1 Question

How does the estimate of grand mean depend on the intra-class correlation in a simple random-effects model?

2 Data Management

dta1 <- data.frame(Score = c(11,13,51,53,55,91,93,95,97),
                   SchoolID = c("S1","S1","S2","S2","S2","S3","S3","S3","S3"),
                   StudentID = c("P1","P2","P3","P4","P5","P6","P7","P8","P9"))
show(dta1)
  Score SchoolID StudentID
1    11       S1        P1
2    13       S1        P2
3    51       S2        P3
4    53       S2        P4
5    55       S2        P5
6    91       S3        P6
7    93       S3        P7
8    95       S3        P8
9    97       S3        P9
dta2 <- data.frame(Score = c(11,51,13,55,91,51,53,95,97),
                   SchoolID = c("S1","S1","S2","S2","S2","S3","S3","S3","S3"),
                   StudentID = c("P1","P2","P3","P4","P5","P6","P7","P8","P9"))
show(dta2)
  Score SchoolID StudentID
1    11       S1        P1
2    51       S1        P2
3    13       S2        P3
4    55       S2        P4
5    91       S2        P5
6    51       S3        P6
7    53       S3        P7
8    95       S3        P8
9    97       S3        P9

3 simple random-effects model

3.1 Data set 1

m1_mlm <- lme4::lmer(Score ~ (1 | SchoolID), data=dta1)
m1_lm <- lm(Score ~ 1, data=dta1)
sjPlot::tab_model(m1_mlm, show.p=FALSE, show.r2=FALSE)
  Score
Predictors Estimates CI
(Intercept) 53.01 -4.91 – 110.93
Random Effects
σ2 5.00
τ00 SchoolID 1679.06
ICC 1.00
N SchoolID 3
Observations 9 
summary(m1_mlm)
Linear mixed model fit by REML ['lmerMod']
Formula: Score ~ (1 | SchoolID)
   Data: dta1

REML criterion at convergence: 51.5

Scaled residuals: 
   Min     1Q Median     3Q    Max 
-1.328 -0.474  0.000  0.461  1.355 

Random effects:
 Groups   Name        Variance Std.Dev.
 SchoolID (Intercept) 1679     40.98   
 Residual                5      2.24   
Number of obs: 9, groups:  SchoolID, 3

Fixed effects:
            Estimate Std. Error t value
(Intercept)     53.0       23.7    2.24
summary(m1_lm)

Call:
lm(formula = Score ~ 1, data = dta1)

Residuals:
   Min     1Q Median     3Q    Max 
-51.11 -11.11  -7.11  30.89  34.89 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)     62.1       11.4    5.44  0.00061

Residual standard error: 34.2 on 8 degrees of freedom

結果顯示：

  • 在此資料中，ICC近乎等於1，表示同組內同質性非常高，顯示組間效果不可忽略，因此若忽略組間差異直接進行迴歸分析，結果顯示在多層次模型及迴歸分析中估計的總平均分數分別為53.0(SE=23.7)與62.1(SE=11.4)，即兩者總平均分數的估計有明顯差異，且在迴歸分析中標準誤會被低估。

3.2 Data set 2

m2_mlm <- lme4::lmer(Score ~ (1 | SchoolID), data=dta2)
m2_lm <- lm(Score ~ 1, data=dta2)
sjPlot::tab_model(m2_mlm, show.p=FALSE, show.r2=FALSE)
  Score
Predictors Estimates CI
(Intercept) 56.36 26.97 – 85.74
Random Effects
σ2 967.76
τ00 SchoolID 104.17
ICC 0.10
N SchoolID 3
Observations 9 
summary(m2_mlm)
Linear mixed model fit by REML ['lmerMod']
Formula: Score ~ (1 | SchoolID)
   Data: dta2

REML criterion at convergence: 80.4

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.3673 -0.3429 -0.0278  1.0715  1.1400 

Random effects:
 Groups   Name        Variance Std.Dev.
 SchoolID (Intercept) 104      10.2    
 Residual             968      31.1    
Number of obs: 9, groups:  SchoolID, 3

Fixed effects:
            Estimate Std. Error t value
(Intercept)     56.4       12.0    4.69
summary(m2_lm)

Call:
lm(formula = Score ~ 1, data = dta2)

Residuals:
   Min     1Q Median     3Q    Max 
-46.44  -6.44  -4.44  33.56  39.56 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)     57.4       10.8    5.34    7e-04

Residual standard error: 32.3 on 8 degrees of freedom

結果顯示：

  • 相對Data set 1 , 在此資料中ICC為0.1，顯示組內異質性較高，若忽略組間差異直接進行迴歸分析，在多層次模型及迴歸分析中估計的總平均分數分別為56.4(SE=12.0)與57.4(SE=10.8)，即兩者的總平均分數的估計與標準誤差異不大。