Linear mixed effects model: R example

When to use mixed effects models:

Independence assumptions will sometimes (often!) break down.
- Think for instance of being interested in relationship between rain and malaria, and collecting data at village levels.
- Or think of asssessing success of a treatment and collecting data at hospital level.
We use mixed effects models to allow us to correctly estimate the uncertainty in our model when independence does not hold.
Remember that in this your “true” sample size is smaller, so uncertainty will be larger.
Mixed effects models let you control for the variables of interested without coding each one as a categorical variable (what would be the issue with doing that?)
We can specify both a model with random slope or random intercept (or both).

Fixed vs Random Effects :

Fixed Effect :
- the variables we want to answer research questions about

- variables that researchers would care about if we were to collect data to replicate the study in a new location/group.

Random effects :

- less related to the scientific question of interest , but we must account for them to address the relationship of interest

- effects where we do not include all levels of the category.

Interpretation

You can interpret the slopes as :

1) Changes in population average.

2) Changes for “typical” individuals, i.e. \(\beta_1\) is the expected change in the response for subpopulations that differ by 1 unit for individuals for which the \(b_1=0\).

(Similar for the intercept)

Generalizing from linear model..

You can also fit mixed effects models with a non-linear link - so fit GLMs with random effects to adjust for break down of independence assumption.
However, interpretation here because more difficult, as we are always looking at the conditional effects (individual specific).

Alternatives?

Another type of model that can be appropriate for correlated data are Generalized Estimating Equation Models (GEE).

R example

library(tidyverse)

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.5.2     ✔ tibble    3.3.0
✔ lubridate 1.9.4     ✔ tidyr     1.3.1
✔ purrr     1.0.4     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(lme4)

Loading required package: Matrix

Attaching package: 'Matrix'

The following objects are masked from 'package:tidyr':

    expand, pack, unpack

data

chart_dat <- read_csv("data/chart_wide_condense.csv")

New names:
Rows: 618 Columns: 11
── Column specification
──────────────────────────────────────────────────────── Delimiter: "," chr
(2): schoolid, schoolName dbl (9): ...1, urban, charter, schPctnonw,
schPctsped, schPctfree, MathAvgSc...
ℹ Use `spec()` to retrieve the full column specification for this data. ℹ
Specify the column types or set `show_col_types = FALSE` to quiet this message.
• `` -> `...1`

head(chart_dat)

# A tibble: 6 × 11
   ...1 schoolid       schoolName urban charter schPctnonw schPctsped schPctfree
  <dbl> <chr>          <chr>      <dbl>   <dbl>      <dbl>      <dbl>      <dbl>
1     1 Dtype 1 Dnum … RIPPLESID…     0       0     0          0.118       0.363
2     4 Dtype 1 Dnum … WRENSHALL…     0       0     0.0303     0.152       0.424
3     7 Dtype 1 Dnum … CENTRAL M…     0       0     0.0769     0.123       0.262
4    10 Dtype 1 Dnum … SANDBURG …     1       0     0.0977     0.0827      0.248
5    13 Dtype 1 Dnum … OAK VIEW …     1       0     0.0538     0.0954      0.142
6    16 Dtype 1 Dnum … ROOSEVELT…     1       0     0.123      0.0886      0.241
# ℹ 3 more variables: MathAvgScore.0 <dbl>, MathAvgScore.1 <dbl>,
#   MathAvgScore.2 <dbl>

dim(chart_dat)

[1] 618  11

Question

Do we see an average increase in the average Math Average Score between the years 2008 and 2010 for sixth grade students?
Is such an increase different for charter vs non-charter school

Data Organization

Key variables in chart_wide_condense.csv which we will examine to address the research question above are:

schoolid = includes district type, district number, and school number
schoolName = name of school
urban = is the school in an urban (1) or rural (0) location?
charter = is the school a charter school (1) or a non-charter public school (0)?
schPctnonw = proportion of non-white students in a school (based on 2010 figures)
schPctsped = proportion of special education students in a school (based on 2010 figures)
schPctfree = proportion of students who receive free or reduced lunches in a school (based on 2010 figures). This serves as a measure of poverty among school families.
MathAvgScore.0 = average MCA-II math score for all sixth grade students in a school in 2008
MathAvgScore.1 = average MCA-II math score for all sixth grade students in a school in 2009
MathAvgScore.2 = average MCA-II math score for all sixth grade students in a school in 2010

This data is stored in WIDE format, with one row per school, as illustrated in Table @ref(tab:table1chp9).

