1 Data

pacman::p_load(mlmRev, tidyverse, lme4, nlme)

dta <- read.table("C:/Users/ASUS/Desktop/data/iq_language.txt", header = T)

head(dta)
##   School Pupil   IQ Language Group_size       IQ_c School_mean Group_mean
## 1      1 17001 15.0       46         29  3.1659379    -1.51406        5.9
## 2      1 17002 14.5       45         29  2.6659379    -1.51406        5.9
## 3      1 17003  9.5       33         29 -2.3340621    -1.51406        5.9
## 4      1 17004 11.0       46         29 -0.8340621    -1.51406        5.9
## 5      1 17005  8.0       20         29 -3.8340621    -1.51406        5.9
## 6      1 17006  9.5       30         29 -2.3340621    -1.51406        5.9
##        SES_c
## 1  -4.811981
## 2 -17.811981
## 3 -12.811981
## 4  -4.811981
## 5 -17.811981
## 6 -17.811981

2 Model

2.1 M1

summary(m1 <- lmer(Language ~ (1|School), data = dta))
## Linear mixed model fit by REML ['lmerMod']
## Formula: Language ~ (1 | School)
##    Data: dta
## 
## REML criterion at convergence: 16253.1
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -3.11618 -0.65703  0.07597  0.74128  2.50639 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  School   (Intercept) 19.63    4.431   
##  Residual             64.56    8.035   
## Number of obs: 2287, groups:  School, 131
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept)  40.3624     0.4282   94.26
library(ICC)

2.2 M2

summary(m2<- lmer(Language ~ IQ_c + (1|School), data = dta))
## Linear mixed model fit by REML ['lmerMod']
## Formula: Language ~ IQ_c + (1 | School)
##    Data: dta
## 
## REML criterion at convergence: 15255.8
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.0939 -0.6375  0.0579  0.7061  3.1448 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  School   (Intercept)  9.602   3.099   
##  Residual             42.245   6.500   
## Number of obs: 2287, groups:  School, 131
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept) 40.60823    0.30819   131.8
## IQ_c         2.48759    0.07008    35.5
## 
## Correlation of Fixed Effects:
##      (Intr)
## IQ_c 0.018

2.3 Compare M1 with M2

sjPlot::tab_model(m1, m2, show.p=FALSE, show.r2=FALSE, show.obs=FALSE, show.ngroups=FALSE, show.se=TRUE, show.ci=FALSE)
  Language Language
Predictors Estimates std. Error Estimates std. Error
(Intercept) 40.36 0.43 40.61 0.31
IQ_c 2.49 0.07
Random Effects
σ2 64.56 42.24
τ00 19.63 School 9.60 School
ICC 0.23 0.19 
# (0.23-0.19)/0.23= 0.17, which means adding IQ_c in M2 model increase the explained variance by 17% (??)