Intercept) |       62.31
school | 328   | female1     |       -1.22
school | 329   | (Intercept) |       61.86
school | 329   | female1     |       -0.98
school | 330   | (Intercept) |       54.60
school | 330   | female1     |       -1.05
school | 331   | (Intercept) |       58.68
school | 331   | female1     |       -1.09
school | 332   | (Intercept) |       58.55
school | 332   | female1     |       -1.45
school | 333   | (Intercept) |       60.35
school | 333   | female1     |       -0.48
school | 334   | (Intercept) |       60.78
school | 334   | female1     |       -1.28
school | 335   | (Intercept) |       62.57
school | 335   | female1     |       -1.20
school | 336   | (Intercept) |       57.42
school | 336   | female1     |       -0.41
school | 337   | (Intercept) |       58.35
school | 337   | female1     |       -1.55
school | 338   | (Intercept) |       57.11
school | 338   | female1     |       -1.67
school | 339   | (Intercept) |       53.61
school | 339   | female1     |       -1.32
school | 340   | (Intercept) |       53.71
school | 340   | female1     |       -0.77
school | 341   | (Intercept) |       60.74
school | 341   | female1     |       -1.46
school | 342   | (Intercept) |       55.42
school | 342   | female1     |       -0.84
school | 343   | (Intercept) |       60.80
school | 343   | female1     |       -1.28
school | 344   | (Intercept) |       58.53
school | 344   | female1     |       -2.66
school | 345   | (Intercept) |       58.59
school | 345   | female1     |       -1.01
school | 346   | (Intercept) |       59.74
school | 346   | female1     |       -1.98
school | 347   | (Intercept) |       58.48
school | 347   | female1     |       -1.52
school | 348   | (Intercept) |       62.11
school | 348   | female1     |       -0.07
school | 349   | (Intercept) |       58.80
school | 349   | female1     |       -1.83
school | 350   | (Intercept) |       58.70
school | 350   | female1     |       -1.26
school | 351   | (Intercept) |       59.43
school | 351   | female1     |       -1.07
school | 352   | (Intercept) |       61.04
school | 352   | female1     |       -0.10
school | 353   | (Intercept) |       59.96
school | 353   | female1     |       -2.10
school | 354   | (Intercept) |       61.40
school | 354   | female1     |       -1.33
school | 355   | (Intercept) |       58.20
school | 355   | female1     |       -1.64
school | 356   | (Intercept) |       59.79
school | 356   | female1     |       -1.28
school | 357   | (Intercept) |       58.88
school | 357   | female1     |       -2.22
school | 358   | (Intercept) |       57.92
school | 358   | female1     |       -1.21
school | 359   | (Intercept) |       58.51
school | 359   | female1     |       -2.12
school | 360   | (Intercept) |       55.78
school | 360   | female1     |       -1.25
school | 361   | (Intercept) |       56.52
school | 361   | female1     |       -0.80
school | 362   | (Intercept) |       52.41
school | 362   | female1     |       -1.83
school | 363   | (Intercept) |       53.27
school | 363   | female1     |        0.30
school | 364   | (Intercept) |       55.48
school | 364   | female1     |       -1.34
school | 365   | (Intercept) |       60.77
school | 365   | female1     |       -1.52
school | 366   | (Intercept) |       61.08
school | 366   | female1     |       -2.11
school | 367   | (Intercept) |       60.43
school | 367   | female1     |       -1.84
school | 368   | (Intercept) |       63.27
school | 368   | female1     |       -1.60
school | 369   | (Intercept) |       62.95
school | 369   | female1     |       -1.25
school | 370   | (Intercept) |       55.84
school | 370   | female1     |       -0.51
school | 371   | (Intercept) |       54.25
school | 371   | female1     |       -0.88
school | 372   | (Intercept) |       57.37
school | 372   | female1     |       -1.23
school | 373   | (Intercept) |       58.92
school | 373   | female1     |       -0.25
school | 374   | (Intercept) |       60.14
school | 374   | female1     |       -1.94
school | 375   | (Intercept) |       55.25
school | 375   | female1     |       -1.14
school | 376   | (Intercept) |       59.23
school | 376   | female1     |       -0.95
school | 377   | (Intercept) |       55.94
school | 377   | female1     |       -1.24
school | 378   | (Intercept) |       56.50
school | 378   | female1     |       -1.36
school | 379   | (Intercept) |       58.81
school | 379   | female1     |       -1.16
school | 380   | (Intercept) |       58.84
school | 380   | female1     |       -0.90
