Multilevel Modeling

Cross-level Interactions

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

Roadmap

1. Conceptual Foundations
   • Cross-level Interactions
2. Application and Interpretation
   • Fitting and Interpreting Models

Conceptual Foundations

Interactions

• Interaction (moderation) effects allow the effect (i.e., slope) of a predictor to depend on the values of other predictors.
  • We simply estimate a new slope for the product of the predictors.
  • In week 02, we did this for predictors in single-level models.

• We also used tools to aid interpretation.
  • Spotlight plots via estimate_relation().
  • Contrast analyses via estimate_contrasts().
  • Simple slopes analysis via estimate_slopes().

Interactions Equation

• Add a slope for the product of two predictors.
  • \(y_i = \beta_0 + \beta_1 x_i + \beta_2 z_i + \color{red}{\beta_3 x_i z_i} + e_{i}\)

• We can reorganize this equation using algebra.
  • \(y_i = \beta_0 + (\beta_1 + \color{red}{\beta_3 z_i}) \color{red}{x_i} + \beta_2 z_i + e_i\)
  • \(\beta_1\) is the simple slope of \(x_i\) and \(\beta_3\) is the interaction effect.

• The total slope of \(x_i\) is now \((\beta_1 + \beta_3 m_i)\).
  • It thus depends on the value of \(z_i\).

• Moderation is symmetrical: the slope of \(z_i\) also depends on \(x_i\).

Cross-level Interactions

• But what if your predictors are on different levels?

• We can make this work using cross-level interactions.
  • Do L1 relationships depend on L2 predictors?
  • How do aspects of the cluster (e.g., school) influence
    within-cluster (e.g., student-level) relationships?

• This allows us to treat the cluster as a moderator.
  • Why are relationships stronger in some clusters than others?

Example Cross-level Questions

• Is the difference between male and female smiling greater in nations that are more religious (vs. those that are less religious)?
  • L1: \(x_{ij}\) = participant’s sex, \(y_{ij}\) = participant’s smiling
  • L2: \(z_j\) = nation’s religiousness

• Is the relationship between homework and math scores stronger in schools with after-school programs (vs. those without)?
  • L1: \(x_{ij}\) = student’s homework time, \(y_{ij}\) = student’s math score
  • L2: \(z_j\) = school’s after-school program status

Same-Level Interactions

• Multilevel models are not restricted to cross-level interactions.
• You can also include interactions between predictors at the same level.
• Level 1 Interaction:
  • Fixed effects: math ~ 1 + homework * motivation
• Random Slopes for L1 Interactions:
  • Maximal syntax: + (1 + homework * motivation | school)
  • Warning: This estimates a massive random effects matrix and may cause convergence issues.
• Level 2 Interaction:
  • math ~ 1 + public * funding + (1 | school)

A Note on Predictors Today

1. For today’s examples, we are strictly using categorical (dummy-coded) predictors.
2. Because dummy codes (0 and 1) have an inherently meaningful zero (the reference group), our model intercepts remain easy to interpret.
3. If we were using continuous predictors, we would need to scale them carefully so that their zero values make sense.
4. We will learn how to do that next week (using centering and degrouping) before we apply cross-level interactions to continuous variables.

Multilevel Equation Part 1

• We will accomplish this by adding to the random slope equation

Level One (Observation)

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

Level Two (Cluster)

\[ \begin{align} \beta_{0j} &= \color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{01}} z_j + \color{#e41a1c}{u_{0j}} \\ \beta_{1j} &= \color{#377eb8}{\gamma_{10}} + \underbrace{\color{#377eb8}{\gamma_{11}} z_j}_{\text{New}} + \color{#e41a1c}{u_{1j}} \end{align} \]

• The slope \(\beta_{1j}\) thus now depends on the value of \(z_j\)

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 (e.g., \(\tau_{00}\) and \(\tau_{11}\)) and correlations (e.g., \(\tau_{10}\)).

Level One (Observation)

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

Level Two (Cluster)

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

Path Diagram

Mixed (or Reduced) Equation

\[ \begin{alignat}{2} &\text{L1:} \quad & y_{ij} &= \beta_{0j} + \beta_{1j} x_{ij} + \color{#4daf4a}{e_{ij}} \\ &\text{L2:} \quad & \beta_{0j} &= \color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{01}} z_j + \color{#e41a1c}{u_{0j}} \\ & & \beta_{1j} &= \color{#377eb8}{\gamma_{10}} + \color{#377eb8}{\gamma_{11}} z_j + \color{#e41a1c}{u_{1j}} \end{alignat} \]

Substitute the L2 equations directly into the L1 equation:

\[ y_{ij} = (\color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{01}} z_j + \color{#e41a1c}{u_{0j}}) + (\color{#377eb8}{\gamma_{10}} + \color{#377eb8}{\gamma_{11}} z_j + \color{#e41a1c}{u_{1j}}) x_{ij} + \color{#4daf4a}{e_{ij}} \]

Rearrange to group the fixed and random parts:

\[ y_{ij} = \underbrace{\color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{01}} z_j + \color{#377eb8}{\gamma_{10}} x_{ij} + \color{#377eb8}{\gamma_{11}} z_j x_{ij}}_{\text{Fixed}} + \underbrace{\color{#e41a1c}{u_{0j}} + \color{#e41a1c}{u_{1j}} x_{ij}}_{\text{Random}} + \color{#4daf4a}{e_{ij}} \]

Formula

\[ y_{ij} = \underbrace{\color{#377eb8}{\gamma_{00}} + \color{#377eb8}{\gamma_{01}} z_j + \color{#377eb8}{\gamma_{10}} x_{ij} + \color{#377eb8}{\gamma_{11}} z_j x_{ij}}_{\text{Fixed}} + \underbrace{\color{#e41a1c}{u_{0j}} + \color{#e41a1c}{u_{1j}} x_{ij}}_{\text{(Random)}} + \color{#4daf4a}{e_{ij}} \]

Generic

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

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

Example

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

The Importance of Random Slopes

• When testing a cross-level interaction, you must include the random slope for the Level 1 predictor.
• If you omit the random slope (e.g., using (1 | school) instead of (1 + female | school)), the model assumes the Level 2 predictor perfectly explains all slope variance.
• This results in standard errors that are too small and severely inflates your Type I error rate.
• Rule of Thumb: If a Level 1 predictor \(x_{ij}\) interacts with a Level 2 predictor \(z_j\), \(x_{ij}\) needs a random slope.

Application and Interpretation

Setup

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

dat <- read_csv("heck2011.csv") |>
  mutate(school = factor(school), female = factor(female), public = factor(public))
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  <fct> 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(math ~ 1 + public * female + (1 + female | school), data = dat)
model_parameters(fit)

# Fixed Effects

Parameter               | Coefficient |   SE |         95% CI | t(6863) |      p
--------------------------------------------------------------------------------
(Intercept)             |       57.79 | 0.42 | [56.97, 58.62] |  137.47 | < .001
public [1]              |        0.65 | 0.49 | [-0.31,  1.61] |    1.33 | 0.185 
female [1]              |       -0.61 | 0.42 | [-1.44,  0.21] |   -1.45 | 0.146 
public [1] × female [1] |       -0.81 | 0.49 | [-1.78,  0.16] |   -1.64 | 0.102 

# Random Effects

Parameter                       | Coefficient
---------------------------------------------
SD (Intercept: school)          |        3.34
SD (female1: school)            |        1.69
Cor (Intercept~female1: school) |       -0.24
SD (Residual)                   |        8.10

Interpreting the Fixed Effects

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

• The fixed simple slope of public \((\gamma_{01})\) is…
  • The difference between public and private schools’ average intercepts (i.e., average male student’s math scores)
  • Not significantly different from zero

Interpreting the Fixed Effects

• The fixed simple slope of female \((\gamma_{10})\) is…
  • The difference between female and male students’ average math scores at the average private school
  • Not significantly different from zero

• The fixed public-by-female interaction \((\gamma_{11})\) is…
  • The difference in the female-male sex difference in math scores between the average public and private school
  • Not significantly different from zero

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 private schools?

• The random slope (of female) SD \((\tau_{11})\) is…
  • The spread of the random slope deviations
  • How much do sex differences vary between schools (after accounting for public status)?

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 students’ math scores within schools
    (after accounting for sex and public status)