(Data and some code sourced from Beyond Multiple Linear Regression, data originally from Minnesota Department of Education (2018))

Fitting a simple model of time with a random intercept for school

Writing out the equation for this model:

For Year \(i\) and School \(j\):

\[ MathAvgScore_{ij} | Year_{ij} = \beta_0 + b_{0j} + \beta_1 Year_i + \epsilon_{ij} \]

model1 <- lmer(MathAvgScore ~ year08 + (1|schoolid), 
  REML=T, 
  data= chart_long)
summary(model1)

Linear mixed model fit by REML ['lmerMod']
Formula: MathAvgScore ~ year08 + (1 | schoolid)
   Data: chart_long

REML criterion at convergence: 10345.6

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.1676 -0.4609  0.0141  0.4708  3.6022 

Random effects:
 Groups   Name        Variance Std.Dev.
 schoolid (Intercept) 42.857   6.547   
 Residual              8.906   2.984   
Number of obs: 1733, groups:  schoolid, 618

Fixed effects:
            Estimate Std. Error t value
(Intercept) 651.3898     0.2895  2250.1
year08        1.2799     0.0895    14.3

Correlation of Fixed Effects:
       (Intr)
year08 -0.326

plot the fitted values

Lets look at the predicted values fro three schools:

predict_dat = chart_long %>%
              filter(schoolid %in% c(
                                     "Dtype 7 Dnum 4008 Snum 10",
                                     "Dtype 1 Dnum 23 Snum 10",
                                     "Dtype 1 Dnum 2134 Snum 10")) 
predict_vals = predict(model1, 
                       predict_dat %>% dplyr::select(schoolid, year08))

predict_dat = cbind(predict_dat, predict_vals)  
plot_fit = predict_dat %>% 
            ggplot(aes(x = year08, y = predict_vals, color = schoolid)) +
            geom_line() +
            geom_point(aes(x = year08, y = MathAvgScore, color = schoolid)) +
            xlab("Years from 2008") +
            theme_bw()
            
plot_fit

Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_point()`).

Fitting a model of time with a random intercept and random slope for school

Writing the equation for this model:

For Year \(i\) and School \(j\):

\[ MathAvgScore_{ij} | Year_{ij} = \beta_0 + b_{0j} + \beta_1 Year_i + b_{1j} Year_i + \epsilon_{ij} \]

model2 <- lmer(MathAvgScore~ year08 + (1 + year08|schoolid), 
  REML=T, data= chart_long)
summary(model2)

Linear mixed model fit by REML ['lmerMod']
Formula: MathAvgScore ~ year08 + (1 + year08 | schoolid)
   Data: chart_long

REML criterion at convergence: 10339.5

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.1579 -0.4668  0.0180  0.4596  3.4972 

Random effects:
 Groups   Name        Variance Std.Dev. Corr
 schoolid (Intercept) 39.4410  6.2802       
          year08       0.1105  0.3325   0.72
 Residual              8.8200  2.9699       
Number of obs: 1733, groups:  schoolid, 618

Fixed effects:
             Estimate Std. Error t value
(Intercept) 651.40766    0.27934 2331.96
year08        1.26495    0.08997   14.06

Correlation of Fixed Effects:
       (Intr)
year08 -0.234

Visualizing fitted values

predict_dat2 = chart_long %>%
              filter(schoolid %in% c(
                                     "Dtype 7 Dnum 4008 Snum 10",
                                     "Dtype 1 Dnum 23 Snum 10",
                                     "Dtype 1 Dnum 2134 Snum 10")) 

predict_vals2 = predict(model2, predict_dat2 %>% dplyr::select(schoolid, year08))

predict_dat2 = cbind(predict_dat2, predict_vals2)  
plot_fit2 = predict_dat2 %>% 
            ggplot(aes(x = year08, y = predict_vals2, color = schoolid)) +
            geom_line() +
            geom_point(aes(x = year08, y = MathAvgScore, color = schoolid))+
            theme_bw()
            
plot_fit2

Warning: Removed 1 row containing missing values or values outside the scale range
(`geom_point()`).

