The R Book

Random Effects in Designed Experiments

The rats example, studied by aov with an Error term on p. 476, can be repeated as a linear mixed-effects model. This example works much better with lmer than with lme.

dd<-read.table("c:\\temp\\rats.txt",h=T)
attach(dd)
names(dd)

[1]  "Glycogen"  "Treatment"  "Rat"  "Liver"

Treatment<-factor(Treatment)
Liver<-factor(Liver)
Rat<-factor(Rat)

There is a single fixed effect (Treatment), and pseudoreplication enters the dataframe because each rat's liver is cut into three pieces and each separate liver bit produces two readings.

The rats are numbered 1 and 2 within each treatment, so we need Treatment as the largest scale of the random effects.

model<-lmer(Glycogen~Treatment+(1|Treatment/Rat/Liver))
summary(model)

Linear mixed-effects model fit by REML
Formula: Glycogen ~ Treatment + (1 | Treatment/Rat/Liver)
    AIC    BIC   logLik    MLdeviance    REMLdeviance
231.6    241.1   -109.8         234.9           219.6

Random effects:
Groups                     Name           Variance  Std.Dev.
Liver:(Rat:Treatment)      (Intercept)     14.1617    3.7632
Rat:Treatment              (Intercept)     36.0843    6.0070
Treatment                  (Intercept)      4.7039    2.1689
Residual                                   21.1678    4.6008

number of obs: 36, groups: Liver:(Rat:Treatment), 18; Rat:Treatment, 6;
Treatment, 3

Fixed effects: Estimate Std. Error t value (Intercept) 140.500 5.184 27.104 Treatment2 10.500 7.331 1.432 Treatment3 -5.333 7.331 -0.728 Correlation of Fixed Effects: (Intr) Trtmn2 Treatment2 -0.707 ...