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) |        1.15 | 2.05 | [ -2.87,  5.17]
school | 1     | female [1]  |       -0.31 | 1.59 | [ -3.43,  2.80]
school | 2     | (Intercept) |        4.03 | 1.86 | [  0.37,  7.68]
school | 2     | female1     |       -0.50 | 1.65 | [ -3.74,  2.75]
school | 3     | (Intercept) |       -7.16 | 1.87 | [-10.83, -3.49]
school | 3     | female1     |       -0.26 | 1.55 | [ -3.30,  2.78]
school | 4     | (Intercept) |        5.50 | 2.20 | [  1.19,  9.80]
school | 4     | female1     |        0.83 | 1.65 | [ -2.40,  4.06]
school | 5     | (Intercept) |       -6.47 | 1.83 | [-10.06, -2.87]
school | 5     | female1     |       -0.86 | 1.55 | [ -3.90,  2.18]
school | 6     | (Intercept) |        1.93 | 1.88 | [ -1.76,  5.62]
school | 6     | female1     |        1.25 | 1.56 | [ -1.80,  4.31]
school | 7     | (Intercept) |        0.12 | 1.88 | [ -3.56,  3.81]
school | 7     | female1     |       -0.77 | 1.56 | [ -3.83,  2.29]
school | 8     | (Intercept) |        2.66 | 1.61 | [ -0.49,  5.81]
school | 8     | female1     |        0.71 | 1.49 | [ -2.21,  3.64]
school | 9     | (Intercept) |       -1.47 | 1.92 | [ -5.25,  2.30]
school | 9     | female1     |       -0.84 | 1.56 | [ -3.89,  2.22]
school | 10    | (Intercept) |        2.77 | 1.78 | [ -0.72,  6.26]
school | 10    | female1     |        0.41 | 1.53 | [ -2.59,  3.41]
school | 11    | (Intercept) |       -1.26 | 1.72 | [ -4.64,  2.12]
school | 11    | female1     |        0.14 | 1.57 | [ -2.93,  3.21]
school | 12    | (Intercept) |       -1.97 | 2.09 | [ -6.06,  2.12]
school | 12    | female1     |        0.45 | 1.59 | [ -2.66,  3.57]
school | 13    | (Intercept) |        0.77 | 1.98 | [ -3.12,  4.65]
school | 13    | female1     |        0.01 | 1.57 | [ -3.07,  3.09]
school | 14    | (Intercept) |       -0.45 | 2.03 | [ -4.43,  3.52]
school | 14    | female1     |        1.20 | 1.58 | [ -1.90,  4.29]
school | 15    | (Intercept) |        4.33 | 1.73 | [  0.94,  7.72]
school | 15    | female1     |        0.08 | 1.53 | [ -2.92,  3.08]
school | 16    | (Intercept) |       -1.09 | 2.06 | [ -5.12,  2.94]
school | 16    | female1     |       -0.56 | 1.58 | [ -3.67,  2.54]
school | 17    | (Intercept) |       -2.07 | 1.80 | [ -5.59,  1.45]
school | 17    | female1     |       -0.48 | 1.54 | [ -3.50,  2.53]
school | 18    | (Intercept) |        3.71 | 1.97 | [ -0.14,  7.57]
school | 18    | female1     |       -0.67 | 1.57 | [ -3.75,  2.40]
school | 19    | (Intercept) |        0.12 | 1.91 | [ -3.62,  3.85]
school | 19    | female1     |        0.13 | 1.57 | [ -2.94,  3.20]
school | 20    | (Intercept) |        1.51 | 2.10 | [ -2.60,  5.62]
school | 20    | female1     |       -0.32 | 1.59 | [ -3.44,  2.80]
school | 21    | (Intercept) |        1.37 | 1.81 | [ -2.18,  4.91]
school | 21    | female1     |        0.79 | 1.53 | [ -2.22,  3.79]
school | 22    | (Intercept) |       -1.24 | 2.05 | [ -5.26,  2.78]
school | 22    | female1     |       -0.09 | 1.59 | [ -3.21,  3.02]
school | 23    | (Intercept) |        1.50 | 1.93 | [ -2.29,  5.28]
school | 23    | female1     |        0.55 | 1.58 | [ -2.54,  3.64]
school | 24    | (Intercept) |        0.63 | 2.00 | [ -3.30,  4.56]
school | 24    | female1     |       -0.07 | 1.57 | [ -3.16,  3.01]
school | 25    | (Intercept) |       -1.14 | 1.77 | [ -4.61,  2.33]
school | 25    | female1     |       -1.36 | 1.54 | [ -4.37,  1.66]
school | 26    | (Intercept) |        2.02 | 1.76 | [ -1.44,  5.48]
school | 26    | female1     |        0.91 | 1.55 | [ -2.13,  3.95]
school | 27    | (Intercept) |       -3.03 | 1.80 | [ -6.55,  0.49]
school | 27    | female1     |       -0.11 | 1.54 | [ -3.13,  2.90]
school | 28    | (Intercept) |        5.63 | 1.71 | [  2.27,  8.99]
school | 28    | female1     |       -1.06 | 1.52 | [ -4.05,  1.92]
school | 29    | (Intercept) |        1.64 | 1.94 | [ -2.15,  5.44]
school | 29    | female1     |       -0.05 | 1.57 | [ -3.12,  3.02]
school | 30    | (Intercept) |       -1.04 | 1.81 | [ -4.58,  2.50]
school | 30    | female1     |       -1.00 | 1.53 | [ -4.01,  2.00]
school | 31    | (Intercept) |       -0.08 | 1.87 | [ -3.74,  3.59]
school | 31    | female1     |       -1.72 | 1.55 | [ -4.75,  1.31]
school | 32    | (Intercept) |        3.51 | 1.42 | [  0.73,  6.29]
school | 32    | female1     |        0.39 | 1.43 | [ -2.42,  3.20]
school | 33    | (Intercept) |       -4.49 | 1.67 | [ -7.76, -1.21]
school | 33    | female1     |       -0.06 | 1.53 | [ -3.07,  2.94]
school | 34    | (Intercept) |       -0.35 | 1.79 | [ -3.86,  3.15]
school | 34    | female1     |       -0.44 | 1.55 | [ -3.47,  2.59]
school | 35    | (Intercept) |        2.34 | 1.96 | [ -1.51,  6.18]
school | 35    | female1     |       -0.34 | 1.59 | [ -3.44,  2.77]
school | 36    | (Intercept) |       -3.50 | 2.13 | [ -7.66,  0.67]
school | 36    | female1     |       -0.47 | 1.60 | [ -3.60,  2.65]
school | 37    | (Intercept) |        2.33 | 1.91 | [ -1.40,  6.07]
school | 37    | female1     |        0.66 | 1.57 | [ -2.41,  3.73]
school | 38    | (Intercept) |       -1.20 | 2.09 | [ -5.29,  2.89]
school | 38    | female1     |       -0.82 | 1.59 | [ -3.94,  2.30]
school | 39    | (Intercept) |        0.33 | 1.78 | [ -3.15,  3.82]
school | 39    | female1     |       -0.89 | 1.53 | [ -3.90,  2.11]
school | 40    | (Intercept) |       -3.43 | 1.98 | [ -7.31,  0.45]
school | 40    | female1     |       -0.31 | 1.57 | [ -3.39,  2.77]
school | 41    | (Intercept) |        1.66 | 1.69 | [ -1.64,  4.97]
school | 41    | female1     |        0.56 | 1.51 | [ -2.40,  3.53]
school | 42    | (Intercept) |        0.29 | 2.10 | [ -3.82,  4.40]
school | 42    | female1     |       -0.02 | 1.59 | [ -3.13,  3.10]
school | 43    | (Intercept) |        0.65 | 1.97 | [ -3.21,  4.51]
school | 43    | female1     |        0.75 | 1.57 | [ -2.33,  3.82]
school | 44    | (Intercept) |        3.17 | 1.89 | [ -0.53,  6.87]
school | 44    | female1     |        0.27 | 1.55 | [ -2.77,  3.31]
school | 45    | (Intercept) |        2.13 | 1.72 | [ -1.25,  5.51]
school | 45    | female1     |    7.48e-03 | 1.55 | [ -3.02,  3.04]
school | 46    | (Intercept) |       -0.66 | 1.84 | [ -4.27,  2.95]
school | 46    | female1     |        0.54 | 1.55 | [ -2.49,  3.57]
school | 47    | (Intercept) |       -4.54 | 1.97 | [ -8.40, -0.69]
school | 47    | female1     |        0.69 | 1.57 | [ -2.38,  3.77]
school | 48    | (Intercept) |       -0.73 | 1.99 | [ -4.64,  3.17]
school | 48    | female1     |        0.39 | 1.60 | [ -2.74,  3.52]
school | 49    | (Intercept) |        0.08 | 1.81 | [ -3.47,  3.64]
school | 49    | female1     |       -0.55 | 1.55 | [ -3.58,  2.48]
school | 50    | (Intercept) |       -4.04 | 1.99 | [ -7.95, -0.13]
school | 50    | female1     |       -0.52 | 1.60 | [ -3.65,  2.61]
school | 51    | (Intercept) |        1.95 | 2.22 | [ -2.39,  6.30]
school | 51    | female1     |        0.35 | 1.62 | [ -2.81,  3.52]
school | 52    | (Intercept) |        3.00 | 1.99 | [ -0.89,  6.90]
school | 52    | female1     |       -0.09 | 1.58 | [ -3.19,  3.01]
school | 53    | (Intercept) |        2.29 | 1.79 | [ -1.21,  5.80]
school | 53    | female1     |       -1.04 | 1.55 | [ -4.07,  2.00]
school | 54    | (Intercept) |        4.85 | 1.88 | [  1.16,  8.54]
school | 54    | female1     |       -0.49 | 1.59 | [ -3.60,  2.63]
school | 55    | (Intercept) |       -6.41 | 1.92 | [-10.18, -2.64]
school | 55    | female1     |       -1.50 | 1.56 | [ -4.56,  1.55]
school | 56    | (Intercept) |       -0.91 | 2.07 | [ -4.97,  3.14]
school | 56    | female1     |        0.06 | 1.59 | [ -3.05,  3.17]
school | 57    | (Intercept) |        0.48 | 1.88 | [ -3.20,  4.17]
school | 57    | female1     |       -0.26 | 1.57 | [ -3.33,  2.82]
school | 58    | (Intercept) |        0.25 | 1.94 | [ -3.54,  4.04]
school | 58    | female1     |        0.45 | 1.57 | [ -2.62,  3.52]
school | 59    | (Intercept) |        0.93 | 1.79 | [ -2.58,  4.43]
school | 59    | female1     |       -0.01 | 1.55 | [ -3.05,  3.02]
school | 60    | (Intercept) |       -3.84 | 1.99 | [ -7.75,  0.07]
school | 60    | female1     |        0.60 | 1.58 | [ -2.49,  3.69]
school | 61    | (Intercept) |        1.97 | 1.73 | [ -1.42,  5.36]
school | 61    | female1     |       -1.10 | 1.53 | [ -4.10,  1.90]
school | 62    | (Intercept) |       -8.75 | 2.04 | [-12.76, -4.75]
school | 62    | female1     |        0.59 | 1.58 | [ -2.51,  3.69]
school | 63    | (Intercept) |        1.06 | 1.79 | [ -2.45,  4.56]
school | 63    | female1     |        0.17 | 1.55 | [ -2.86,  3.20]
school | 64    | (Intercept) |        0.87 | 1.88 | [ -2.82,  4.56]
school | 64    | female1     |       -0.13 | 1.56 | [ -3.18,  2.93]
school | 65    | (Intercept) |        3.79 | 1.93 | [  0.01,  7.58]
school | 65    | female1     |       -0.21 | 1.58 | [ -3.30,  2.88]
school | 66    | (Intercept) |       -1.46 | 1.81 | [ -5.00,  2.09]
school | 66    | female1     |       -0.22 | 1.56 | [ -3.27,  2.83]
school | 67    | (Intercept) |        1.89 | 1.99 | [ -2.01,  5.79]
school | 67    | female1     |       -0.21 | 1.58 | [ -3.31,  2.89]
school | 68    | (Intercept) |       -0.52 | 2.03 | [ -4.49,  3.46]
school | 68    | female1     |        0.70 | 1.58 | [ -2.39,  3.80]
school | 69    | (Intercept) |       -3.47 | 2.00 | [ -7.40,  0.45]
school | 69    | female1     |       -0.31 | 1.57 | [ -3.39,  2.78]
school | 70    | (Intercept) |        4.77 | 2.00 | [  0.85,  8.70]
school | 70    | female1     |        0.13 | 1.57 | [ -2.96,  3.21]
school | 71    | (Intercept) |       -1.54 | 1.94 | [ -5.34,  2.25]
school | 71    | female1     |       -0.63 | 1.57 | [ -3.71,  2.44]
school | 72    | (Intercept) |       -5.51 | 1.82 | [ -9.08, -1.93]
school | 72    | female1     |       -0.02 | 1.54 | [ -3.03,  3.00]
school | 73    | (Intercept) |       -0.79 | 1.74 | [ -4.20,  2.63]
school | 73    | female1     |        0.57 | 1.56 | [ -2.48,  3.62]
school | 74    | (Intercept) |        5.19 | 1.89 | [  1.49,  8.90]
school | 74    | female1     |        0.06 | 1.55 | [ -2.99,  3.10]
school | 75    | (Intercept) |       -0.47 | 1.81 | [ -4.01,  3.07]
school | 75    | female1     |        0.93 | 1.57 | [ -2.15,  4.01]
school | 76    | (Intercept) |       -0.53 | 1.91 | [ -4.26,  3.21]
school | 76    | female1     |        1.26 | 1.57 | [ -1.81,  4.33]
school | 77    | (Intercept) |        4.37 | 1.57 | [  1.30,  7.44]
school | 77    | female1     |        0.11 | 1.50 | [ -2.82,  3.04]
school | 78    | (Intercept) |       -1.12 | 2.09 | [ -5.21,  2.97]
school | 78    | female1     |       -0.06 | 1.59 | [ -3.19,  3.06]
school | 79    | (Intercept) |       -2.62 | 2.05 | [ -6.64,  1.40]
school | 79    | female1     |       -0.23 | 1.59 | [ -3.35,  2.88]
school | 80    | (Intercept) |        4.35 | 1.74 | [  0.94,  7.77]
school | 80    | female1     |        0.14 | 1.56 | [ -2.92,  3.19]
school | 81    | (Intercept) |       -2.51 | 2.06 | [ -6.54,  1.52]
school | 81    | female1     |       -0.13 | 1.58 | [ -3.23,  2.98]
school | 82    | (Intercept) |        1.69 | 1.75 | [ -1.74,  5.13]
school | 82    | female1     |       -0.78 | 1.53 | [ -3.78,  2.22]
school | 83    | (Intercept) |        1.81 | 1.55 | [ -1.23,  4.85]
school | 83    | female1     |        1.40 | 1.47 | [ -1.47,  4.28]
school | 84    | (Intercept) |        3.11 | 2.05 | [ -0.91,  7.14]
school | 84    | female1     |        0.12 | 1.59 | [ -3.00,  3.23]
school | 85    | (Intercept) |        0.33 | 1.97 | [ -3.53,  4.18]
school | 85    | female1     |        0.78 | 1.57 | [ -2.29,  3.86]
school | 86    | (Intercept) |       -1.39 | 1.79 | [ -4.89,  2.10]
school | 86    | female1     |        0.12 | 1.56 | [ -2.94,  3.17]
school | 87    | (Intercept) |        2.16 | 1.96 | [ -1.68,  6.01]
school | 87    | female1     |       -0.24 | 1.57 | [ -3.32,  2.85]
school | 88    | (Intercept) |        1.96 | 1.73 | [ -1.43,  5.35]
school | 88    | female1     |       -0.57 | 1.53 | [ -3.58,  2.43]
school | 89    | (Intercept) |        0.42 | 2.17 | [ -3.83,  4.68]
school | 89    | female1     |        0.04 | 1.60 | [ -3.10,  3.19]
school | 90    | (Intercept) |        3.59 | 1.78 | [  0.10,  7.07]
school | 90    | female1     |       -0.41 | 1.53 | [ -3.40,  2.58]
school | 91    | (Intercept) |        0.19 | 1.83 | [ -3.40,  3.79]
school | 91    | female1     |       -0.21 | 1.55 | [ -3.25,  2.83]
school | 92    | (Intercept) |        2.50 | 1.94 | [ -1.29,  6.30]
school | 92    | female1     |        0.06 | 1.57 | [ -3.02,  3.13]
school | 93    | (Intercept) |       -2.46 | 1.57 | [ -5.53,  0.61]
school | 93    | female1     |        0.71 | 1.48 | [ -2.19,  3.61]
school | 94    | (Intercept) |        1.94 | 2.06 | [ -2.09,  5.97]
school | 94    | female1     |        0.16 | 1.58 | [ -2.95,  3.26]
school | 95    | (Intercept) |       -0.32 | 1.75 | [ -3.75,  3.12]
school | 95    | female1     |        0.23 | 1.53 | [ -2.77,  3.23]
school | 96    | (Intercept) |        1.70 | 1.99 | [ -2.21,  5.60]
school | 96    | female1     |    7.79e-03 | 1.60 | [ -3.12,  3.14]
school | 97    | (Intercept) |        3.30 | 1.75 | [ -0.14,  6.73]
school | 97    | female1     |        0.07 | 1.53 | [ -2.93,  3.07]
school | 98    | (Intercept) |        0.21 | 1.88 | [ -3.48,  3.90]
school | 98    | female1     |        0.17 | 1.59 | [ -2.94,  3.29]
school | 99    | (Intercept) |       -0.94 | 2.13 | [ -5.11,  3.22]
school | 99    | female1     |        0.38 | 1.60 | [ -2.74,  3.51]
school | 100   | (Intercept) |        0.99 | 1.72 | [ -2.39,  4.36]
school | 100   | female1     |       -0.84 | 1.55 | [ -3.87,  2.19]
school | 101   | (Intercept) |       -0.04 | 1.82 | [ -3.62,  3.53]
school | 101   | female1     |       -0.08 | 1.54 | [ -3.10,  2.93]
school | 102   | (Intercept) |       -0.13 | 1.98 | [ -4.01,  3.76]
school | 102   | female1     |        0.35 | 1.57 | [ -2.73,  3.43]
school | 103   | (Intercept) |        7.01 | 1.93 | [  3.21, 10.80]
school | 103   | female1     |       -0.38 | 1.59 | [ -3.50,  2.74]
school | 104   | (Intercept) |        0.21 | 2.05 | [ -3.81,  4.23]
school | 104   | female1     |        0.33 | 1.59 | [ -2.78,  3.45]
school | 105   | (Intercept) |        1.34 | 1.86 | [ -2.29,  4.98]