Adding in charter school as a covariate

And interaction between charter and year.

For Year \(i\) and School \(j\):

\[ MathAvgScore_{ij} | Year_{ij} = \beta_0 + b_{0j} + \beta_1 Year_i + b_{1j} Year_i + \beta_2 \mathbb{1}(Charter_{j}) + \beta_3\mathbb{1}(Charter_{j})\times Year_i + \epsilon_{ij} \]

model3 <- lmer(MathAvgScore~ charter*year08  + (year08|schoolid), 
  REML=T, data=chart_long)
summary(model3)

Linear mixed model fit by REML ['lmerMod']
Formula: MathAvgScore ~ charter * year08 + (year08 | schoolid)
   Data: chart_long

REML criterion at convergence: 10291.6

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.1935 -0.4710  0.0129  0.4660  3.4592 

Random effects:
 Groups   Name        Variance Std.Dev. Corr
 schoolid (Intercept) 35.8317  5.9860       
          year08       0.1311  0.3621   0.88
 Residual              8.7845  2.9639       
Number of obs: 1733, groups:  schoolid, 618

Fixed effects:
                Estimate Std. Error  t value
(Intercept)    652.05844    0.28449 2292.001
charter         -6.01843    0.86562   -6.953
year08           1.19709    0.09427   12.698
charter:year08   0.85571    0.31430    2.723

Correlation of Fixed Effects:
            (Intr) chartr year08
charter     -0.329              
year08      -0.203  0.067       
chartr:yr08  0.061 -0.298 -0.300

Visualizing fitted values

predict_dat3 = chart_long %>%
              filter(schoolid %in% c(
                                     "Dtype 7 Dnum 4008 Snum 10",
                                     "Dtype 1 Dnum 23 Snum 10",
                                     "Dtype 1 Dnum 2134 Snum 10",
                                     "Dtype 1 Dnum 500 Snum 50",
                                     "Dtype 7 Dnum 4188 Snum 10")) 

predict_vals3 = predict(model3, predict_dat3 %>% dplyr::select(schoolid, charter,year08))

predict_dat3 = cbind(predict_dat3, predict_vals3)  
plot_fit3 = predict_dat3 %>% 
            ggplot(aes(x = year08, y = predict_vals3, color = schoolid)) +
            geom_line(aes(linetype=as.factor(charter))) +
            geom_point(aes(x = year08, y = MathAvgScore, color = schoolid))+
            scale_linetype_discrete(name="Charter School:")+
            theme_bw()
            
plot_fit3

Warning: Removed 3 rows containing missing values or values outside the scale range
(`geom_point()`).

How to get estimates for \(b_{0j}\) and \(b_{1j}\)?

Each row for the specific schoolid :

school_random_effects <- ranef(model3)
head(school_random_effects)

