[R] how to analyze this design using lmer

[Adaikalavan Ramasamy]

Dear all,

A friend of mine requested me to analyze some data she has generated. I 
am hoping for some advice on best way of properly analyzing the data as 
I have never worked with such complicated or nested designs.

Here is the setup. She has taken material from 5 animals and each 
material is subdivided into 6 plate (30 plates in total). Each plate is 
then assigned as either a control or a treated with a chemical AND kept 
at one of three concentrations. A sample is taken daily from each plate 
for six continuous days and measured (180 measurement in total). Her 
main question is whether treatment has an effect.

Here is a simulated dataset:

  df <- expand.grid( animal=LETTERS[1:5], group=c("Control", "Treated"),
                     conc=c("X", "Y", "Z"), day=1:6 )
  df$plate <- as.numeric(factor(apply(df[ ,1:3], 1, paste, collapse="")))
  df <- df[ order(df$plate), ]
  df$plate <- as.factor(df$plate)
  rownames(df) <- NULL

  set.seed(1066)
  df$value <- runif(90, 1, 2)*(df$group=="Control") +
              c(0, -0.5, -0.20)[as.numeric(df$conc)] +
              rnorm(30)[ as.numeric(df$plate) ] +
              runif(180, 0.9, 1.1)*df$day + rnorm(180, sd=0.5)

  df[1:10, ]
      animal   group conc day plate     value
  1        A Control    X   1     1 3.3403510
  2        A Control    X   2     1 5.1042965
  3        A Control    X   3     1 5.4003462
  ...
  ...
  178      E Treated    Z   4    30 2.8558186
  179      E Treated    Z   5    30 4.4567206
  180      E Treated    Z   6    30 5.4542460


I have tried analyzing the data as follows:
library(lme4)
lmer( value ~ group + day + conc + (1 | animal/plate), data=df )
lmer( value ~ group + day + conc + (1 | animal), data=df )
lmer( value ~ group + day + conc + (1 | plate), data=df )

BUT I am not sure which of the models above is appropriate. Any advice 
would be very useful. Many thanks in advance.

Regards, Adai