school | 105   | female1     |        0.14 | 1.56 | [ -2.92,  3.20]
school | 106   | (Intercept) |       -4.74 | 1.74 | [ -8.14, -1.33]
school | 106   | female1     |        0.72 | 1.52 | [ -2.27,  3.71]
school | 107   | (Intercept) |        5.41 | 1.85 | [  1.77,  9.04]
school | 107   | female1     |        0.66 | 1.58 | [ -2.43,  3.75]
school | 108   | (Intercept) |        1.00 | 1.81 | [ -2.55,  4.54]
school | 108   | female1     |       -1.12 | 1.56 | [ -4.16,  1.93]
school | 109   | (Intercept) |        4.08 | 1.91 | [  0.34,  7.81]
school | 109   | female1     |        0.07 | 1.57 | [ -3.00,  3.14]
school | 110   | (Intercept) |       -1.59 | 1.88 | [ -5.28,  2.09]
school | 110   | female1     |       -0.29 | 1.57 | [ -3.37,  2.78]
school | 111   | (Intercept) |       -1.47 | 1.86 | [ -5.12,  2.17]
school | 111   | female1     |        0.42 | 1.55 | [ -2.62,  3.47]
school | 112   | (Intercept) |        0.50 | 1.74 | [ -2.91,  3.91]
school | 112   | female1     |        1.57 | 1.52 | [ -1.42,  4.56]
school | 113   | (Intercept) |        1.05 | 1.66 | [ -2.21,  4.31]
school | 113   | female1     |        0.26 | 1.50 | [ -2.68,  3.20]
school | 114   | (Intercept) |       -1.61 | 1.86 | [ -5.26,  2.03]
school | 114   | female1     |       -0.52 | 1.60 | [ -3.66,  2.61]
school | 115   | (Intercept) |        3.18 | 2.09 | [ -0.91,  7.27]
school | 115   | female1     |       -0.12 | 1.59 | [ -3.23,  3.00]
school | 116   | (Intercept) |       -8.40 | 1.82 | [-11.98, -4.83]
school | 116   | female1     |       -0.10 | 1.54 | [ -3.12,  2.92]
school | 117   | (Intercept) |       -2.48 | 2.05 | [ -6.50,  1.54]
school | 117   | female1     |        0.54 | 1.59 | [ -2.58,  3.65]
school | 118   | (Intercept) |       -0.88 | 1.99 | [ -4.78,  3.02]
school | 118   | female1     |        0.84 | 1.58 | [ -2.26,  3.94]
school | 119   | (Intercept) |        1.88 | 2.05 | [ -2.14,  5.90]
school | 119   | female1     |        0.33 | 1.59 | [ -2.79,  3.44]
school | 120   | (Intercept) |        3.60 | 1.83 | [  0.01,  7.19]
school | 120   | female1     |        0.40 | 1.55 | [ -2.65,  3.44]
school | 121   | (Intercept) |        0.68 | 1.91 | [ -3.07,  4.43]
school | 121   | female1     |       -0.46 | 1.56 | [ -3.52,  2.60]
school | 122   | (Intercept) |       -3.28 | 2.06 | [ -7.31,  0.75]
school | 122   | female1     |       -0.25 | 1.58 | [ -3.35,  2.86]
school | 123   | (Intercept) |        0.09 | 1.84 | [ -3.52,  3.70]
school | 123   | female1     |        0.70 | 1.55 | [ -2.33,  3.73]
school | 124   | (Intercept) |        1.42 | 2.06 | [ -2.61,  5.45]
school | 124   | female1     |        0.72 | 1.58 | [ -2.39,  3.82]
school | 125   | (Intercept) |       -1.62 | 1.86 | [ -5.27,  2.03]
school | 125   | female1     |        0.82 | 1.55 | [ -2.22,  3.86]
school | 126   | (Intercept) |       -0.27 | 1.65 | [ -3.50,  2.97]
school | 126   | female1     |       -0.41 | 1.50 | [ -3.36,  2.54]
school | 127   | (Intercept) |        2.90 | 2.02 | [ -1.06,  6.86]
school | 127   | female1     |    7.65e-03 | 1.59 | [ -3.11,  3.13]
school | 128   | (Intercept) |        0.39 | 1.77 | [ -3.08,  3.86]
school | 128   | female1     |        0.11 | 1.54 | [ -2.91,  3.12]
school | 129   | (Intercept) |       -2.81 | 1.79 | [ -6.32,  0.69]
school | 129   | female1     |        0.52 | 1.55 | [ -2.51,  3.55]
school | 130   | (Intercept) |       -1.18 | 2.02 | [ -5.14,  2.78]
school | 130   | female1     |       -0.14 | 1.58 | [ -3.23,  2.96]
school | 131   | (Intercept) |       -1.13 | 1.79 | [ -4.63,  2.37]
school | 131   | female1     |        0.33 | 1.58 | [ -2.77,  3.43]
school | 132   | (Intercept) |       -3.06 | 2.03 | [ -7.04,  0.92]
school | 132   | female1     |       -0.30 | 1.58 | [ -3.39,  2.80]
school | 133   | (Intercept) |        4.17 | 1.73 | [  0.78,  7.56]
school | 133   | female1     |       -0.35 | 1.59 | [ -3.47,  2.77]
school | 134   | (Intercept) |        0.90 | 1.96 | [ -2.94,  4.74]
school | 134   | female1     |       -0.61 | 1.57 | [ -3.68,  2.46]
school | 135   | (Intercept) |       -4.98 | 1.69 | [ -8.28, -1.67]
school | 135   | female1     |        0.86 | 1.54 | [ -2.16,  3.88]
school | 136   | (Intercept) |        0.15 | 1.99 | [ -3.76,  4.06]
school | 136   | female1     |       -1.04 | 1.58 | [ -4.13,  2.05]
school | 137   | (Intercept) |        0.54 | 1.81 | [ -3.01,  4.10]
school | 137   | female1     |       -0.51 | 1.55 | [ -3.54,  2.52]
school | 138   | (Intercept) |       -0.03 | 2.17 | [ -4.29,  4.22]
school | 138   | female1     |       -0.23 | 1.60 | [ -3.37,  2.91]
school | 139   | (Intercept) |       -1.89 | 2.00 | [ -5.82,  2.04]
school | 139   | female1     |        1.37 | 1.57 | [ -1.72,  4.45]
school | 140   | (Intercept) |        6.10 | 1.79 | [  2.58,  9.61]
school | 140   | female1     |       -0.66 | 1.53 | [ -3.66,  2.33]
school | 141   | (Intercept) |        1.68 | 2.06 | [ -2.35,  5.71]
school | 141   | female1     |       -0.20 | 1.58 | [ -3.30,  2.91]
school | 142   | (Intercept) |        2.70 | 1.77 | [ -0.77,  6.17]
school | 142   | female1     |       -0.16 | 1.54 | [ -3.18,  2.85]
school | 143   | (Intercept) |        1.85 | 2.03 | [ -2.13,  5.82]
school | 143   | female1     |        0.86 | 1.58 | [ -2.24,  3.95]
school | 144   | (Intercept) |       -1.53 | 1.88 | [ -5.21,  2.15]
school | 144   | female1     |        0.38 | 1.57 | [ -2.70,  3.46]
school | 145   | (Intercept) |        0.04 | 1.82 | [ -3.54,  3.61]
school | 145   | female1     |        0.04 | 1.54 | [ -2.98,  3.06]
school | 146   | (Intercept) |        0.23 | 1.69 | [ -3.09,  3.54]
school | 146   | female1     |        0.85 | 1.53 | [ -2.15,  3.84]
school | 147   | (Intercept) |        0.23 | 1.82 | [ -3.34,  3.81]
school | 147   | female1     |        0.05 | 1.54 | [ -2.97,  3.07]
school | 148   | (Intercept) |       -3.66 | 1.85 | [ -7.30, -0.03]
school | 148   | female1     |       -0.51 | 1.58 | [ -3.60,  2.58]
school | 149   | (Intercept) |       -0.99 | 1.84 | [ -4.60,  2.62]
school | 149   | female1     |        1.30 | 1.55 | [ -1.73,  4.33]
school | 150   | (Intercept) |        3.17 | 1.62 | [ -0.01,  6.35]
school | 150   | female1     |       -0.39 | 1.65 | [ -3.63,  2.84]
school | 151   | (Intercept) |       -2.92 | 1.89 | [ -6.62,  0.79]
school | 151   | female1     |        0.35 | 1.55 | [ -2.69,  3.39]
school | 152   | (Intercept) |        3.21 | 1.98 | [ -0.68,  7.09]
school | 152   | female1     |        0.58 | 1.57 | [ -2.50,  3.65]
school | 153   | (Intercept) |        6.72 | 1.91 | [  2.99, 10.46]
school | 153   | female1     |       -0.08 | 1.57 | [ -3.15,  2.99]
school | 154   | (Intercept) |       -1.66 | 1.97 | [ -5.52,  2.20]
school | 154   | female1     |        0.40 | 1.61 | [ -2.75,  3.55]
school | 155   | (Intercept) |       -3.58 | 1.84 | [ -7.19,  0.04]
school | 155   | female1     |        1.38 | 1.55 | [ -1.65,  4.41]
school | 156   | (Intercept) |        3.71 | 1.88 | [  0.02,  7.40]
school | 156   | female1     |       -0.25 | 1.59 | [ -3.36,  2.86]
school | 157   | (Intercept) |        1.41 | 1.85 | [ -2.22,  5.05]
school | 157   | female1     |        0.25 | 1.58 | [ -2.84,  3.34]
school | 158   | (Intercept) |        1.82 | 1.81 | [ -1.73,  5.38]
school | 158   | female1     |       -0.68 | 1.55 | [ -3.71,  2.35]
school | 159   | (Intercept) |       -0.11 | 1.83 | [ -3.70,  3.47]
school | 159   | female1     |        0.45 | 1.57 | [ -2.61,  3.52]
school | 160   | (Intercept) |        1.57 | 2.17 | [ -2.68,  5.83]
school | 160   | female1     |        0.49 | 1.60 | [ -2.65,  3.64]
school | 161   | (Intercept) |       -2.65 | 1.93 | [ -6.44,  1.13]
school | 161   | female1     |        0.69 | 1.58 | [ -2.40,  3.78]
school | 162   | (Intercept) |       -0.06 | 2.02 | [ -4.02,  3.90]
school | 162   | female1     |       -0.77 | 1.58 | [ -3.87,  2.33]
school | 163   | (Intercept) |       -2.49 | 1.93 | [ -6.28,  1.29]
school | 163   | female1     |        0.15 | 1.58 | [ -2.94,  3.24]
school | 164   | (Intercept) |    7.95e-03 | 1.79 | [ -3.50,  3.51]
school | 164   | female1     |        0.04 | 1.55 | [ -2.99,  3.07]
school | 165   | (Intercept) |       -1.00 | 2.13 | [ -5.17,  3.16]
school | 165   | female1     |       -0.65 | 1.60 | [ -3.78,  2.47]
school | 166   | (Intercept) |       -3.60 | 1.97 | [ -7.45,  0.26]
school | 166   | female1     |        0.04 | 1.61 | [ -3.11,  3.19]
school | 167   | (Intercept) |        0.68 | 2.13 | [ -3.50,  4.86]
school | 167   | female1     |        0.12 | 1.61 | [ -3.03,  3.27]
school | 168   | (Intercept) |        1.55 | 1.81 | [ -2.00,  5.09]
school | 168   | female1     |        0.11 | 1.56 | [ -2.94,  3.16]
school | 169   | (Intercept) |       -0.42 | 2.05 | [ -4.44,  3.60]
school | 169   | female1     |       -0.59 | 1.59 | [ -3.71,  2.52]
school | 170   | (Intercept) |        0.28 | 1.93 | [ -3.52,  4.07]
school | 170   | female1     |       -0.49 | 1.59 | [ -3.61,  2.63]
school | 171   | (Intercept) |       -0.69 | 1.50 | [ -3.64,  2.26]
school | 171   | female1     |        0.03 | 1.44 | [ -2.79,  2.85]
school | 172   | (Intercept) |        0.33 | 2.02 | [ -3.63,  4.30]
school | 172   | female1     |       -0.30 | 1.58 | [ -3.40,  2.81]
school | 173   | (Intercept) |        4.22 | 1.83 | [  0.63,  7.81]
school | 173   | female1     |       -1.15 | 1.58 | [ -4.25,  1.96]
school | 174   | (Intercept) |       -1.85 | 2.02 | [ -5.81,  2.11]
school | 174   | female1     |        0.28 | 1.59 | [ -2.84,  3.40]
school | 175   | (Intercept) |        1.17 | 2.02 | [ -2.79,  5.14]
school | 175   | female1     |       -0.18 | 1.58 | [ -3.28,  2.92]
school | 176   | (Intercept) |        1.49 | 1.91 | [ -2.26,  5.24]
school | 176   | female1     |       -0.34 | 1.56 | [ -3.40,  2.72]
school | 177   | (Intercept) |       -0.35 | 2.00 | [ -4.28,  3.58]
school | 177   | female1     |        0.14 | 1.57 | [ -2.95,  3.22]
school | 178   | (Intercept) |        4.98 | 1.97 | [  1.12,  8.84]
school | 178   | female1     |       -0.62 | 1.66 | [ -3.86,  2.63]
school | 179   | (Intercept) |       -1.90 | 1.79 | [ -5.40,  1.60]
school | 179   | female1     |       -0.83 | 1.56 | [ -3.89,  2.23]
school | 180   | (Intercept) |        1.03 | 1.87 | [ -2.64,  4.70]
school | 180   | female1     |       -0.06 | 1.55 | [ -3.10,  2.97]
school | 181   | (Intercept) |        0.33 | 1.69 | [ -2.98,  3.63]
school | 181   | female1     |        0.26 | 1.54 | [ -2.76,  3.28]
school | 182   | (Intercept) |        3.01 | 1.86 | [ -0.63,  6.65]
school | 182   | female1     |       -0.08 | 1.56 | [ -3.14,  2.98]
school | 183   | (Intercept) |        1.25 | 2.13 | [ -2.92,  5.42]
school | 183   | female1     |        0.11 | 1.60 | [ -3.02,  3.24]
school | 184   | (Intercept) |       -0.87 | 1.81 | [ -4.41,  2.68]
school | 184   | female1     |        0.85 | 1.56 | [ -2.20,  3.90]
school | 185   | (Intercept) |       -5.61 | 1.99 | [ -9.51, -1.71]
school | 185   | female1     |        0.20 | 1.58 | [ -2.90,  3.30]
school | 186   | (Intercept) |       -5.05 | 1.93 | [ -8.83, -1.26]
school | 186   | female1     |       -0.25 | 1.58 | [ -3.34,  2.84]
school | 187   | (Intercept) |       -0.84 | 1.98 | [ -4.72,  3.05]
school | 187   | female1     |       -0.63 | 1.57 | [ -3.70,  2.45]
school | 188   | (Intercept) |        2.74 | 2.02 | [ -1.22,  6.70]
school | 188   | female1     |        0.86 | 1.58 | [ -2.24,  3.96]
school | 189   | (Intercept) |       -3.15 | 1.69 | [ -6.46,  0.15]
school | 189   | female1     |       -1.36 | 1.50 | [ -4.30,  1.59]
school | 190   | (Intercept) |       -1.72 | 2.03 | [ -5.70,  2.25]
school | 190   | female1     |        1.52 | 1.58 | [ -1.58,  4.62]
school | 191   | (Intercept) |       -0.80 | 2.03 | [ -4.78,  3.18]
school | 191   | female1     |       -0.15 | 1.58 | [ -3.25,  2.94]
school | 192   | (Intercept) |        3.48 | 1.59 | [  0.36,  6.60]
school | 192   | female1     |       -0.43 | 1.65 | [ -3.66,  2.81]
school | 193   | (Intercept) |       -2.59 | 1.99 | [ -6.49,  1.30]
school | 193   | female1     |       -0.50 | 1.58 | [ -3.60,  2.60]
school | 194   | (Intercept) |        2.81 | 2.09 | [ -1.28,  6.90]
school | 194   | female1     |        0.21 | 1.59 | [ -2.91,  3.33]
school | 195   | (Intercept) |        0.30 | 1.76 | [ -3.16,  3.76]
school | 195   | female1     |        0.73 | 1.53 | [ -2.26,  3.72]
school | 196   | (Intercept) |       -1.50 | 1.99 | [ -5.40,  2.41]
school | 196   | female1     |        0.85 | 1.58 | [ -2.24,  3.93]
school | 197   | (Intercept) |       -8.27 | 1.91 | [-12.00, -4.53]
school | 197   | female1     |        0.41 | 1.55 | [ -2.64,  3.46]
school | 198   | (Intercept) |       -1.60 | 1.76 | [ -5.06,  1.86]
school | 198   | female1     |       -0.07 | 1.53 | [ -3.07,  2.92]
school | 199   | (Intercept) |        0.26 | 1.84 | [ -3.35,  3.87]
school | 199   | female1     |       -0.92 | 1.55 | [ -3.95,  2.11]
school | 200   | (Intercept) |       -3.45 | 1.76 | [ -6.91,  0.00]
school | 200   | female1     |        1.93 | 1.57 | [ -1.14,  5.01]
school | 201   | (Intercept) |        0.30 | 1.93 | [ -3.49,  4.08]
school | 201   | female1     |        1.32 | 1.58 | [ -1.76,  4.41]
school | 202   | (Intercept) |        2.88 | 1.59 | [ -0.23,  5.98]
school | 202   | female1     |        0.57 | 1.47 | [ -2.31,  3.44]
school | 203   | (Intercept) |        0.12 | 1.99 | [ -3.78,  4.02]
school | 203   | female1     |       -0.31 | 1.58 | [ -3.41,  2.79]
school | 204   | (Intercept) |        0.85 | 2.06 | [ -3.20,  4.89]
school | 204   | female1     |       -0.01 | 1.59 | [ -3.13,  3.11]
school | 205   | (Intercept) |       -1.83 | 1.62 | [ -5.00,  1.34]
school | 205   | female1     |       -1.17 | 1.50 | [ -4.11,  1.76]
school | 206   | (Intercept) |       -1.56 | 2.00 | [ -5.49,  2.37]
school | 206   | female1     |        0.02 | 1.57 | [ -3.06,  3.11]
school | 207   | (Intercept) |       -0.33 | 1.83 | [ -3.93,  3.26]
school | 207   | female1     |        0.17 | 1.58 | [ -2.94,  3.28]
school | 208   | (Intercept) |       -2.76 | 1.69 | [ -6.07,  0.56]
school | 208   | female1     |       -0.21 | 1.53 | [ -3.20,  2.78]
school | 209   | (Intercept) |        2.80 | 1.82 | [ -0.78,  6.37]
school | 209   | female1     |        0.49 | 1.54 | [ -2.53,  3.51]
school | 210   | (Intercept) |       -4.57 | 1.67 | [ -7.84, -1.30]
