1 Install package

library(pacman)
pacman::p_load(mlmRev, tidyverse, lme4, merTools)

2 Import data

dta <- read.csv("/Users/sudewei/Desktop/報告、作業/Multilevel/pts.csv")
head(dta)

  School Teacher Pupil      Read_0 Read_1
1      1      11    18 -0.16407000 4.2426
2      1      11    19 -0.98126000 1.0000
3      1      11    20 -1.25370000 2.2361
4      1      11    15 -0.87230000 3.1623
5      1      11    17 -0.00063104 3.4641
6      1      11    16 -0.92678000 4.0000

3 Compute

# coerce variables to factor type and
# compute mean Read_0 by school
dta <- dta %>%
  mutate(School = factor(School), Teacher = factor(Teacher),
         Pupil = factor(Pupil)) %>%
  group_by( School ) %>%
  mutate(msRead_0 = mean(Read_0))

#
# compute mean Read_0 by teacher and
# centered teacher mean of Read_0 (from respective school means)
#
dta <- dta %>%
  group_by( Teacher ) %>%
  mutate(mtRead_0 = mean(Read_0), ctRead_0 = mean(Read_0) - msRead_0)

4 Model 1

# no predictor
summary(m0 <- lmer(Read_1 ~  (1 | School) + (1 | Teacher), data = dta))

Linear mixed model fit by REML ['lmerMod']
Formula: Read_1 ~ (1 | School) + (1 | Teacher)
   Data: dta

REML criterion at convergence: 2486.8

Scaled residuals: 
   Min     1Q Median     3Q    Max 
-3.054 -0.657  0.065  0.625  2.472 

Random effects:
 Groups   Name        Variance Std.Dev.
 Teacher  (Intercept) 0.137    0.370   
 School   (Intercept) 0.128    0.357   
 Residual             1.325    1.151   
Number of obs: 777, groups:  Teacher, 46; School, 20

Fixed effects:
            Estimate Std. Error t value
(Intercept)    3.375      0.107    31.7

4.1 Explain

Model中有學校層次與學生層次的隨機截距

Teacher截距的變異數為0.137

School截距的變異數為0.1277

5 Model 2

# add a school-level predictor
summary(m0_s <- update(m0, . ~ . + msRead_0))

Linear mixed model fit by REML ['lmerMod']
Formula: Read_1 ~ (1 | School) + (1 | Teacher) + msRead_0
   Data: dta

REML criterion at convergence: 2467.4

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.0611 -0.6560  0.0682  0.6277  2.4539 

Random effects:
 Groups   Name        Variance Std.Dev.
 Teacher  (Intercept) 0.123    0.351   
 School   (Intercept) 0.000    0.000   
 Residual             1.322    1.150   
Number of obs: 777, groups:  Teacher, 46; School, 20

Fixed effects:
            Estimate Std. Error t value
(Intercept)   3.3982     0.0681   49.92
msRead_0      1.0016     0.1781    5.62

Correlation of Fixed Effects:
         (Intr)
msRead_0 0.086 
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')

5.1 Explain

Model中有學校層次與學生層次的隨機截距以及固定效果的school-level msRead_0

Teacher截距的變異數變為0.1234

School截距的變異數變為0

msRead_0的變異數為0.1787

6 Model 3

# add a teacher-level predictor
summary(m0_t <- update(m0, . ~ . + mtRead_0))

Linear mixed model fit by REML ['lmerMod']
Formula: Read_1 ~ (1 | School) + (1 | Teacher) + mtRead_0
   Data: dta

REML criterion at convergence: 2434.9

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-3.0754 -0.6598  0.0671  0.6326  2.4997 

Random effects:
 Groups   Name        Variance Std.Dev.
 Teacher  (Intercept) 3.30e-15 5.74e-08
 School   (Intercept) 4.47e-02 2.11e-01
 Residual             1.31e+00 1.14e+00
Number of obs: 777, groups:  Teacher, 46; School, 20

Fixed effects:
            Estimate Std. Error t value
(Intercept)   3.4065     0.0631   54.01
mtRead_0      1.0732     0.1149    9.34

Correlation of Fixed Effects:
         (Intr)
mtRead_0 0.024 
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')

6.1 Explain

Model中有學校層次與學生層次的隨機截距以及固定效果的teacher-level mtRead_0

Teacher截距的變異數變為0

School截距的變異數變為0.04

mtRead_0的變異數為0.013

7 Model 4

# add a teacher-level predictor away from respective school means
summary(m0_ct <- update(m0, . ~ . + ctRead_0))

Linear mixed model fit by REML ['lmerMod']
Formula: Read_1 ~ (1 | School) + (1 | Teacher) + ctRead_0
   Data: dta

REML criterion at convergence: 2454.1

Scaled residuals: 
   Min     1Q Median     3Q    Max 
-3.180 -0.646  0.079  0.642  2.527 

Random effects:
 Groups   Name        Variance Std.Dev.
 Teacher  (Intercept) 2.46e-10 1.57e-05
 School   (Intercept) 1.76e-01 4.20e-01
 Residual             1.31e+00 1.14e+00
Number of obs: 777, groups:  Teacher, 46; School, 20

Fixed effects:
            Estimate Std. Error t value
(Intercept)    3.390      0.103   32.93
ctRead_0       1.174      0.158    7.43

Correlation of Fixed Effects:
         (Intr)
ctRead_0 0.000 
optimizer (nloptwrap) convergence code: 0 (OK)
boundary (singular) fit: see help('isSingular')

7.1 Explain

Model中有學校層次與學生層次的隨機截距以及固定效果的ctRead_0