school | 381   | (Intercept) |       61.18
school | 381   | female1     |       -1.93
school | 382   | (Intercept) |       53.60
school | 382   | female1     |        0.47
school | 383   | (Intercept) |       51.99
school | 383   | female1     |       -0.23
school | 384   | (Intercept) |       58.55
school | 384   | female1     |       -1.66
school | 385   | (Intercept) |       54.44
school | 385   | female1     |       -0.36
school | 386   | (Intercept) |       59.28
school | 386   | female1     |       -1.24
school | 387   | (Intercept) |       54.67
school | 387   | female1     |       -1.50
school | 388   | (Intercept) |       58.51
school | 388   | female1     |       -0.20
school | 389   | (Intercept) |       57.83
school | 389   | female1     |       -1.17
school | 390   | (Intercept) |       62.31
school | 390   | female1     |       -1.72
school | 391   | (Intercept) |       56.29
school | 391   | female1     |       -1.03
school | 392   | (Intercept) |       60.34
school | 392   | female1     |       -1.47
school | 393   | (Intercept) |       56.90
school | 393   | female1     |       -1.38
school | 394   | (Intercept) |       55.15
school | 394   | female1     |       -1.75
school | 395   | (Intercept) |       61.65
school | 395   | female1     |       -1.62
school | 396   | (Intercept) |       56.45
school | 396   | female1     |       -1.23
school | 397   | (Intercept) |       58.88
school | 397   | female1     |       -1.56
school | 398   | (Intercept) |       59.35
school | 398   | female1     |       -1.34
school | 399   | (Intercept) |       54.30
school | 399   | female1     |       -0.31
school | 400   | (Intercept) |       57.98
school | 400   | female1     |       -1.75
school | 401   | (Intercept) |       58.99
school | 401   | female1     |       -1.13
school | 402   | (Intercept) |       55.75
school | 402   | female1     |       -1.86
school | 403   | (Intercept) |       60.43
school | 403   | female1     |       -0.64
school | 404   | (Intercept) |       59.76
school | 404   | female1     |       -1.46
school | 405   | (Intercept) |       58.07
school | 405   | female1     |       -1.11
school | 406   | (Intercept) |       58.64
school | 406   | female1     |       -1.49
school | 407   | (Intercept) |       58.06
school | 407   | female1     |       -2.40
school | 408   | (Intercept) |       56.37
school | 408   | female1     |       -1.57
school | 409   | (Intercept) |       51.84
school | 409   | female1     |       -0.03
school | 410   | (Intercept) |       57.86
school | 410   | female1     |       -1.73
school | 411   | (Intercept) |       59.23
school | 411   | female1     |        0.16
school | 412   | (Intercept) |       59.00
school | 412   | female1     |       -1.71
school | 413   | (Intercept) |       60.03
school | 413   | female1     |       -1.51
school | 414   | (Intercept) |       57.81
school | 414   | female1     |       -1.15
school | 415   | (Intercept) |       61.93
school | 415   | female1     |       -0.65
school | 416   | (Intercept) |       59.49
school | 416   | female1     |       -1.71
school | 417   | (Intercept) |       52.54
school | 417   | female1     |       -1.77
school | 418   | (Intercept) |       56.78
school | 418   | female1     |       -0.11
school | 419   | (Intercept) |       55.74
school | 419   | female1     |       -1.01

Plotting the Fixed Effects

These are estimates for the average school in the population.

rel <- estimate_relation(fit, by = "female")
plot(rel)

Plotting the Random Effects

Because we estimated a random intercept and a random slope, our model allows the effect of being female on math scores to vary across schools.
Notice how the lines (one per cluster/school) are no longer parallel!

Visualizing the Slope Distribution

Sample-based slope distribution (blue)
Population-level slope estimate (red)

The Concept of Shrinkage

The histogram is narrower than the curve. This is not an error!

• The Red Curve represents the estimated true distribution of slopes in the population.
• The Blue Histogram represents our specific predictions for the schools in this sample.
• Shrinkage (Partial Pooling): Because we have uncertainty about each specific school, the model is conservative. It pulls extreme, uncertain estimates toward the overall mean.
• As a result, the spread of our sample predictions is always smaller than the estimated population spread.

Plotting the RE Correlation

This is a visualization of \(u_{0j}\), \(u_{1j}\), and \(\color{#e41a1c}{\tau_{10}}\).