school | 210   | female1     |        1.31 | 1.53 | [ -1.69,  4.31]
school | 211   | (Intercept) |       -2.41 | 1.94 | [ -6.21,  1.38]
school | 211   | female1     |        1.71 | 1.57 | [ -1.37,  4.78]
school | 212   | (Intercept) |       -3.65 | 1.81 | [ -7.19, -0.10]
school | 212   | female1     |        0.05 | 1.56 | [ -2.99,  3.10]
school | 213   | (Intercept) |        2.82 | 1.87 | [ -0.85,  6.49]
school | 213   | female1     |       -0.26 | 1.55 | [ -3.29,  2.78]
school | 214   | (Intercept) |       -3.99 | 1.96 | [ -7.83, -0.15]
school | 214   | female1     |        0.67 | 1.57 | [ -2.42,  3.75]
school | 215   | (Intercept) |        0.72 | 1.81 | [ -2.83,  4.26]
school | 215   | female1     |       -0.47 | 1.56 | [ -3.52,  2.58]
school | 216   | (Intercept) |        1.28 | 1.99 | [ -2.62,  5.18]
school | 216   | female1     |        0.49 | 1.58 | [ -2.61,  3.59]
school | 217   | (Intercept) |        2.87 | 1.99 | [ -1.04,  6.77]
school | 217   | female1     |        0.34 | 1.58 | [ -2.74,  3.43]
school | 218   | (Intercept) |       -0.38 | 1.92 | [ -4.15,  3.40]
school | 218   | female1     |       -0.84 | 1.56 | [ -3.90,  2.22]
school | 219   | (Intercept) |        0.23 | 2.09 | [ -3.86,  4.32]
school | 219   | female1     |       -1.07 | 1.59 | [ -4.19,  2.05]
school | 220   | (Intercept) |       -5.54 | 2.09 | [ -9.63, -1.45]
school | 220   | female1     |        0.19 | 1.59 | [ -2.93,  3.31]
school | 221   | (Intercept) |        3.28 | 2.03 | [ -0.70,  7.25]
school | 221   | female1     |        0.04 | 1.58 | [ -3.05,  3.14]
school | 222   | (Intercept) |       -2.80 | 1.84 | [ -6.41,  0.80]
school | 222   | female1     |       -0.66 | 1.54 | [ -3.68,  2.36]
school | 223   | (Intercept) |       -5.25 | 1.97 | [ -9.11, -1.40]
school | 223   | female1     |       -0.80 | 1.57 | [ -3.88,  2.27]
school | 224   | (Intercept) |        1.90 | 2.00 | [ -2.02,  5.82]
school | 224   | female1     |        0.15 | 1.58 | [ -2.94,  3.24]
school | 225   | (Intercept) |       -3.51 | 1.79 | [ -7.01,  0.00]
school | 225   | female1     |       -0.48 | 1.55 | [ -3.51,  2.55]
school | 226   | (Intercept) |        3.32 | 2.02 | [ -0.64,  7.28]
school | 226   | female1     |        0.35 | 1.58 | [ -2.75,  3.45]
school | 227   | (Intercept) |       -1.00 | 2.04 | [ -5.01,  3.00]
school | 227   | female1     |       -0.51 | 1.58 | [ -3.61,  2.60]
school | 228   | (Intercept) |        0.52 | 1.99 | [ -3.39,  4.43]
school | 228   | female1     |       -0.60 | 1.58 | [ -3.69,  2.49]
school | 229   | (Intercept) |        2.54 | 1.74 | [ -0.88,  5.96]
school | 229   | female1     |       -0.15 | 1.56 | [ -3.20,  2.90]
school | 230   | (Intercept) |        1.75 | 1.99 | [ -2.16,  5.65]
school | 230   | female1     |       -0.57 | 1.60 | [ -3.70,  2.56]
school | 231   | (Intercept) |       -2.33 | 1.99 | [ -6.24,  1.58]
school | 231   | female1     |       -0.09 | 1.58 | [ -3.18,  3.00]
school | 232   | (Intercept) |       -0.51 | 1.86 | [ -4.15,  3.12]
school | 232   | female1     |        0.78 | 1.56 | [ -2.28,  3.84]
school | 233   | (Intercept) |        0.42 | 1.99 | [ -3.49,  4.32]
school | 233   | female1     |        0.23 | 1.60 | [ -2.90,  3.35]
school | 234   | (Intercept) |        0.80 | 1.94 | [ -3.00,  4.59]
school | 234   | female1     |        0.38 | 1.57 | [ -2.69,  3.45]
school | 235   | (Intercept) |        2.98 | 1.78 | [ -0.51,  6.47]
school | 235   | female1     |        0.08 | 1.53 | [ -2.92,  3.08]
school | 236   | (Intercept) |       -0.97 | 1.83 | [ -4.57,  2.62]
school | 236   | female1     |        0.78 | 1.55 | [ -2.26,  3.82]
school | 237   | (Intercept) |       -1.76 | 1.82 | [ -5.33,  1.82]
school | 237   | female1     |       -0.67 | 1.54 | [ -3.68,  2.34]
school | 238   | (Intercept) |        0.33 | 2.09 | [ -3.76,  4.42]
school | 238   | female1     |        0.46 | 1.59 | [ -2.65,  3.58]
school | 239   | (Intercept) |       -2.84 | 1.65 | [ -6.07,  0.40]
school | 239   | female1     |       -0.09 | 1.50 | [ -3.03,  2.86]
school | 240   | (Intercept) |        3.30 | 2.02 | [ -0.66,  7.26]
school | 240   | female1     |       -0.31 | 1.58 | [ -3.41,  2.79]
school | 241   | (Intercept) |        4.56 | 2.10 | [  0.45,  8.67]
school | 241   | female1     |       -0.20 | 1.59 | [ -3.32,  2.92]
school | 242   | (Intercept) |       -0.59 | 2.00 | [ -4.52,  3.34]
school | 242   | female1     |       -0.12 | 1.57 | [ -3.21,  2.97]
school | 243   | (Intercept) |       -5.00 | 2.09 | [ -9.09, -0.91]
school | 243   | female1     |       -0.17 | 1.59 | [ -3.29,  2.95]
school | 244   | (Intercept) |        0.88 | 2.02 | [ -3.08,  4.84]
school | 244   | female1     |        1.11 | 1.58 | [ -1.99,  4.21]
school | 245   | (Intercept) |        2.09 | 1.76 | [ -1.37,  5.55]
school | 245   | female1     |        0.10 | 1.55 | [ -2.94,  3.14]
school | 246   | (Intercept) |       -6.49 | 2.02 | [-10.45, -2.53]
school | 246   | female1     |       -0.21 | 1.59 | [ -3.33,  2.91]
school | 247   | (Intercept) |       -1.76 | 1.68 | [ -5.04,  1.53]
school | 247   | female1     |        0.58 | 1.52 | [ -2.40,  3.56]
school | 248   | (Intercept) |        1.33 | 1.99 | [ -2.56,  5.23]
school | 248   | female1     |        0.17 | 1.58 | [ -2.93,  3.27]
school | 249   | (Intercept) |       -4.29 | 1.99 | [ -8.19, -0.39]
school | 249   | female1     |        0.74 | 1.58 | [ -2.37,  3.84]
school | 250   | (Intercept) |       -3.27 | 1.42 | [ -6.05, -0.50]
school | 250   | female1     |       -1.26 | 1.45 | [ -4.10,  1.58]
school | 251   | (Intercept) |        2.79 | 1.81 | [ -0.75,  6.33]
school | 251   | female1     |       -1.02 | 1.57 | [ -4.10,  2.06]
school | 252   | (Intercept) |        4.49 | 1.89 | [  0.78,  8.19]
school | 252   | female1     |        0.15 | 1.55 | [ -2.90,  3.20]
school | 253   | (Intercept) |       -2.49 | 1.86 | [ -6.14,  1.16]
school | 253   | female1     |        1.42 | 1.55 | [ -1.62,  4.46]
school | 254   | (Intercept) |       -2.45 | 1.75 | [ -5.88,  0.97]
school | 254   | female1     |        0.21 | 1.58 | [ -2.89,  3.30]
school | 255   | (Intercept) |        0.47 | 2.06 | [ -3.56,  4.50]
school | 255   | female1     |        0.20 | 1.58 | [ -2.91,  3.31]
school | 256   | (Intercept) |        0.48 | 1.95 | [ -3.34,  4.30]
school | 256   | female1     |        0.26 | 1.62 | [ -2.91,  3.43]
school | 257   | (Intercept) |        5.61 | 2.20 | [  1.30,  9.92]
school | 257   | female1     |        0.84 | 1.65 | [ -2.38,  4.07]
school | 258   | (Intercept) |        0.53 | 1.99 | [ -3.37,  4.44]
school | 258   | female1     |        0.64 | 1.58 | [ -2.45,  3.72]
school | 259   | (Intercept) |       -3.43 | 1.91 | [ -7.16,  0.31]
school | 259   | female1     |       -1.25 | 1.58 | [ -4.35,  1.85]
school | 260   | (Intercept) |       -0.98 | 1.99 | [ -4.87,  2.92]
school | 260   | female1     |       -0.78 | 1.58 | [ -3.89,  2.32]
school | 261   | (Intercept) |       -0.91 | 1.99 | [ -4.82,  2.99]
school | 261   | female1     |       -0.38 | 1.58 | [ -3.47,  2.71]
school | 262   | (Intercept) |        3.98 | 1.80 | [  0.46,  7.50]
school | 262   | female1     |       -0.77 | 1.54 | [ -3.78,  2.25]
school | 263   | (Intercept) |       -0.53 | 1.86 | [ -4.18,  3.11]
school | 263   | female1     |       -0.57 | 1.55 | [ -3.62,  2.47]
school | 264   | (Intercept) |       -3.63 | 1.93 | [ -7.42,  0.15]
school | 264   | female1     |       -0.99 | 1.58 | [ -4.08,  2.10]
school | 265   | (Intercept) |       -1.95 | 1.91 | [ -5.70,  1.79]
school | 265   | female1     |       -0.74 | 1.56 | [ -3.80,  2.32]
school | 266   | (Intercept) |       -2.69 | 1.83 | [ -6.27,  0.90]
school | 266   | female1     |       -0.86 | 1.57 | [ -3.92,  2.21]
school | 267   | (Intercept) |        0.64 | 2.05 | [ -3.38,  4.67]
school | 267   | female1     |       -0.01 | 1.59 | [ -3.13,  3.10]
school | 268   | (Intercept) |        2.27 | 2.05 | [ -1.75,  6.29]
school | 268   | female1     |       -0.25 | 1.59 | [ -3.36,  2.87]
school | 269   | (Intercept) |       -2.46 | 1.99 | [ -6.37,  1.45]
school | 269   | female1     |        0.82 | 1.58 | [ -2.27,  3.90]
school | 270   | (Intercept) |        2.05 | 1.61 | [ -1.10,  5.20]
school | 270   | female1     |        0.13 | 1.49 | [ -2.80,  3.05]
school | 271   | (Intercept) |        2.50 | 1.83 | [ -1.10,  6.09]
school | 271   | female1     |       -0.73 | 1.55 | [ -3.78,  2.31]
school | 272   | (Intercept) |       -1.06 | 2.17 | [ -5.32,  3.19]
school | 272   | female1     |        0.52 | 1.60 | [ -2.62,  3.66]
school | 273   | (Intercept) |       -0.37 | 1.99 | [ -4.28,  3.54]
school | 273   | female1     |       -0.03 | 1.60 | [ -3.16,  3.10]
school | 274   | (Intercept) |        3.83 | 1.82 | [  0.25,  7.40]
school | 274   | female1     |       -0.96 | 1.54 | [ -3.98,  2.06]
school | 275   | (Intercept) |        4.57 | 1.79 | [  1.07,  8.08]
school | 275   | female1     |       -0.04 | 1.55 | [ -3.07,  2.99]
school | 276   | (Intercept) |       -1.31 | 1.97 | [ -5.17,  2.55]
school | 276   | female1     |    5.84e-03 | 1.61 | [ -3.14,  3.15]
school | 277   | (Intercept) |       -0.22 | 1.91 | [ -3.96,  3.51]
school | 277   | female1     |        0.49 | 1.55 | [ -2.56,  3.54]
school | 278   | (Intercept) |        1.65 | 1.91 | [ -2.09,  5.38]
school | 278   | female1     |        0.06 | 1.55 | [ -2.99,  3.10]
school | 279   | (Intercept) |        0.89 | 1.80 | [ -2.63,  4.41]
school | 279   | female1     |       -0.26 | 1.54 | [ -3.28,  2.75]
school | 280   | (Intercept) |        0.42 | 1.59 | [ -2.69,  3.52]
school | 280   | female1     |       -0.64 | 1.49 | [ -3.57,  2.28]
school | 281   | (Intercept) |        5.08 | 1.76 | [  1.62,  8.54]
school | 281   | female1     |       -0.09 | 1.55 | [ -3.13,  2.95]
school | 282   | (Intercept) |        5.42 | 1.27 | [  2.94,  7.90]
school | 282   | female1     |       -0.67 | 1.65 | [ -3.90,  2.56]
school | 283   | (Intercept) |       -2.10 | 1.81 | [ -5.65,  1.46]
school | 283   | female1     |        0.52 | 1.55 | [ -2.51,  3.55]
school | 284   | (Intercept) |        2.19 | 1.49 | [ -0.74,  5.11]
school | 284   | female1     |       -1.58 | 1.47 | [ -4.46,  1.30]
school | 285   | (Intercept) |       -1.45 | 2.06 | [ -5.49,  2.58]
school | 285   | female1     |       -0.43 | 1.58 | [ -3.54,  2.67]
school | 286   | (Intercept) |       -1.42 | 1.78 | [ -4.91,  2.07]
school | 286   | female1     |    4.79e-03 | 1.53 | [ -3.00,  3.01]
school | 287   | (Intercept) |       -3.30 | 1.85 | [ -6.93,  0.34]
school | 287   | female1     |       -0.80 | 1.54 | [ -3.82,  2.23]
school | 288   | (Intercept) |        3.51 | 1.81 | [ -0.03,  7.06]
school | 288   | female1     |        0.37 | 1.56 | [ -2.68,  3.42]
school | 289   | (Intercept) |       -0.84 | 2.02 | [ -4.80,  3.12]
school | 289   | female1     |       -0.20 | 1.58 | [ -3.30,  2.90]
school | 290   | (Intercept) |       -2.34 | 1.75 | [ -5.76,  1.08]
school | 290   | female1     |       -0.75 | 1.54 | [ -3.77,  2.27]
school | 291   | (Intercept) |       -0.64 | 1.80 | [ -4.16,  2.88]
school | 291   | female1     |       -1.00 | 1.54 | [ -4.02,  2.01]
school | 292   | (Intercept) |       -1.40 | 1.97 | [ -5.26,  2.46]
school | 292   | female1     |        0.15 | 1.57 | [ -2.93,  3.22]
school | 293   | (Intercept) |       -2.12 | 1.81 | [ -5.67,  1.42]
school | 293   | female1     |        0.49 | 1.56 | [ -2.56,  3.54]
school | 294   | (Intercept) |        0.43 | 2.13 | [ -3.74,  4.59]
school | 294   | female1     |        1.25 | 1.60 | [ -1.87,  4.38]
school | 295   | (Intercept) |       -5.36 | 1.99 | [ -9.26, -1.46]
school | 295   | female1     |        0.21 | 1.58 | [ -2.89,  3.32]
school | 296   | (Intercept) |        1.57 | 1.88 | [ -2.11,  5.25]
school | 296   | female1     |       -0.14 | 1.57 | [ -3.22,  2.94]
school | 297   | (Intercept) |       -6.76 | 1.93 | [-10.55, -2.97]
school | 297   | female1     |       -0.13 | 1.59 | [ -3.25,  2.99]
school | 298   | (Intercept) |        0.53 | 2.05 | [ -3.49,  4.55]
school | 298   | female1     |       -0.43 | 1.59 | [ -3.55,  2.68]
school | 299   | (Intercept) |       -0.02 | 1.96 | [ -3.87,  3.82]
school | 299   | female1     |        1.65 | 1.57 | [ -1.44,  4.73]
school | 300   | (Intercept) |       -0.98 | 1.52 | [ -3.95,  1.99]
school | 300   | female1     |        0.57 | 1.51 | [ -2.39,  3.54]
school | 301   | (Intercept) |       -2.60 | 1.99 | [ -6.50,  1.30]
school | 301   | female1     |       -0.30 | 1.58 | [ -3.40,  2.80]
school | 302   | (Intercept) |       -2.33 | 2.10 | [ -6.44,  1.78]
school | 302   | female1     |       -1.06 | 1.59 | [ -4.18,  2.05]
school | 303   | (Intercept) |        4.53 | 1.97 | [  0.67,  8.39]
school | 303   | female1     |        0.19 | 1.57 | [ -2.88,  3.27]
school | 304   | (Intercept) |        2.32 | 1.93 | [ -1.47,  6.11]
school | 304   | female1     |       -0.70 | 1.59 | [ -3.82,  2.42]
school | 305   | (Intercept) |       -2.18 | 1.96 | [ -6.02,  1.66]
school | 305   | female1     |        0.69 | 1.59 | [ -2.42,  3.80]
school | 306   | (Intercept) |       -2.68 | 1.77 | [ -6.15,  0.79]
school | 306   | female1     |       -0.06 | 1.54 | [ -3.07,  2.96]
school | 307   | (Intercept) |        1.94 | 1.81 | [ -1.60,  5.48]
school | 307   | female1     |       -0.06 | 1.57 | [ -3.14,  3.02]
school | 308   | (Intercept) |        1.39 | 1.91 | [ -2.36,  5.14]
school | 308   | female1     |       -0.61 | 1.60 | [ -3.75,  2.53]
school | 309   | (Intercept) |        0.66 | 1.89 | [ -3.05,  4.37]
school | 309   | female1     |        0.71 | 1.55 | [ -2.34,  3.76]
school | 310   | (Intercept) |       -1.92 | 1.99 | [ -5.82,  1.97]
school | 310   | female1     |        0.65 | 1.58 | [ -2.45,  3.75]
school | 311   | (Intercept) |        4.37 | 1.87 | [  0.70,  8.04]
school | 311   | female1     |        0.07 | 1.55 | [ -2.97,  3.10]
school | 312   | (Intercept) |       -1.32 | 2.09 | [ -5.41,  2.77]
school | 312   | female1     |        1.32 | 1.59 | [ -1.80,  4.43]
school | 313   | (Intercept) |       -3.41 | 1.96 | [ -7.26,  0.43]
school | 313   | female1     |        0.24 | 1.57 | [ -2.85,  3.32]
school | 314   | (Intercept) |        0.54 | 1.84 | [ -3.07,  4.15]
school | 314   | female1     |       -0.55 | 1.55 | [ -3.58,  2.48]