$schoolid
                             (Intercept)        year08
Dtype 1 Dnum 1 Snum 2        0.625917263  0.0254036967
Dtype 1 Dnum 100 Snum 1     -4.577508611 -0.2375916039
Dtype 1 Dnum 108 Snum 30     3.894795355  0.2174232651
Dtype 1 Dnum 11 Snum 121     3.935646266  0.2148689330
Dtype 1 Dnum 11 Snum 193     4.683721476  0.2506134504
Dtype 1 Dnum 11 Snum 195     4.593626873  0.2530555078
Dtype 1 Dnum 11 Snum 3       0.182167525  0.0164495407
Dtype 1 Dnum 11 Snum 4       3.690712738  0.1991820803
Dtype 1 Dnum 11 Snum 6       1.968898260  0.1025948715
Dtype 1 Dnum 11 Snum 90      2.308079799  0.1249261816
Dtype 1 Dnum 110 Snum 220    1.327703218  0.0697415104
Dtype 1 Dnum 111 Snum 841   -0.308898500 -0.0153051084
Dtype 1 Dnum 112 Snum 61     3.732120149  0.2045714315
Dtype 1 Dnum 112 Snum 62     4.872859967  0.2563396258
Dtype 1 Dnum 112 Snum 63     3.791991606  0.2076272574
Dtype 1 Dnum 113 Snum 10    -0.915852061 -0.0483536128
Dtype 1 Dnum 115 Snum 20    -2.376531829 -0.1285656071
Dtype 1 Dnum 116 Snum 10    -4.286928297 -0.2229353080
Dtype 1 Dnum 118 Snum 10     8.954729526  0.4736618336
Dtype 1 Dnum 12 Snum 166     3.097562420  0.1660600502
Dtype 1 Dnum 129 Snum 30    -2.346001106 -0.1110983278
Dtype 1 Dnum 13 Snum 17     -6.995028232 -0.3921183300
Dtype 1 Dnum 138 Snum 20     1.168068419  0.0500888250
Dtype 1 Dnum 139 Snum 10    -0.839616466 -0.0441244816
Dtype 1 Dnum 14 Snum 23      0.851581321  0.0505018707
Dtype 1 Dnum 146 Snum 10     1.605651375  0.0881511073
Dtype 1 Dnum 15 Snum 123     0.877402136  0.0509389519
Dtype 1 Dnum 150 Snum 10     1.571351809  0.1005538420
Dtype 1 Dnum 152 Snum 380    1.858507153  0.0830544608
Dtype 1 Dnum 16 Snum 777     3.942247315  0.2158543803
Dtype 1 Dnum 16 Snum 92      0.572434195  0.0317113299
Dtype 1 Dnum 162 Snum 10    -3.572510184 -0.1823507954
Dtype 1 Dnum 166 Snum 40     1.007062713  0.0610680413
Dtype 1 Dnum 173 Snum 10    -1.706561534 -0.0865695938
Dtype 1 Dnum 177 Snum 60     3.788643624  0.1914629465
Dtype 1 Dnum 181 Snum 22    -0.293225936 -0.0258592612
Dtype 1 Dnum 182 Snum 6     -4.395007434 -0.2262074083
Dtype 1 Dnum 186 Snum 15     0.233166688  0.0248864426
Dtype 1 Dnum 191 Snum 482    4.989245918  0.2777847557
Dtype 1 Dnum 191 Snum 483   -3.084312629 -0.1748081400
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Dtype 1 Dnum 194 Snum 104    2.087125649  0.1168582057
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Dtype 1 Dnum 194 Snum 89     3.705100365  0.2037534064
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Dtype 1 Dnum 2 Snum 1        4.055158592  0.2140067883
Dtype 1 Dnum 200 Snum 130    0.917610578  0.0559473592
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Dtype 1 Dnum 213 Snum 10     4.872859967  0.2563396258
Dtype 1 Dnum 2134 Snum 10   -3.634832293 -0.1962188149
Dtype 1 Dnum 2135 Snum 30    4.222629674  0.2283823973
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Dtype 1 Dnum 2142 Snum 100  -1.266320815 -0.0625844565
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Dtype 1 Dnum 2142 Snum 142  -2.987752649 -0.1518953721
