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m4 <- lmer(Obs ~ Treatment * Day + (1 + Treatment/Subject) + (1 + Day*Treatment/Subject), mydata)

How do I deal with the fact that subject is nested within treatment? How does m1 differ from

mydata <- data.frame(
  Subject = c(13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65), 
  Day = c(rep(c("Day1", "Day3", "Day6"), each=28)), 
  Treatment = c(rep(c("B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C", "A", "A"), each = 4)), 
  Obs = c(6.472687, 7.017110, 6.200715, 6.613928, 6.829968, 7.387583, 7.367293, 8.018853, 7.527408, 6.746739, 7.296910, 6.983360, 6.816621, 6.571689, 5.911261, 6.954988, 7.624122, 7.669865, 7.676225, 7.263593, 7.704737, 7.328716, 7.295610, 5.964180, 6.880814, 6.926342, 6.926342, 7.562293, 6.677607, 7.023526, 6.441864, 7.020875, 7.478931, 7.495336, 7.427709, 7.633020, 7.382091, 7.359731, 7.285889, 7.496863, 6.632403, 6.171196, 6.306012, 7.253833, 7.594852, 6.915225, 7.220147, 7.298227, 7.573612, 7.366550, 7.560513, 7.289078, 7.287802, 7.155336, 7.394452, 7.465383, 6.976048, 7.222966, 6.584153, 7.013223, 7.569905, 7.459185, 7.504068, 7.801867, 7.598728, 7.475841, 7.511873, 7.518384, 6.618589, 5.854754, 6.125749, 6.962720, 7.540600, 7.379861, 7.344189, 7.362815, 7.805802, 7.764172, 7.789844, 7.616437, NA, NA, NA, NA))

Any help would be immensely appreciated.
~A Baffled Biologist

m4 <- lmer(Obs~Treatment*Day + (1+Treatment/Subject) + (1+Day*Treatment/Subject), mydata)

How do I deal with the fact that subject is nested within treatment? How does m1 differ from:

mydata <- data.frame(
  Subject  = c(13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 
               34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 
               19, 20, 21, 22, 23, 24, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 
               40, 62, 63, 64, 65, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 
               29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 62, 63, 64, 65), 
  Day       = c(rep(c("Day1", "Day3", "Day6"), each=28)), 
  Treatment = c(rep(c("B", "A", "C", "B", "C", "A", "A", "B", "A", "C", "B", "C", 
                      "A", "A", "B", "A", "C", "B", "C", "A", "A"), each = 4)), 
  Obs       = c(6.472687, 7.017110, 6.200715, 6.613928, 6.829968, 7.387583, 7.367293, 
                8.018853, 7.527408, 6.746739, 7.296910, 6.983360, 6.816621, 6.571689, 
                5.911261, 6.954988, 7.624122, 7.669865, 7.676225, 7.263593, 7.704737, 
                7.328716, 7.295610, 5.964180, 6.880814, 6.926342, 6.926342, 7.562293, 
                6.677607, 7.023526, 6.441864, 7.020875, 7.478931, 7.495336, 7.427709, 
                7.633020, 7.382091, 7.359731, 7.285889, 7.496863, 6.632403, 6.171196, 
                6.306012, 7.253833, 7.594852, 6.915225, 7.220147, 7.298227, 7.573612, 
                7.366550, 7.560513, 7.289078, 7.287802, 7.155336, 7.394452, 7.465383, 
                6.976048, 7.222966, 6.584153, 7.013223, 7.569905, 7.459185, 7.504068, 
                7.801867, 7.598728, 7.475841, 7.511873, 7.518384, 6.618589, 5.854754, 
                6.125749, 6.962720, 7.540600, 7.379861, 7.344189, 7.362815, 7.805802, 
                7.764172, 7.789844, 7.616437, NA, NA, NA, NA))
                

So, is this the correct syntax to use?

UPDATE OK, so I went and talked to a statistician, but he's an SAS user. He thinks that the model should be:

Treatment + Day + Subject(Treatment) + Day*Subject(Treatment)

Obviously his notation is different from the R syntax, but this model is supposed to account for:

• Treatment (fixed)
• Day (fixed)
• the Treatment*Day interaction
• Subject nested within Treatment (random)
• Day crossed with "Subject within Treatment" (random)

So, is this the correct syntax to use?

m4 <- lmer(Obs ~ Treatment * Day + (1 + Treatment/Subject) + (1 + Day*Treatment/Subject), mydata)

I'm particularly concerned about whether the Day crossed with "Subject within Treatment" part is right. Is anyone familiar with SAS, or confident that they understand what's going on in his model, able to comment on whether my sad attempt at R syntax matches?

Here are my previous attempts at building a model and writing syntax (discussed in answers & comments):