school | 315   | (Intercept) |        1.28 | 2.10 | [ -2.82,  5.39]
school | 315   | female1     |       -0.86 | 1.59 | [ -3.98,  2.26]
school | 316   | (Intercept) |       -4.43 | 1.83 | [ -8.02, -0.85]
school | 316   | female1     |        0.87 | 1.57 | [ -2.20,  3.94]
school | 317   | (Intercept) |        1.40 | 1.99 | [ -2.50,  5.31]
school | 317   | female1     |       -1.06 | 1.58 | [ -4.15,  2.03]
school | 318   | (Intercept) |    9.58e-03 | 2.05 | [ -4.01,  4.03]
school | 318   | female1     |       -0.28 | 1.59 | [ -3.39,  2.84]
school | 319   | (Intercept) |       -0.58 | 2.14 | [ -4.77,  3.62]
school | 319   | female1     |        0.33 | 1.60 | [ -2.80,  3.47]
school | 320   | (Intercept) |        2.23 | 2.09 | [ -1.86,  6.32]
school | 320   | female1     |       -0.58 | 1.59 | [ -3.70,  2.53]
school | 321   | (Intercept) |        1.68 | 1.56 | [ -1.37,  4.73]
school | 321   | female1     |       -1.06 | 1.47 | [ -3.95,  1.82]
school | 322   | (Intercept) |       -0.26 | 1.91 | [ -4.00,  3.48]
school | 322   | female1     |       -0.07 | 1.56 | [ -3.12,  2.98]
school | 323   | (Intercept) |        0.95 | 1.96 | [ -2.89,  4.79]
school | 323   | female1     |        0.11 | 1.59 | [ -3.00,  3.22]
school | 324   | (Intercept) |        0.59 | 1.81 | [ -2.95,  4.13]
school | 324   | female1     |        1.54 | 1.53 | [ -1.47,  4.55]
school | 325   | (Intercept) |        2.95 | 1.80 | [ -0.57,  6.47]
school | 325   | female1     |        0.26 | 1.54 | [ -2.75,  3.28]
school | 326   | (Intercept) |        0.65 | 1.96 | [ -3.19,  4.50]
school | 326   | female1     |       -1.55 | 1.57 | [ -4.62,  1.52]
school | 327   | (Intercept) |        2.79 | 2.06 | [ -1.24,  6.82]
school | 327   | female1     |        0.70 | 1.58 | [ -2.41,  3.80]
school | 328   | (Intercept) |        3.98 | 1.82 | [  0.41,  7.56]
school | 328   | female1     |        0.03 | 1.54 | [ -2.99,  3.05]
school | 329   | (Intercept) |        3.53 | 1.93 | [ -0.26,  7.31]
school | 329   | female1     |        0.25 | 1.58 | [ -2.83,  3.34]
school | 330   | (Intercept) |       -3.74 | 1.83 | [ -7.33, -0.14]
school | 330   | female1     |        0.18 | 1.55 | [ -2.86,  3.23]
school | 331   | (Intercept) |        0.49 | 2.07 | [ -3.57,  4.54]
school | 331   | female1     |        0.05 | 1.59 | [ -3.06,  3.16]
school | 332   | (Intercept) |        0.22 | 1.91 | [ -3.52,  3.97]
school | 332   | female1     |       -0.21 | 1.56 | [ -3.27,  2.85]
school | 333   | (Intercept) |        1.99 | 1.53 | [ -1.01,  4.98]
school | 333   | female1     |        0.77 | 1.52 | [ -2.22,  3.75]
school | 334   | (Intercept) |        2.47 | 2.09 | [ -1.62,  6.56]
school | 334   | female1     |       -0.05 | 1.59 | [ -3.17,  3.07]
school | 335   | (Intercept) |        4.23 | 1.86 | [  0.59,  7.86]
school | 335   | female1     |        0.04 | 1.56 | [ -3.02,  3.10]
school | 336   | (Intercept) |       -0.91 | 1.74 | [ -4.32,  2.50]
school | 336   | female1     |        0.82 | 1.52 | [ -2.17,  3.81]
school | 337   | (Intercept) |        0.03 | 1.81 | [ -3.52,  3.57]
school | 337   | female1     |       -0.31 | 1.53 | [ -3.32,  2.70]
school | 338   | (Intercept) |       -1.23 | 1.93 | [ -5.02,  2.55]
school | 338   | female1     |       -0.43 | 1.58 | [ -3.52,  2.66]
school | 339   | (Intercept) |       -4.74 | 1.81 | [ -8.28, -1.19]
school | 339   | female1     |       -0.09 | 1.56 | [ -3.13,  2.96]
school | 340   | (Intercept) |       -4.65 | 1.91 | [ -8.39, -0.90]
school | 340   | female1     |        0.45 | 1.60 | [ -2.69,  3.59]
school | 341   | (Intercept) |        2.41 | 1.82 | [ -1.16,  5.99]
school | 341   | female1     |       -0.21 | 1.54 | [ -3.23,  2.81]
school | 342   | (Intercept) |       -2.92 | 1.93 | [ -6.70,  0.87]
school | 342   | female1     |        0.39 | 1.58 | [ -2.70,  3.47]
school | 343   | (Intercept) |        2.48 | 1.97 | [ -1.38,  6.34]
school | 343   | female1     |       -0.04 | 1.57 | [ -3.11,  3.04]
school | 344   | (Intercept) |        0.19 | 1.86 | [ -3.45,  3.84]
school | 344   | female1     |       -1.41 | 1.55 | [ -4.45,  1.64]
school | 345   | (Intercept) |        0.26 | 1.86 | [ -3.39,  3.91]
school | 345   | female1     |        0.22 | 1.55 | [ -2.82,  3.27]
school | 346   | (Intercept) |        1.41 | 1.86 | [ -2.24,  5.05]
school | 346   | female1     |       -0.73 | 1.55 | [ -3.78,  2.31]
school | 347   | (Intercept) |        0.16 | 2.02 | [ -3.80,  4.12]
school | 347   | female1     |       -0.28 | 1.58 | [ -3.39,  2.82]
school | 348   | (Intercept) |        3.80 | 1.97 | [ -0.05,  7.66]
school | 348   | female1     |        1.16 | 1.57 | [ -1.92,  4.23]
school | 349   | (Intercept) |        0.47 | 2.05 | [ -3.55,  4.49]
school | 349   | female1     |       -0.59 | 1.59 | [ -3.70,  2.52]
school | 350   | (Intercept) |        0.30 | 1.69 | [ -3.02,  3.62]
school | 350   | female1     |       -0.04 | 1.65 | [ -3.28,  3.20]
school | 351   | (Intercept) |        1.09 | 1.86 | [ -2.55,  4.72]
school | 351   | female1     |        0.16 | 1.56 | [ -2.90,  3.22]
school | 352   | (Intercept) |        2.74 | 2.09 | [ -1.35,  6.83]
school | 352   | female1     |        1.12 | 1.59 | [ -1.99,  4.24]
school | 353   | (Intercept) |        1.61 | 1.91 | [ -2.12,  5.34]
school | 353   | female1     |       -0.86 | 1.58 | [ -3.96,  2.24]
school | 354   | (Intercept) |        3.05 | 1.56 | [ -0.00,  6.10]
school | 354   | female1     |       -0.07 | 1.49 | [ -2.99,  2.84]
school | 355   | (Intercept) |       -0.16 | 1.61 | [ -3.31,  2.99]
school | 355   | female1     |       -0.38 | 1.51 | [ -3.34,  2.57]
school | 356   | (Intercept) |        1.44 | 1.91 | [ -2.29,  5.18]
school | 356   | female1     |       -0.04 | 1.58 | [ -3.14,  3.06]
school | 357   | (Intercept) |        0.52 | 1.72 | [ -2.86,  3.90]
school | 357   | female1     |       -0.96 | 1.55 | [ -3.99,  2.07]
school | 358   | (Intercept) |       -0.42 | 1.99 | [ -4.33,  3.48]
school | 358   | female1     |        0.02 | 1.60 | [ -3.11,  3.15]
school | 359   | (Intercept) |        0.17 | 1.96 | [ -3.68,  4.01]
school | 359   | female1     |       -0.87 | 1.59 | [ -3.98,  2.23]
school | 360   | (Intercept) |       -2.51 | 2.02 | [ -6.47,  1.45]
school | 360   | female1     |       -0.02 | 1.58 | [ -3.12,  3.07]
school | 361   | (Intercept) |       -1.84 | 1.76 | [ -5.30,  1.62]
school | 361   | female1     |        0.42 | 1.57 | [ -2.65,  3.50]
school | 362   | (Intercept) |       -5.93 | 1.81 | [ -9.48, -2.37]
school | 362   | female1     |       -0.60 | 1.55 | [ -3.62,  2.43]
school | 363   | (Intercept) |       -5.06 | 2.02 | [ -9.02, -1.10]
school | 363   | female1     |        1.50 | 1.59 | [ -1.61,  4.62]
school | 364   | (Intercept) |       -2.85 | 2.05 | [ -6.87,  1.17]
school | 364   | female1     |       -0.12 | 1.59 | [ -3.23,  3.00]
school | 365   | (Intercept) |        2.38 | 1.92 | [ -1.37,  6.14]
school | 365   | female1     |       -0.29 | 1.66 | [ -3.54,  2.95]
school | 366   | (Intercept) |        2.74 | 1.78 | [ -0.75,  6.23]
school | 366   | female1     |       -0.85 | 1.53 | [ -3.85,  2.15]
school | 367   | (Intercept) |        2.09 | 1.75 | [ -1.35,  5.53]
school | 367   | female1     |       -0.59 | 1.53 | [ -3.59,  2.41]
school | 368   | (Intercept) |        4.96 | 2.00 | [  1.03,  8.89]
school | 368   | female1     |       -0.36 | 1.57 | [ -3.44,  2.73]
school | 369   | (Intercept) |        4.64 | 2.06 | [  0.61,  8.67]
school | 369   | female1     |       -0.01 | 1.58 | [ -3.12,  3.09]
school | 370   | (Intercept) |       -2.49 | 1.83 | [ -6.08,  1.11]
school | 370   | female1     |        0.71 | 1.55 | [ -2.33,  3.75]
school | 371   | (Intercept) |       -4.08 | 1.91 | [ -7.82, -0.35]
school | 371   | female1     |        0.35 | 1.57 | [ -2.73,  3.42]
school | 372   | (Intercept) |       -0.96 | 1.96 | [ -4.80,  2.89]
school | 372   | female1     |   -4.09e-04 | 1.57 | [ -3.09,  3.08]
school | 373   | (Intercept) |        0.79 | 1.85 | [ -2.85,  4.42]
school | 373   | female1     |        0.85 | 1.54 | [ -2.17,  3.88]
school | 374   | (Intercept) |        1.97 | 1.91 | [ -1.76,  5.71]
school | 374   | female1     |       -0.81 | 1.55 | [ -3.86,  2.24]
school | 375   | (Intercept) |       -2.92 | 2.03 | [ -6.89,  1.06]
school | 375   | female1     |       -0.01 | 1.58 | [ -3.11,  3.08]
school | 376   | (Intercept) |        1.17 | 1.91 | [ -2.56,  4.91]
school | 376   | female1     |        0.18 | 1.58 | [ -2.92,  3.28]
school | 377   | (Intercept) |       -2.15 | 1.74 | [ -5.55,  1.26]
school | 377   | female1     |       -0.14 | 1.52 | [ -3.13,  2.84]
school | 378   | (Intercept) |       -1.65 | 2.09 | [ -5.74,  2.44]
school | 378   | female1     |       -0.22 | 1.59 | [ -3.34,  2.90]
school | 379   | (Intercept) |        0.61 | 2.04 | [ -3.40,  4.62]
school | 379   | female1     |       -0.02 | 1.58 | [ -3.12,  3.08]
school | 380   | (Intercept) |        0.74 | 1.78 | [ -2.75,  4.23]
school | 380   | female1     |        0.20 | 1.53 | [ -2.80,  3.20]
school | 381   | (Intercept) |        3.14 | 1.72 | [ -0.24,  6.52]
school | 381   | female1     |       -0.82 | 1.55 | [ -3.85,  2.21]
school | 382   | (Intercept) |       -4.50 | 1.75 | [ -7.93, -1.07]
school | 382   | female1     |        1.54 | 1.52 | [ -1.44,  4.52]
school | 383   | (Intercept) |       -6.15 | 2.09 | [-10.24, -2.06]
school | 383   | female1     |        0.89 | 1.59 | [ -2.23,  4.00]
school | 384   | (Intercept) |        0.47 | 1.86 | [ -3.17,  4.10]
school | 384   | female1     |       -0.54 | 1.56 | [ -3.60,  2.52]
school | 385   | (Intercept) |       -3.61 | 1.81 | [ -7.16, -0.07]
school | 385   | female1     |        0.74 | 1.56 | [ -2.31,  3.79]
school | 386   | (Intercept) |        1.22 | 1.81 | [ -2.32,  4.77]
school | 386   | female1     |       -0.13 | 1.56 | [ -3.18,  2.92]
school | 387   | (Intercept) |       -3.63 | 1.98 | [ -7.52,  0.25]
school | 387   | female1     |       -0.27 | 1.57 | [ -3.35,  2.81]
school | 388   | (Intercept) |        0.22 | 1.89 | [ -3.49,  3.92]
school | 388   | female1     |        1.02 | 1.55 | [ -2.02,  4.06]
school | 389   | (Intercept) |       -0.48 | 1.99 | [ -4.39,  3.42]
school | 389   | female1     |        0.06 | 1.58 | [ -3.03,  3.15]
school | 390   | (Intercept) |        3.92 | 1.77 | [  0.45,  7.39]
school | 390   | female1     |       -0.48 | 1.65 | [ -3.72,  2.76]
school | 391   | (Intercept) |       -2.04 | 1.78 | [ -5.53,  1.45]
school | 391   | female1     |        0.21 | 1.53 | [ -2.80,  3.21]
school | 392   | (Intercept) |        1.94 | 1.73 | [ -1.45,  5.34]
school | 392   | female1     |       -0.24 | 1.65 | [ -3.48,  3.00]
school | 393   | (Intercept) |       -1.46 | 1.79 | [ -4.95,  2.04]
school | 393   | female1     |       -0.14 | 1.56 | [ -3.20,  2.91]
school | 394   | (Intercept) |       -3.20 | 1.85 | [ -6.84,  0.43]
school | 394   | female1     |       -0.52 | 1.58 | [ -3.60,  2.57]
school | 395   | (Intercept) |        3.32 | 1.96 | [ -0.52,  7.16]
school | 395   | female1     |       -0.37 | 1.57 | [ -3.46,  2.71]
school | 396   | (Intercept) |       -1.88 | 1.78 | [ -5.37,  1.60]
school | 396   | female1     |        0.01 | 1.53 | [ -2.99,  3.01]
school | 397   | (Intercept) |        0.53 | 1.99 | [ -3.38,  4.43]
school | 397   | female1     |       -0.32 | 1.60 | [ -3.45,  2.81]
school | 398   | (Intercept) |        0.96 | 1.77 | [ -2.52,  4.43]
school | 398   | female1     |       -0.12 | 1.65 | [ -3.36,  3.12]
school | 399   | (Intercept) |       -3.94 | 2.22 | [ -8.29,  0.41]
school | 399   | female1     |        0.84 | 1.62 | [ -2.33,  4.01]
school | 400   | (Intercept) |       -0.11 | 1.75 | [ -3.54,  3.33]
school | 400   | female1     |       -0.64 | 1.53 | [ -3.65,  2.36]
school | 401   | (Intercept) |        0.64 | 1.85 | [ -2.99,  4.28]
school | 401   | female1     |        0.11 | 1.58 | [ -2.98,  3.20]
school | 402   | (Intercept) |       -2.54 | 2.14 | [ -6.73,  1.65]
school | 402   | female1     |       -0.63 | 1.60 | [ -3.76,  2.51]
school | 403   | (Intercept) |        2.40 | 1.88 | [ -1.29,  6.09]
school | 403   | female1     |        0.49 | 1.59 | [ -2.62,  3.60]
school | 404   | (Intercept) |        1.68 | 1.93 | [ -2.11,  5.46]
school | 404   | female1     |       -0.33 | 1.58 | [ -3.42,  2.76]
school | 405   | (Intercept) |   -1.85e-03 | 1.96 | [ -3.84,  3.84]
school | 405   | female1     |        0.02 | 1.59 | [ -3.08,  3.13]
school | 406   | (Intercept) |        0.57 | 1.77 | [ -2.90,  4.04]
school | 406   | female1     |       -0.38 | 1.54 | [ -3.40,  2.63]
school | 407   | (Intercept) |       -0.28 | 1.99 | [ -4.18,  3.62]
school | 407   | female1     |       -1.15 | 1.58 | [ -4.25,  1.95]
school | 408   | (Intercept) |       -1.96 | 2.02 | [ -5.92,  2.00]
school | 408   | female1     |       -0.34 | 1.58 | [ -3.44,  2.76]
school | 409   | (Intercept) |       -6.48 | 1.94 | [-10.27, -2.68]
school | 409   | female1     |        1.18 | 1.57 | [ -1.89,  4.25]
school | 410   | (Intercept) |       -0.44 | 2.04 | [ -4.44,  3.57]
school | 410   | female1     |       -0.50 | 1.58 | [ -3.60,  2.61]
school | 411   | (Intercept) |        0.88 | 1.67 | [ -2.39,  4.15]
school | 411   | female1     |        1.39 | 1.53 | [ -1.61,  4.39]
school | 412   | (Intercept) |        0.66 | 1.81 | [ -2.90,  4.21]
school | 412   | female1     |       -0.47 | 1.55 | [ -3.49,  2.56]
school | 413   | (Intercept) |        1.70 | 1.82 | [ -1.88,  5.28]
school | 413   | female1     |       -0.26 | 1.54 | [ -3.28,  2.76]
school | 414   | (Intercept) |       -0.52 | 1.88 | [ -4.21,  3.17]
school | 414   | female1     |        0.09 | 1.56 | [ -2.97,  3.14]
school | 415   | (Intercept) |        3.70 | 2.02 | [ -0.26,  7.66]
school | 415   | female1     |        0.56 | 1.64 | [ -2.66,  3.77]
school | 416   | (Intercept) |        1.16 | 1.96 | [ -2.68,  5.00]
school | 416   | female1     |       -0.47 | 1.57 | [ -3.56,  2.61]
school | 417   | (Intercept) |       -5.77 | 2.07 | [ -9.82, -1.71]
school | 417   | female1     |       -0.55 | 1.59 | [ -3.66,  2.56]
school | 418   | (Intercept) |       -1.49 | 1.57 | [ -4.57,  1.59]
school | 418   | female1     |        1.08 | 1.52 | [ -1.90,  4.06]
school | 419   | (Intercept) |       -2.44 | 1.71 | [ -5.80,  0.92]
school | 419   | female1     |        0.15 | 1.52 | [ -2.83,  3.14]