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Dtype 1 Dnum 2142 Snum 96    2.814089951  0.1691958401
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Dtype 1 Dnum 261 Snum 10    -1.997612380 -0.1246759958
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Dtype 1 Dnum 2683 Snum 21    1.117539788  0.0651020292
Dtype 1 Dnum 2687 Snum 86    1.912546722  0.0846905109
Dtype 1 Dnum 2689 Snum 7     2.612491240  0.1362101202
Dtype 1 Dnum 270 Snum 561    3.286700911  0.1717862256
Dtype 1 Dnum 270 Snum 563    7.796091143  0.4006730589
Dtype 1 Dnum 270 Snum 566    1.469525421  0.0526671927
Dtype 1 Dnum 270 Snum 567    2.823767671  0.1567369681
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Dtype 1 Dnum 270 Snum 570    6.959120298  0.3677515426
Dtype 1 Dnum 271 Snum 13     3.092766545  0.1645362748
Dtype 1 Dnum 271 Snum 345   -6.570633933 -0.3582464299
Dtype 1 Dnum 271 Snum 346    6.368367918  0.3353914002
Dtype 1 Dnum 2711 Snum 10   -0.909713163 -0.0741629077
Dtype 1 Dnum 272 Snum 880    5.378956087  0.2851028616
Dtype 1 Dnum 273 Snum 19     5.392787213  0.2817305044
Dtype 1 Dnum 273 Snum 20     8.261892853  0.4399343101
Dtype 1 Dnum 2752 Snum 50    0.870850791  0.0410905494
Dtype 1 Dnum 2753 Snum 12   -6.343128438 -0.3403300516
Dtype 1 Dnum 2754 Snum 2    -0.790807501 -0.0432999884
Dtype 1 Dnum 2759 Snum 10    3.759696434  0.2133331398
Dtype 1 Dnum 276 Snum 175   10.037190399  0.5302138857
Dtype 1 Dnum 276 Snum 75     7.352897906  0.3996625861
Dtype 1 Dnum 277 Snum 63     7.259292364  0.3854553893
Dtype 1 Dnum 278 Snum 31     4.949765912  0.2497071862
Dtype 1 Dnum 279 Snum 665    3.808854159  0.2289518373
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Dtype 1 Dnum 840 Snum 21    -3.965731292 -0.2063181556
Dtype 1 Dnum 846 Snum 10     1.300421735  0.0798024191
Dtype 1 Dnum 85 Snum 2       6.396586672  0.3365903691
Dtype 1 Dnum 850 Snum 10     1.305565278  0.0549408380
Dtype 1 Dnum 852 Snum 10    -2.323199018 -0.1203199257
Dtype 1 Dnum 857 Snum 20    -1.766509979 -0.1056167859
Dtype 1 Dnum 858 Snum 2      2.616002178  0.1528593745
Dtype 1 Dnum 861 Snum 6     -3.137709729 -0.1840069296
Dtype 1 Dnum 876 Snum 2      5.228270570  0.2760604664
Dtype 1 Dnum 877 Snum 30     0.545414410  0.0308933049
Dtype 1 Dnum 879 Snum 15     7.162560447  0.3935554669
Dtype 1 Dnum 88 Snum 30      1.312673123  0.0727329238
Dtype 1 Dnum 881 Snum 10     2.156281281  0.0999964197
Dtype 1 Dnum 882 Snum 30     0.664284268  0.0375938997
Dtype 1 Dnum 883 Snum 3      2.616174115  0.1218465290
Dtype 1 Dnum 885 Snum 30     2.393378527  0.1209603488
Dtype 1 Dnum 885 Snum 90     1.913795251  0.1047882222
Dtype 1 Dnum 891 Snum 20     5.950438996  0.3268743254
Dtype 1 Dnum 91 Snum 1      -2.071563888 -0.1093379851
Dtype 1 Dnum 911 Snum 301    0.111342392  0.0094802768
Dtype 1 Dnum 911 Snum 302   -1.932868060 -0.1041050283
Dtype 1 Dnum 911 Snum 303    6.044130507  0.3255750995
Dtype 1 Dnum 912 Snum 10    -0.533363588 -0.0400223385
Dtype 1 Dnum 914 Snum 1     -4.429863500 -0.2217483569
Dtype 1 Dnum 93 Snum 1      -7.354869977 -0.4057574121
