```

References

Load packages

library(magrittr)
library(nlme)
library(lme4)
library(ggplot2)

Examine data

data(BodyWeight)
ggplot(data = BodyWeight, mapping = aes(x = Time, y = weight, group = Rat, color = Diet)) +
    geom_line() +
    theme_bw() + theme(legend.key = element_blank())

Models

## Random intercepts model
lmer1  <- lmer(formula = weight ~ Diet + Time + (1 | Rat),
               data = BodyWeight)
## Print
lmer1
## Linear mixed model fit by REML ['lmerMod']
## Formula: weight ~ Diet + Time + (1 | Rat)
##    Data: BodyWeight
## REML criterion at convergence: 1304.284
## Random effects:
##  Groups   Name        Std.Dev.
##  Rat      (Intercept) 36.577  
##  Residual              8.176  
## Number of obs: 176, groups:  Rat, 16
## Fixed Effects:
## (Intercept)        Diet2        Diet3         Time  
##    244.0689     220.9886     262.0795       0.5857
## Summary
lmer1
## Linear mixed model fit by REML ['lmerMod']
## Formula: weight ~ Diet + Time + (1 | Rat)
##    Data: BodyWeight
## REML criterion at convergence: 1304.284
## Random effects:
##  Groups   Name        Std.Dev.
##  Rat      (Intercept) 36.577  
##  Residual              8.176  
## Number of obs: 176, groups:  Rat, 16
## Fixed Effects:
## (Intercept)        Diet2        Diet3         Time  
##    244.0689     220.9886     262.0795       0.5857
## Estimated variance-covariance matrix for fixed effects estimates.
## Fixed effects are fixed quantities, thus, their true values do not have
## variance. Their estimates, however, are functions of data, which are
## considered random. The estimates, thus, have variability from data to data.
## The true variance of the fixed effects estimates is another set of unknown
## quantities, thus, they are estimated.
vcov(lmer1)
## 4 x 4 Matrix of class "dpoMatrix"
##               (Intercept)         Diet2         Diet3          Time
## (Intercept)  169.12400730 -1.679945e+02 -1.679945e+02 -3.367172e-02
## Diet2       -167.99447412  5.039834e+02  1.679945e+02  2.240254e-16
## Diet3       -167.99447412  1.679945e+02  5.039834e+02  2.240254e-16
## Time          -0.03367172  2.240254e-16  2.240254e-16  1.003764e-03
## Estimated variance for the random intercepts. The true intercepts vary across
## individuals. Thus, their true values have variance. The estimate of this true
## variance, has its own uncertainty (variance), but lmer does not seem to give it.
summary(lmer1)$varcor$Rat
##             (Intercept)
## (Intercept)    1337.878
## attr(,"stddev")
## (Intercept) 
##    36.57702 
## attr(,"correlation")
##             (Intercept)
## (Intercept)           1