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) |       58.94
school | 1     | female1     |       -0.93
school | 2     | (Intercept) |       61.82
school | 2     | female1     |       -1.11
school | 3     | (Intercept) |       50.63
school | 3     | female1     |       -0.87
school | 4     | (Intercept) |       63.29
school | 4     | female1     |        0.21
school | 5     | (Intercept) |       51.33
school | 5     | female1     |       -1.47
school | 6     | (Intercept) |       59.72
school | 6     | female1     |        0.64
school | 7     | (Intercept) |       57.92
school | 7     | female1     |       -1.38
school | 8     | (Intercept) |       60.46
school | 8     | female1     |        0.10
school | 9     | (Intercept) |       56.32
school | 9     | female1     |       -1.45
school | 10    | (Intercept) |       60.56
school | 10    | female1     |       -0.20
school | 11    | (Intercept) |       56.53
school | 11    | female1     |       -0.47
school | 12    | (Intercept) |       55.83
school | 12    | female1     |       -0.16
school | 13    | (Intercept) |       58.56
school | 13    | female1     |       -0.60
school | 14    | (Intercept) |       57.34
school | 14    | female1     |        0.58
school | 15    | (Intercept) |       62.12
school | 15    | female1     |       -0.53
school | 16    | (Intercept) |       56.70
school | 16    | female1     |       -1.18
school | 17    | (Intercept) |       55.72
school | 17    | female1     |       -1.09
school | 18    | (Intercept) |       61.51
school | 18    | female1     |       -1.29
school | 19    | (Intercept) |       57.91
school | 19    | female1     |       -0.48
school | 20    | (Intercept) |       59.31
school | 20    | female1     |       -0.93
school | 21    | (Intercept) |       59.16
school | 21    | female1     |        0.17
school | 22    | (Intercept) |       56.55
school | 22    | female1     |       -0.70
school | 23    | (Intercept) |       59.29
school | 23    | female1     |       -0.07
school | 24    | (Intercept) |       58.42
school | 24    | female1     |       -0.69
school | 25    | (Intercept) |       56.65
school | 25    | female1     |       -1.97
school | 26    | (Intercept) |       59.82
school | 26    | female1     |        0.29
school | 27    | (Intercept) |       54.76
school | 27    | female1     |       -0.73
school | 28    | (Intercept) |       63.43
school | 28    | female1     |       -1.68
school | 29    | (Intercept) |       59.44
school | 29    | female1     |       -0.66
school | 30    | (Intercept) |       56.75
school | 30    | female1     |       -1.62
school | 31    | (Intercept) |       57.72
school | 31    | female1     |       -2.33
school | 32    | (Intercept) |       61.31
school | 32    | female1     |       -0.22
school | 33    | (Intercept) |       53.31
school | 33    | female1     |       -0.68
school | 34    | (Intercept) |       57.44
school | 34    | female1     |       -1.05
school | 35    | (Intercept) |       60.13
school | 35    | female1     |       -0.95
school | 36    | (Intercept) |       54.30
school | 36    | female1     |       -1.09
school | 37    | (Intercept) |       60.12
school | 37    | female1     |        0.04
school | 38    | (Intercept) |       56.59
school | 38    | female1     |       -1.43
school | 39    | (Intercept) |       58.13
school | 39    | female1     |       -1.51
school | 40    | (Intercept) |       54.36
school | 40    | female1     |       -0.92
school | 41    | (Intercept) |       59.45
school | 41    | female1     |       -0.05
school | 42    | (Intercept) |       58.08
school | 42    | female1     |       -0.63
school | 43    | (Intercept) |       58.44
school | 43    | female1     |        0.13
school | 44    | (Intercept) |       60.96
school | 44    | female1     |       -0.34
school | 45    | (Intercept) |       59.93
school | 45    | female1     |       -0.61
school | 46    | (Intercept) |       57.14
school | 46    | female1     |       -0.08
school | 47    | (Intercept) |       53.25
school | 47    | female1     |        0.08
school | 48    | (Intercept) |       57.06
school | 48    | female1     |       -0.22
school | 49    | (Intercept) |       57.87
school | 49    | female1     |       -1.17
school | 50    | (Intercept) |       53.75
school | 50    | female1     |       -1.13
school | 51    | (Intercept) |       59.75
school | 51    | female1     |       -0.26
school | 52    | (Intercept) |       60.80
school | 52    | female1     |       -0.70
school | 53    | (Intercept) |       60.08
school | 53    | female1     |       -1.65
school | 54    | (Intercept) |       62.65
school | 54    | female1     |       -1.10
school | 55    | (Intercept) |       51.39
school | 55    | female1     |       -2.12
school | 56    | (Intercept) |       56.88
school | 56    | female1     |       -0.56
school | 57    | (Intercept) |       58.28
school | 57    | female1     |       -0.87
school | 58    | (Intercept) |       58.05
school | 58    | female1     |       -0.17
school | 59    | (Intercept) |       58.72
school | 59    | female1     |       -0.63
school | 60    | (Intercept) |       53.95
school | 60    | female1     |       -0.01
school | 61    | (Intercept) |       59.76
school | 61    | female1     |       -1.71
school | 62    | (Intercept) |       49.04
school | 62    | female1     |       -0.02
school | 63    | (Intercept) |       58.85
school | 63    | female1     |       -0.44
school | 64    | (Intercept) |       58.66
school | 64    | female1     |       -0.74
school | 65    | (Intercept) |       61.59
school | 65    | female1     |       -0.82
school | 66    | (Intercept) |       56.34
school | 66    | female1     |       -0.83
school | 67    | (Intercept) |       59.69
school | 67    | female1     |       -0.83
school | 68    | (Intercept) |       57.28
school | 68    | female1     |        0.09
school | 69    | (Intercept) |       54.32
school | 69    | female1     |       -0.92
school | 70    | (Intercept) |       62.57
school | 70    | female1     |       -0.48
school | 71    | (Intercept) |       56.25
school | 71    | female1     |       -1.25
school | 72    | (Intercept) |       52.29
school | 72    | female1     |       -0.63
school | 73    | (Intercept) |       57.01
school | 73    | female1     |       -0.04
school | 74    | (Intercept) |       62.99
school | 74    | female1     |       -0.56
school | 75    | (Intercept) |       57.32
school | 75    | female1     |        0.32
school | 76    | (Intercept) |       57.27
school | 76    | female1     |        0.65
school | 77    | (Intercept) |       62.17
school | 77    | female1     |       -0.50
school | 78    | (Intercept) |       56.67
school | 78    | female1     |       -0.68
school | 79    | (Intercept) |       55.17
school | 79    | female1     |       -0.85
school | 80    | (Intercept) |       62.15
school | 80    | female1     |       -0.48
school | 81    | (Intercept) |       55.29
school | 81    | female1     |       -0.74
school | 82    | (Intercept) |       59.49
school | 82    | female1     |       -1.39
school | 83    | (Intercept) |       59.60
school | 83    | female1     |        0.79
school | 84    | (Intercept) |       60.91
school | 84    | female1     |       -0.49
school | 85    | (Intercept) |       58.12
school | 85    | female1     |        0.17
school | 86    | (Intercept) |       56.40
school | 86    | female1     |       -0.50
school | 87    | (Intercept) |       59.96
school | 87    | female1     |       -0.85
school | 88    | (Intercept) |       59.75
school | 88    | female1     |       -1.18
school | 89    | (Intercept) |       58.22
school | 89    | female1     |       -0.57
school | 90    | (Intercept) |       61.38
school | 90    | female1     |       -1.02
school | 91    | (Intercept) |       57.99
school | 91    | female1     |       -0.82
school | 92    | (Intercept) |       60.30
school | 92    | female1     |       -0.56
school | 93    | (Intercept) |       55.33
school | 93    | female1     |        0.10
school | 94    | (Intercept) |       59.73
school | 94    | female1     |       -0.45
school | 95    | (Intercept) |       57.48
school | 95    | female1     |       -0.39
school | 96    | (Intercept) |       59.49
school | 96    | female1     |       -0.61
school | 97    | (Intercept) |       61.09
school | 97    | female1     |       -0.55
school | 98    | (Intercept) |       58.01
school | 98    | female1     |       -0.44
school | 99    | (Intercept) |       56.85
school | 99    | female1     |       -0.23
school | 100   | (Intercept) |       58.78
school | 100   | female1     |       -1.45
school | 101   | (Intercept) |       57.75
school | 101   | female1     |       -0.70
school | 102   | (Intercept) |       57.67
school | 102   | female1     |       -0.26
school | 103   | (Intercept) |       64.80
school | 103   | female1     |       -1.00
school | 104   | (Intercept) |       58.00
school | 104   | female1     |       -0.28
school | 105   | (Intercept) |       59.14
school | 105   | female1     |       -0.47
school | 106   | (Intercept) |       53.06
school | 106   | female1     |        0.11
school | 107   | (Intercept) |       63.20
school | 107   | female1     |        0.05
school | 108   | (Intercept) |       58.79
school | 108   | female1     |       -1.73
school | 109   | (Intercept) |       61.87
school | 109   | female1     |       -0.54
school | 110   | (Intercept) |       56.20
school | 110   | female1     |       -0.91
school | 111   | (Intercept) |       56.32
school | 111   | female1     |       -0.19
school | 112   | (Intercept) |       58.29
school | 112   | female1     |        0.95
school | 113   | (Intercept) |       58.84
school | 113   | female1     |       -0.36
school | 114   | (Intercept) |       56.18
school | 114   | female1     |       -1.14
school | 115   | (Intercept) |       60.98
school | 115   | female1     |       -0.73
school | 116   | (Intercept) |       49.39
school | 116   | female1     |       -0.71
school | 117   | (Intercept) |       55.31
school | 117   | female1     |       -0.08
school | 118   | (Intercept) |       56.92
school | 118   | female1     |        0.23
school | 119   | (Intercept) |       59.67
school | 119   | female1     |       -0.29
school | 120   | (Intercept) |       61.39
school | 120   | female1     |       -0.22
school | 121   | (Intercept) |       58.48
school | 121   | female1     |       -1.07
school | 122   | (Intercept) |       54.52
school | 122   | female1     |       -0.86
school | 123   | (Intercept) |       57.88
school | 123   | female1     |        0.09
school | 124   | (Intercept) |       59.21
school | 124   | female1     |        0.10
school | 125   | (Intercept) |       56.18
school | 125   | female1     |        0.21
school | 126   | (Intercept) |       57.53
school | 126   | female1     |       -1.02
school | 127   | (Intercept) |       60.69
school | 127   | female1     |       -0.61
school | 128   | (Intercept) |       58.19
school | 128   | female1     |       -0.50
school | 129   | (Intercept) |       54.98
school | 129   | female1     |       -0.09
school | 130   | (Intercept) |       56.61
school | 130   | female1     |       -0.75
school | 131   | (Intercept) |       56.66
school | 131   | female1     |       -0.29
school | 132   | (Intercept) |       54.74
school | 132   | female1     |       -0.91
school | 133   | (Intercept) |       61.97
school | 133   | female1     |       -0.96
school | 134   | (Intercept) |       58.69
school | 134   | female1     |       -1.22
school | 135   | (Intercept) |       52.81
school | 135   | female1     |        0.25
school | 136   | (Intercept) |       57.95
school | 136   | female1     |       -1.65
school | 137   | (Intercept) |       58.34
school | 137   | female1     |       -1.12
school | 138   | (Intercept) |       57.76
school | 138   | female1     |       -0.84
school | 139   | (Intercept) |       55.90
school | 139   | female1     |        0.76
school | 140   | (Intercept) |       63.89
school | 140   | female1     |       -1.28
school | 141   | (Intercept) |       59.47
school | 141   | female1     |       -0.81
school | 142   | (Intercept) |       60.50
school | 142   | female1     |       -0.78
school | 143   | (Intercept) |       59.64
school | 143   | female1     |        0.24
school | 144   | (Intercept) |       56.26
school | 144   | female1     |       -0.23
school | 145   | (Intercept) |       57.83
school | 145   | female1     |       -0.57
school | 146   | (Intercept) |       58.02
school | 146   | female1     |        0.23
school | 147   | (Intercept) |       58.03
school | 147   | female1     |       -0.56
school | 148   | (Intercept) |       54.13
school | 148   | female1     |       -1.13
school | 149   | (Intercept) |       56.80
school | 149   | female1     |        0.69
school | 150   | (Intercept) |       60.97
school | 150   | female1     |       -1.00
school | 151   | (Intercept) |       54.88
school | 151   | female1     |       -0.27
school | 152   | (Intercept) |       61.00
school | 152   | female1     |       -0.04
school | 153   | (Intercept) |       64.52
school | 153   | female1     |       -0.69
school | 154   | (Intercept) |       56.13
school | 154   | female1     |       -0.21
school | 155   | (Intercept) |       54.22
school | 155   | female1     |        0.77
school | 156   | (Intercept) |       61.50
school | 156   | female1     |       -0.86
school | 157   | (Intercept) |       59.21
school | 157   | female1     |       -0.36
school | 158   | (Intercept) |       59.62
school | 158   | female1     |       -1.29
school | 159   | (Intercept) |       57.68
school | 159   | female1     |       -0.16
school | 160   | (Intercept) |       59.37
school | 160   | female1     |       -0.12
school | 161   | (Intercept) |       55.14
school | 161   | female1     |        0.08
school | 162   | (Intercept) |       57.73
school | 162   | female1     |       -1.38
school | 163   | (Intercept) |       55.30
school | 163   | female1     |       -0.47
school | 164   | (Intercept) |       57.80
school | 164   | female1     |       -0.57
school | 165   | (Intercept) |       56.79
school | 165   | female1     |       -1.27
school | 166   | (Intercept) |       54.20
school | 166   | female1     |       -0.57
school | 167   | (Intercept) |       58.47
school | 167   | female1     |       -0.49
school | 168   | (Intercept) |       59.34
school | 168   | female1     |       -0.50
school | 169   | (Intercept) |       57.37
school | 169   | female1     |       -1.21
school | 170   | (Intercept) |       58.07
school | 170   | female1     |       -1.10
school | 171   | (Intercept) |       57.10
school | 171   | female1     |       -0.58
school | 172   | (Intercept) |       58.13
school | 172   | female1     |       -0.91
school | 173   | (Intercept) |       62.01
school | 173   | female1     |       -1.76
school | 174   | (Intercept) |       55.94
school | 174   | female1     |       -0.33
school | 175   | (Intercept) |       58.97
school | 175   | female1     |       -0.80
school | 176   | (Intercept) |       59.28
school | 176   | female1     |       -0.95
school | 177   | (Intercept) |       57.45
school | 177   | female1     |       -0.47
school | 178   | (Intercept) |       62.77
school | 178   | female1     |       -1.23
school | 179   | (Intercept) |       55.89
school | 179   | female1     |       -1.44
school | 180   | (Intercept) |       58.82
school | 180   | female1     |       -0.68
school | 181   | (Intercept) |       58.12
school | 181   | female1     |       -0.35
school | 182   | (Intercept) |       60.80
school | 182   | female1     |       -0.69
school | 183   | (Intercept) |       59.04
school | 183   | female1     |       -0.50
school | 184   | (Intercept) |       56.93
school | 184   | female1     |        0.24
school | 185   | (Intercept) |       52.18
school | 185   | female1     |       -0.41
school | 186   | (Intercept) |       52.74
school | 186   | female1     |       -0.87
school | 187   | (Intercept) |       56.96
school | 187   | female1     |       -1.24
school | 188   | (Intercept) |       60.54
school | 188   | female1     |        0.25
school | 189   | (Intercept) |       54.64
school | 189   | female1     |       -1.97
school | 190   | (Intercept) |       56.07
school | 190   | female1     |        0.91
school | 191   | (Intercept) |       56.99
school | 191   | female1     |       -0.76
school | 192   | (Intercept) |       61.27
school | 192   | female1     |       -1.04
school | 193   | (Intercept) |       55.20
school | 193   | female1     |       -1.11
school | 194   | (Intercept) |       60.61
school | 194   | female1     |       -0.40
school | 195   | (Intercept) |       58.09
school | 195   | female1     |        0.12
school | 196   | (Intercept) |       56.30
school | 196   | female1     |        0.23
school | 197   | (Intercept) |       49.52
school | 197   | female1     |       -0.20
school | 198   | (Intercept) |       56.20
school | 198   | female1     |       -0.69
school | 199   | (Intercept) |       58.05
school | 199   | female1     |       -1.53
school | 200   | (Intercept) |       54.34
school | 200   | female1     |        1.32
school | 201   | (Intercept) |       58.09
school | 201   | female1     |        0.71
school | 202   | (Intercept) |       60.67
school | 202   | female1     |       -0.05
school | 203   | (Intercept) |       57.91
school | 203   | female1     |       -0.92
school | 204   | (Intercept) |       58.64
school | 204   | female1     |       -0.63
school | 205   | (Intercept) |       55.96
school | 205   | female1     |       -1.79
school | 206   | (Intercept) |       56.23
school | 206   | female1     |       -0.59
school | 207   | (Intercept) |       57.46
school | 207   | female1     |       -0.44
school | 208   | (Intercept) |       55.04
school | 208   | female1     |       -0.82
school | 209   | (Intercept) |       60.59
school | 209   | female1     |       -0.12
school | 210   | (Intercept) |       53.22