Dtype 1 Dnum 94 Snum 201     0.890034293  0.0471856508
Dtype 1 Dnum 95 Snum 10      1.723150327  0.1254836038
Dtype 1 Dnum 97 Snum 20     -1.603834773 -0.0927649523
Dtype 1 Dnum 99 Snum 30      6.631842480  0.3647360937
Dtype 3 Dnum 1 Snum 106      1.142247603  0.0496517438
Dtype 3 Dnum 1 Snum 119      4.019746025  0.2105221564
Dtype 3 Dnum 1 Snum 121     10.573903211  0.5609379781
Dtype 3 Dnum 1 Snum 124    -10.975284031 -0.6315232628
Dtype 3 Dnum 1 Snum 134      0.195912684  0.0285836063
Dtype 3 Dnum 1 Snum 160     -0.275797902 -0.0280887869
Dtype 3 Dnum 1 Snum 161    -11.583079880 -0.6226973508
Dtype 3 Dnum 1 Snum 170    -11.409442017 -0.6374298681
Dtype 3 Dnum 1 Snum 175     -7.685380998 -0.4405903885
Dtype 3 Dnum 1 Snum 179     -7.357076148 -0.4061811503
Dtype 3 Dnum 1 Snum 190    -14.322438974 -0.7862784898
Dtype 3 Dnum 1 Snum 193    -15.042937900 -0.8132612989
Dtype 3 Dnum 1 Snum 225    -15.533533393 -0.8215658420
Dtype 3 Dnum 1 Snum 226     -2.279243412 -0.1287220014
Dtype 3 Dnum 1 Snum 256    -14.489910056 -0.8006540988
Dtype 3 Dnum 1 Snum 260     -8.457992087 -0.4918974661
Dtype 3 Dnum 1 Snum 282    -20.200631020 -1.0927935791
Dtype 3 Dnum 1 Snum 288    -10.808627354 -0.5785720688
Dtype 3 Dnum 1 Snum 290    -17.183237800 -0.9263092659
Dtype 3 Dnum 1 Snum 291     -7.562443702 -0.4092968562
Dtype 3 Dnum 1 Snum 300      1.955709602  0.0984044892
Dtype 3 Dnum 1 Snum 301     -8.260076251 -0.4445481550
Dtype 3 Dnum 1 Snum 304    -11.231180215 -0.6196257645
Dtype 3 Dnum 1 Snum 311    -25.748504841 -1.4030940712
Dtype 3 Dnum 1 Snum 316     -7.325260488 -0.4038393499
Dtype 3 Dnum 1 Snum 318    -15.603716057 -0.8360978454
Dtype 3 Dnum 1 Snum 324     -4.059979302 -0.2129626130
Dtype 3 Dnum 1 Snum 405    -20.248168006 -1.0943798056
Dtype 3 Dnum 1 Snum 417    -14.163565553 -0.7755139190
Dtype 3 Dnum 1 Snum 625    -26.184022228 -1.4336837442
Dtype 3 Dnum 1 Snum 689    -24.681465144 -1.3514125161
Dtype 3 Dnum 6 Snum 735     -0.323286126 -0.0198764345
Dtype 3 Dnum 6 Snum 737     -2.782385094 -0.1487796662
Dtype 7 Dnum 4001 Snum 10   -0.579113077 -0.0225727232
Dtype 7 Dnum 4008 Snum 10    7.727475278  0.4263911640
Dtype 7 Dnum 4011 Snum 10  -10.747410706 -0.5737377198
Dtype 7 Dnum 4015 Snum 10    3.864333867  0.1853621988
Dtype 7 Dnum 4016 Snum 10    5.156229733  0.2696228944
Dtype 7 Dnum 4017 Snum 11  -14.162906877 -0.7502067459
Dtype 7 Dnum 4017 Snum 17  -16.335264233 -0.8701360079
Dtype 7 Dnum 4017 Snum 18    0.450593165  0.0172177999
Dtype 7 Dnum 4018 Snum 10    0.595840338  0.0322991592
Dtype 7 Dnum 4020 Snum 20    3.428334443  0.1866373891
Dtype 7 Dnum 4025 Snum 10    4.335573921  0.2185844860
Dtype 7 Dnum 4026 Snum 10    5.183806018  0.2783846028
Dtype 7 Dnum 4027 Snum 10    2.099655104  0.1295239632
Dtype 7 Dnum 4029 Snum 20   -8.511438485 -0.4480447500
Dtype 7 Dnum 4030 Snum 10   -4.253287934 -0.2268626674
Dtype 7 Dnum 4032 Snum 10   -1.645815418 -0.1051853509
Dtype 7 Dnum 4035 Snum 10   -5.801808422 -0.2761567026
Dtype 7 Dnum 4038 Snum 10   -2.851000960 -0.1230615612
Dtype 7 Dnum 4042 Snum 10    9.845637237  0.5246384585
