load data and see it
## 'data.frame': 1190 obs. of 12 variables:
## $ sex : int 1 0 1 0 0 1 0 0 1 0 ...
## $ minority: int 1 1 1 1 1 1 1 1 1 1 ...
## $ mathkind: int 448 460 511 449 425 450 452 443 422 480 ...
## $ mathgain: int 32 109 56 83 53 65 51 66 88 -7 ...
## $ ses : num 0.46 -0.27 -0.03 -0.38 -0.03 0.76 -0.03 0.2 0.64 0.13 ...
## $ yearstea: num 1 1 1 2 2 2 2 2 2 2 ...
## $ mathknow: num NA NA NA -0.11 -0.11 -0.11 -0.11 -0.11 -0.11 -0.11 ...
## $ housepov: num 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 0.082 ...
## $ mathprep: num 2 2 2 3.25 3.25 3.25 3.25 3.25 3.25 3.25 ...
## $ classid : int 160 160 160 217 217 217 217 217 217 217 ...
## $ schoolid: int 1 1 1 1 1 1 1 1 1 1 ...
## $ childid : int 1 2 3 4 5 6 7 8 9 10 ...
## sex minority mathkind mathgain ses yearstea mathknow housepov mathprep
## 1 1 1 448 32 0.46 1 NA 0.082 2.00
## 2 0 1 460 109 -0.27 1 NA 0.082 2.00
## 3 1 1 511 56 -0.03 1 NA 0.082 2.00
## 4 0 1 449 83 -0.38 2 -0.11 0.082 3.25
## 5 0 1 425 53 -0.03 2 -0.11 0.082 3.25
## 6 1 1 450 65 0.76 2 -0.11 0.082 3.25
## classid schoolid childid
## 1 160 1 1
## 2 160 1 2
## 3 160 1 3
## 4 217 1 4
## 5 217 1 5
## 6 217 1 6
model
m0
consider the School and Class level mathgainijk(j) = b0j + b1k(j) + εijk(j)
## Linear mixed model fit by REML ['lmerMod']
## Formula: mathgain ~ (1 | schoolid) + (1 | classid)
## Data: classroom
##
## REML criterion at convergence: 11768.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -4.6441 -0.5984 -0.0336 0.5334 5.6335
##
## Random effects:
## Groups Name Variance Std.Dev.
## classid (Intercept) 99.22 9.961
## schoolid (Intercept) 77.50 8.804
## Residual 1028.23 32.066
## Number of obs: 1190, groups: classid, 312; schoolid, 107
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 57.427 1.443 39.79
|
|
mathgain
|
|
Predictors
|
Estimates
|
CI
|
|
(Intercept)
|
57.43
|
54.60 – 60.26
|
|
Random Effects
|
|
σ2
|
1028.23
|
|
τ00 classid
|
99.22
|
|
τ00 schoolid
|
77.50
|
|
ICC
|
0.15
|
|
N schoolid
|
107
|
|
N classid
|
312
|
|
Observations
|
1190
|
m1
mathgainijk(j) = b0j + b1k(j) + β2 × mathkindijk(j) + β3 × sexijk(j) + β4 × minorityijk(j) + β5 × sesijk(j) + εijk(j)
## Linear mixed model fit by REML ['lmerMod']
## Formula: mathgain ~ mathkind + sex + minority + ses + (1 | schoolid) +
## (1 | classid)
## Data: classroom
##
## REML criterion at convergence: 11385.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -5.8257 -0.6110 -0.0337 0.5538 4.2678
##
## Random effects:
## Groups Name Variance Std.Dev.
## classid (Intercept) 83.28 9.126
## schoolid (Intercept) 75.20 8.672
## Residual 734.57 27.103
## Number of obs: 1190, groups: classid, 312; schoolid, 107
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 282.79033 10.85323 26.056
## mathkind -0.46980 0.02227 -21.100
## sex -1.25119 1.65773 -0.755
## minority -8.26213 2.34011 -3.531
## ses 5.34638 1.24109 4.308
##
## Correlation of Fixed Effects:
## (Intr) mthknd sex minrty
## mathkind -0.978
## sex -0.044 -0.031
## minority -0.307 0.163 -0.018
## ses 0.140 -0.168 0.019 0.163
|
|
mathgain
|
|
Predictors
|
Estimates
|
CI
|
|
(Intercept)
|
282.79
|
261.52 – 304.06
|
|
mathkind
|
-0.47
|
-0.51 – -0.43
|
|
sex
|
-1.25
|
-4.50 – 2.00
|
|
minority
|
-8.26
|
-12.85 – -3.68
|
|
ses
|
5.35
|
2.91 – 7.78
|
|
Random Effects
|
|
σ2
|
734.57
|
|
τ00 classid
|
83.28
|
|
τ00 schoolid
|
75.20
|
|
ICC
|
0.18
|
|
N schoolid
|
107
|
|
N classid
|
312
|
|
Observations
|
1190
|
compare m0 & m1
|
|
mathgain
|
mathgain
|
|
Predictors
|
Estimates
|
std. Error
|
Estimates
|
std. Error
|
|
(Intercept)
|
57.43
|
1.44
|
282.79
|
10.85
|
|
mathkind
|
|
|
-0.47
|
0.02
|
|
sex
|
|
|
-1.25
|
1.66
|
|
minority
|
|
|
-8.26
|
2.34
|
|
ses
|
|
|
5.35
|
1.24
|
|
Random Effects
|
|
σ2
|
1028.23
|
734.57
|
|
τ00
|
99.22 classid
|
83.28 classid
|
|
|
77.50 schoolid
|
75.20 schoolid
|
|
ICC
|
0.15
|
0.18
|