school | 210   | female1     |        0.70
school | 211   | (Intercept) |       55.38
school | 211   | female1     |        1.09
school | 212   | (Intercept) |       54.15
school | 212   | female1     |       -0.56
school | 213   | (Intercept) |       60.61
school | 213   | female1     |       -0.87
school | 214   | (Intercept) |       53.80
school | 214   | female1     |        0.06
school | 215   | (Intercept) |       58.51
school | 215   | female1     |       -1.08
school | 216   | (Intercept) |       59.07
school | 216   | female1     |       -0.13
school | 217   | (Intercept) |       60.66
school | 217   | female1     |       -0.27
school | 218   | (Intercept) |       57.42
school | 218   | female1     |       -1.45
school | 219   | (Intercept) |       58.03
school | 219   | female1     |       -1.68
school | 220   | (Intercept) |       52.25
school | 220   | female1     |       -0.42
school | 221   | (Intercept) |       61.07
school | 221   | female1     |       -0.57
school | 222   | (Intercept) |       54.99
school | 222   | female1     |       -1.27
school | 223   | (Intercept) |       52.54
school | 223   | female1     |       -1.42
school | 224   | (Intercept) |       59.69
school | 224   | female1     |       -0.47
school | 225   | (Intercept) |       54.29
school | 225   | female1     |       -1.09
school | 226   | (Intercept) |       61.11
school | 226   | female1     |       -0.26
school | 227   | (Intercept) |       56.79
school | 227   | female1     |       -1.12
school | 228   | (Intercept) |       58.31
school | 228   | female1     |       -1.21
school | 229   | (Intercept) |       60.33
school | 229   | female1     |       -0.77
school | 230   | (Intercept) |       59.54
school | 230   | female1     |       -1.18
school | 231   | (Intercept) |       55.46
school | 231   | female1     |       -0.70
school | 232   | (Intercept) |       57.28
school | 232   | female1     |        0.17
school | 233   | (Intercept) |       58.21
school | 233   | female1     |       -0.39
school | 234   | (Intercept) |       58.59
school | 234   | female1     |       -0.24
school | 235   | (Intercept) |       60.77
school | 235   | female1     |       -0.53
school | 236   | (Intercept) |       56.82
school | 236   | female1     |        0.17
school | 237   | (Intercept) |       56.04
school | 237   | female1     |       -1.28
school | 238   | (Intercept) |       58.13
school | 238   | female1     |       -0.15
school | 239   | (Intercept) |       54.96
school | 239   | female1     |       -0.70
school | 240   | (Intercept) |       61.09
school | 240   | female1     |       -0.92
school | 241   | (Intercept) |       62.35
school | 241   | female1     |       -0.81
school | 242   | (Intercept) |       57.21
school | 242   | female1     |       -0.73
school | 243   | (Intercept) |       52.80
school | 243   | female1     |       -0.78
school | 244   | (Intercept) |       58.68
school | 244   | female1     |        0.50
school | 245   | (Intercept) |       59.88
school | 245   | female1     |       -0.51
school | 246   | (Intercept) |       51.30
school | 246   | female1     |       -0.82
school | 247   | (Intercept) |       56.04
school | 247   | female1     |       -0.03
school | 248   | (Intercept) |       59.13
school | 248   | female1     |       -0.45
school | 249   | (Intercept) |       53.51
school | 249   | female1     |        0.12
school | 250   | (Intercept) |       54.52
school | 250   | female1     |       -1.87
school | 251   | (Intercept) |       60.58
school | 251   | female1     |       -1.63
school | 252   | (Intercept) |       62.28
school | 252   | female1     |       -0.46
school | 253   | (Intercept) |       55.31
school | 253   | female1     |        0.81
school | 254   | (Intercept) |       55.34
school | 254   | female1     |       -0.41
school | 255   | (Intercept) |       58.26
school | 255   | female1     |       -0.41
school | 256   | (Intercept) |       58.27
school | 256   | female1     |       -0.36
school | 257   | (Intercept) |       63.40
school | 257   | female1     |        0.23
school | 258   | (Intercept) |       58.33
school | 258   | female1     |        0.02
school | 259   | (Intercept) |       54.37
school | 259   | female1     |       -1.86
school | 260   | (Intercept) |       56.82
school | 260   | female1     |       -1.40
school | 261   | (Intercept) |       56.88
school | 261   | female1     |       -0.99
school | 262   | (Intercept) |       61.78
school | 262   | female1     |       -1.38
school | 263   | (Intercept) |       57.26
school | 263   | female1     |       -1.19
school | 264   | (Intercept) |       54.16
school | 264   | female1     |       -1.60
school | 265   | (Intercept) |       55.84
school | 265   | female1     |       -1.35
school | 266   | (Intercept) |       55.11
school | 266   | female1     |       -1.47
school | 267   | (Intercept) |       58.44
school | 267   | female1     |       -0.62
school | 268   | (Intercept) |       60.07
school | 268   | female1     |       -0.86
school | 269   | (Intercept) |       55.33
school | 269   | female1     |        0.20
school | 270   | (Intercept) |       59.84
school | 270   | female1     |       -0.49
school | 271   | (Intercept) |       60.29
school | 271   | female1     |       -1.35
school | 272   | (Intercept) |       56.73
school | 272   | female1     |       -0.09
school | 273   | (Intercept) |       57.42
school | 273   | female1     |       -0.64
school | 274   | (Intercept) |       61.62
school | 274   | female1     |       -1.57
school | 275   | (Intercept) |       62.37
school | 275   | female1     |       -0.66
school | 276   | (Intercept) |       56.48
school | 276   | female1     |       -0.61
school | 277   | (Intercept) |       57.57
school | 277   | female1     |       -0.12
school | 278   | (Intercept) |       59.44
school | 278   | female1     |       -0.56
school | 279   | (Intercept) |       58.68
school | 279   | female1     |       -0.87
school | 280   | (Intercept) |       58.21
school | 280   | female1     |       -1.26
school | 281   | (Intercept) |       62.88
school | 281   | female1     |       -0.71
school | 282   | (Intercept) |       63.21
school | 282   | female1     |       -1.28
school | 283   | (Intercept) |       55.69
school | 283   | female1     |       -0.10
school | 284   | (Intercept) |       59.98
school | 284   | female1     |       -2.20
school | 285   | (Intercept) |       56.34
school | 285   | female1     |       -1.05
school | 286   | (Intercept) |       56.37
school | 286   | female1     |       -0.61
school | 287   | (Intercept) |       54.50
school | 287   | female1     |       -1.41
school | 288   | (Intercept) |       61.31
school | 288   | female1     |       -0.25
school | 289   | (Intercept) |       56.96
school | 289   | female1     |       -0.81
school | 290   | (Intercept) |       55.45
school | 290   | female1     |       -1.36
school | 291   | (Intercept) |       57.16
school | 291   | female1     |       -1.62
school | 292   | (Intercept) |       56.40
school | 292   | female1     |       -0.46
school | 293   | (Intercept) |       55.67
school | 293   | female1     |       -0.12
school | 294   | (Intercept) |       58.22
school | 294   | female1     |        0.64
school | 295   | (Intercept) |       52.43
school | 295   | female1     |       -0.40
school | 296   | (Intercept) |       59.36
school | 296   | female1     |       -0.75
school | 297   | (Intercept) |       51.03
school | 297   | female1     |       -0.74
school | 298   | (Intercept) |       58.32
school | 298   | female1     |       -1.05
school | 299   | (Intercept) |       57.77
school | 299   | female1     |        1.03
school | 300   | (Intercept) |       56.81
school | 300   | female1     |       -0.04
school | 301   | (Intercept) |       55.19
school | 301   | female1     |       -0.92
school | 302   | (Intercept) |       55.47
school | 302   | female1     |       -1.68
school | 303   | (Intercept) |       62.32
school | 303   | female1     |       -0.42
school | 304   | (Intercept) |       60.11
school | 304   | female1     |       -1.32
school | 305   | (Intercept) |       55.61
school | 305   | female1     |        0.08
school | 306   | (Intercept) |       55.11
school | 306   | female1     |       -0.67
school | 307   | (Intercept) |       59.73
school | 307   | female1     |       -0.67
school | 308   | (Intercept) |       59.18
school | 308   | female1     |       -1.22
school | 309   | (Intercept) |       58.45
school | 309   | female1     |        0.10
school | 310   | (Intercept) |       55.87
school | 310   | female1     |        0.04
school | 311   | (Intercept) |       62.16
school | 311   | female1     |       -0.55
school | 312   | (Intercept) |       56.48
school | 312   | female1     |        0.70
school | 313   | (Intercept) |       54.38
school | 313   | female1     |       -0.38
school | 314   | (Intercept) |       58.33
school | 314   | female1     |       -1.17
school | 315   | (Intercept) |       59.08
school | 315   | female1     |       -1.47
school | 316   | (Intercept) |       53.36
school | 316   | female1     |        0.26
school | 317   | (Intercept) |       59.20
school | 317   | female1     |       -1.67
school | 318   | (Intercept) |       57.80
school | 318   | female1     |       -0.89
school | 319   | (Intercept) |       57.22
school | 319   | female1     |       -0.28
school | 320   | (Intercept) |       60.02
school | 320   | female1     |       -1.20
school | 321   | (Intercept) |       59.47
school | 321   | female1     |       -1.68
school | 322   | (Intercept) |       57.54
school | 322   | female1     |       -0.69
school | 323   | (Intercept) |       58.74
school | 323   | female1     |       -0.51
school | 324   | (Intercept) |       58.38
school | 324   | female1     |        0.93
school | 325   | (Intercept) |       60.74
school | 325   | female1     |       -0.35
school | 326   | (Intercept) |       58.45
school | 326   | female1     |       -2.16
school | 327   | (Intercept) |       60.59
school | 327   | female1     |        0.08
school | 328   | (Intercept) |       61.77
school | 328   | female1     |       -0.58
school | 329   | (Intercept) |       61.32
school | 329   | female1     |       -0.36
school | 330   | (Intercept) |       54.06
school | 330   | female1     |       -0.43
school | 331   | (Intercept) |       58.28
school | 331   | female1     |       -0.56
school | 332   | (Intercept) |       58.02
school | 332   | female1     |       -0.83
school | 333   | (Intercept) |       59.78
school | 333   | female1     |        0.15
school | 334   | (Intercept) |       60.26
school | 334   | female1     |       -0.66
school | 335   | (Intercept) |       62.02
school | 335   | female1     |       -0.57
school | 336   | (Intercept) |       56.88
school | 336   | female1     |        0.21
school | 337   | (Intercept) |       57.82
school | 337   | female1     |       -0.92
school | 338   | (Intercept) |       56.56
school | 338   | female1     |       -1.04
school | 339   | (Intercept) |       53.05
school | 339   | female1     |       -0.70
school | 340   | (Intercept) |       53.15
school | 340   | female1     |       -0.16
school | 341   | (Intercept) |       60.21
school | 341   | female1     |       -0.82
school | 342   | (Intercept) |       54.87
school | 342   | female1     |       -0.23
school | 343   | (Intercept) |       60.27
school | 343   | female1     |       -0.65
school | 344   | (Intercept) |       57.99
school | 344   | female1     |       -2.02
school | 345   | (Intercept) |       58.06
school | 345   | female1     |       -0.39
school | 346   | (Intercept) |       59.20
school | 346   | female1     |       -1.35
school | 347   | (Intercept) |       57.95
school | 347   | female1     |       -0.90
school | 348   | (Intercept) |       61.60
school | 348   | female1     |        0.54
school | 349   | (Intercept) |       58.26
school | 349   | female1     |       -1.20
school | 350   | (Intercept) |       58.10
school | 350   | female1     |       -0.65
school | 351   | (Intercept) |       58.88
school | 351   | female1     |       -0.45
school | 352   | (Intercept) |       60.53
school | 352   | female1     |        0.51
school | 353   | (Intercept) |       59.40
school | 353   | female1     |       -1.47
school | 354   | (Intercept) |       60.84
school | 354   | female1     |       -0.69
school | 355   | (Intercept) |       57.63
school | 355   | female1     |       -1.00
school | 356   | (Intercept) |       59.24
school | 356   | female1     |       -0.65
school | 357   | (Intercept) |       58.31
school | 357   | female1     |       -1.58
school | 358   | (Intercept) |       57.37
school | 358   | female1     |       -0.60
school | 359   | (Intercept) |       57.96
school | 359   | female1     |       -1.49
school | 360   | (Intercept) |       55.28
school | 360   | female1     |       -0.64
school | 361   | (Intercept) |       55.96
school | 361   | female1     |       -0.19
school | 362   | (Intercept) |       51.87
school | 362   | female1     |       -1.21
school | 363   | (Intercept) |       52.74
school | 363   | female1     |        0.89
school | 364   | (Intercept) |       54.94
school | 364   | female1     |       -0.73
school | 365   | (Intercept) |       60.18
school | 365   | female1     |       -0.91
school | 366   | (Intercept) |       60.53
school | 366   | female1     |       -1.46
school | 367   | (Intercept) |       59.88
school | 367   | female1     |       -1.20
school | 368   | (Intercept) |       62.75
school | 368   | female1     |       -0.97
school | 369   | (Intercept) |       62.44
school | 369   | female1     |       -0.62
school | 370   | (Intercept) |       55.31
school | 370   | female1     |        0.10
school | 371   | (Intercept) |       53.71
school | 371   | female1     |       -0.27
school | 372   | (Intercept) |       56.84
school | 372   | female1     |       -0.61
school | 373   | (Intercept) |       58.58
school | 373   | female1     |        0.24
school | 374   | (Intercept) |       59.77
school | 374   | female1     |       -1.42
school | 375   | (Intercept) |       54.88
school | 375   | female1     |       -0.62
school | 376   | (Intercept) |       58.97
school | 376   | female1     |       -0.44
school | 377   | (Intercept) |       55.65
school | 377   | female1     |       -0.76
school | 378   | (Intercept) |       56.14
school | 378   | female1     |       -0.83
school | 379   | (Intercept) |       58.40
school | 379   | female1     |       -0.64
school | 380   | (Intercept) |       58.54
school | 380   | female1     |       -0.41
school | 381   | (Intercept) |       60.94
school | 381   | female1     |       -1.43
school | 382   | (Intercept) |       53.30
school | 382   | female1     |        0.93
school | 383   | (Intercept) |       51.64
school | 383   | female1     |        0.27
school | 384   | (Intercept) |       58.26
school | 384   | female1     |       -1.16
school | 385   | (Intercept) |       54.18
school | 385   | female1     |        0.13
school | 386   | (Intercept) |       59.02
school | 386   | female1     |       -0.74
school | 387   | (Intercept) |       54.16
school | 387   | female1     |       -0.89
school | 388   | (Intercept) |       58.01
school | 388   | female1     |        0.41
school | 389   | (Intercept) |       57.31
school | 389   | female1     |       -0.55
school | 390   | (Intercept) |       61.71
school | 390   | female1     |       -1.10
school | 391   | (Intercept) |       55.76
school | 391   | female1     |       -0.41
school | 392   | (Intercept) |       59.74
school | 392   | female1     |       -0.85
school | 393   | (Intercept) |       56.34
school | 393   | female1     |       -0.76
school | 394   | (Intercept) |       54.59
school | 394   | female1     |       -1.13
school | 395   | (Intercept) |       61.11
school | 395   | female1     |       -0.99
school | 396   | (Intercept) |       55.91
school | 396   | female1     |       -0.60
school | 397   | (Intercept) |       58.32
school | 397   | female1     |       -0.94
school | 398   | (Intercept) |       58.75
school | 398   | female1     |       -0.73
school | 399   | (Intercept) |       53.85
school | 399   | female1     |        0.23
school | 400   | (Intercept) |       57.69
school | 400   | female1     |       -1.26
school | 401   | (Intercept) |       58.44
school | 401   | female1     |       -0.51
school | 402   | (Intercept) |       55.25
school | 402   | female1     |       -1.24
school | 403   | (Intercept) |       60.19
school | 403   | female1     |       -0.12
school | 404   | (Intercept) |       59.47
school | 404   | female1     |       -0.94
school | 405   | (Intercept) |       57.79
school | 405   | female1     |       -0.59
school | 406   | (Intercept) |       58.36
school | 406   | female1     |       -0.99
school | 407   | (Intercept) |       57.51
school | 407   | female1     |       -1.77
school | 408   | (Intercept) |       55.84
school | 408   | female1     |       -0.95
school | 409   | (Intercept) |       51.32
school | 409   | female1     |        0.57
school | 410   | (Intercept) |       57.35
school | 410   | female1     |       -1.11
school | 411   | (Intercept) |       58.67
school | 411   | female1     |        0.78
school | 412   | (Intercept) |       58.45
school | 412   | female1     |       -1.08
school | 413   | (Intercept) |       59.49
school | 413   | female1     |       -0.87
school | 414   | (Intercept) |       57.27
school | 414   | female1     |       -0.53
school | 415   | (Intercept) |       61.49
school | 415   | female1     |       -0.06
school | 416   | (Intercept) |       58.96
school | 416   | female1     |       -1.08
school | 417   | (Intercept) |       52.03
school | 417   | female1     |       -1.16
school | 418   | (Intercept) |       56.31
school | 418   | female1     |        0.46
school | 419   | (Intercept) |       55.36
school | 419   | female1     |       -0.46