Dtype 7 Dnum 4043 Snum 10   12.597628489  0.6844662965
Dtype 7 Dnum 4050 Snum 10   -6.457010404 -0.3439477425
Dtype 7 Dnum 4053 Snum 10    4.857813137  0.2602436750
Dtype 7 Dnum 4058 Snum 10    2.273292967  0.1147914459
Dtype 7 Dnum 4059 Snum 10    2.546445248  0.1316772675
Dtype 7 Dnum 4067 Snum 10   -4.902319257 -0.2544389521
Dtype 7 Dnum 4068 Snum 10   -7.713702454 -0.4253702989
Dtype 7 Dnum 4070 Snum 10    6.027884708  0.3290982048
Dtype 7 Dnum 4073 Snum 10   -8.622140901 -0.4576983395
Dtype 7 Dnum 4078 Snum 10   -8.223395481 -0.4552763662
Dtype 7 Dnum 4084 Snum 10    5.721161298  0.3015459557
Dtype 7 Dnum 4086 Snum 10   -3.332773830 -0.2062316111
Dtype 7 Dnum 4088 Snum 10  -14.003272077 -0.7305540605
Dtype 7 Dnum 4091 Snum 10    3.366544561  0.1747719927
Dtype 7 Dnum 4097 Snum 10    2.369837611  0.1297584317
Dtype 7 Dnum 4098 Snum 20    9.882623033  0.5411145724
Dtype 7 Dnum 4099 Snum 10   11.447853421  0.6375942347
Dtype 7 Dnum 4099 Snum 20   12.662353639  0.6933163445
Dtype 7 Dnum 4103 Snum 20   -8.824212668 -0.4831622181
Dtype 7 Dnum 4115 Snum 10   -2.263673550 -0.1228570957
Dtype 7 Dnum 4116 Snum 10    6.670450656  0.3313196642
Dtype 7 Dnum 4118 Snum 10    3.436813193  0.1737975733
Dtype 7 Dnum 4120 Snum 20   10.183134450  0.5575688181
Dtype 7 Dnum 4122 Snum 10    9.607983491  0.4957308461
Dtype 7 Dnum 4124 Snum 10    2.067703118  0.1110062919
Dtype 7 Dnum 4126 Snum 10   -6.127654694 -0.3418116322
Dtype 7 Dnum 4127 Snum 10    6.773561981  0.3640808346
Dtype 7 Dnum 4133 Snum 10    7.088121673  0.3863560073
Dtype 7 Dnum 4135 Snum 10    3.888783777  0.2164311817
Dtype 7 Dnum 4137 Snum 10   -0.915550940 -0.0487689595
Dtype 7 Dnum 4139 Snum 10   -9.632491704 -0.5224066067
Dtype 7 Dnum 4140 Snum 10    9.957750887  0.5452281339
Dtype 7 Dnum 4141 Snum 10    3.006338082  0.1535273768
Dtype 7 Dnum 4142 Snum 10    5.764966647  0.3076971945
Dtype 7 Dnum 4143 Snum 10    0.432694600 -0.0040027804
Dtype 7 Dnum 4144 Snum 10    0.341385547  0.0186922737
Dtype 7 Dnum 4149 Snum 10    2.036023785  0.1248403624
Dtype 7 Dnum 4153 Snum 10   -6.295260003 -0.3579456314
Dtype 7 Dnum 4155 Snum 10   -0.433684899 -0.0231012392
Dtype 7 Dnum 4161 Snum 10   -2.557294271 -0.1307125398
Dtype 7 Dnum 4163 Snum 10  -13.378048194 -0.7416094981
Dtype 7 Dnum 4164 Snum 10    2.501687341  0.1452856523
Dtype 7 Dnum 4168 Snum 10    6.426741739  0.3518907477
Dtype 7 Dnum 4171 Snum 10    1.262311392  0.0652364521
Dtype 7 Dnum 4172 Snum 10    1.543431215  0.0845092562
Dtype 7 Dnum 4180 Snum 10  -16.496873059 -0.8893033646
Dtype 7 Dnum 4182 Snum 10   -9.290760769 -0.5108570299
Dtype 7 Dnum 4185 Snum 10   -0.114389218 -0.0011773548
Dtype 7 Dnum 4186 Snum 10   10.681892604  0.6099998302
Dtype 7 Dnum 4188 Snum 10    2.294709757  0.1256448703
Dtype 7 Dnum 4191 Snum 10   -6.645504896 -0.3638689373
Dtype 7 Dnum 4192 Snum 10    2.001666016  0.1029204572
Dtype 7 Dnum 4194 Snum 10    0.942408381  0.0516007649
Dtype 7 Dnum 4196 Snum 10  -14.233418172 -0.7793386394

Remember that to get the specific intercept or slope for a certain school you will need to add these two \(\beta_0\) and \(\beta_1\).