Plotting the Fixed Effects

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

These are estimates for the average school in the population.

Visualizing the Random Slopes

Contrast Analysis

estimate_contrasts(fit, contrast = "female", by = "public")

Marginal Contrasts Analysis

Level1 | Level2 | public | Difference |   SE |         95% CI | t(6863) |      p
--------------------------------------------------------------------------------
1      | 0      | 0      |      -0.61 | 0.42 | [-1.44,  0.21] |   -1.45 |  0.146
1      | 0      | 1      |      -1.42 | 0.26 | [-1.92, -0.92] |   -5.53 | < .001

Variable predicted: math
Predictors contrasted: female
Predictors averaged: school
p-values are uncorrected.

The sex difference is significant in public but not in private schools.

Interpretation Part 1

• Last week, we regressed math on female with random slopes.
  • math ~ 1 + female + (1 + female | school)

• Males tended to score higher than females at the average school.
  • \(\gamma_{10} = -1.20, p<.001\)

• But the sex difference (random slopes) varied between schools.
  • \(\tau_{11} = 1.70\)

• So some schools had larger sex differences than others.
  • Can we explain this variability with L2 predictors?

Interpretation Part 2

• This week, we tested whether a school’s sex difference in math scores was significantly related to its public vs private status.

• We did this using a cross-level public-by-female interaction.
  • math ~ 1 + public * female + (1 + female | school)

• Sex differences did not differ between public and private schools.
  • \(\gamma_{11} = -0.81, p=.102\)

Warning

Don’t misinterpret the simple slope of female as its main/partial effect.