[R-sig-ME] repeated measure in partially crossed design

[matteo dossena]

Dear All,

I'm having a some doubts on how to properly specify the random effect  in lmer for the following experiment:

I have 10 subject assigned to treatment A and 10 assigned to treatment B. For each subject I measured 2 variables, V1 and V2, and these measures were repeated in season A and season B.

I'm interested in assessing the effect of treatment, season and their interaction on the relationship between the two variables. 

I'm a little confused on how I have to specify the random structure.

According to the design  I identified the following random structure

(1 + V2 | treatment/subject) + (1 + V2 | season)

where the first term account for the nested design and temporal pseudorplication
and the second term for the other unmeasured variable that covary with season and could affect the within season relationship between V1 and V2

correct?

Then i performed a model selection procedure, to investigate which would be the best random structure. To do so I subsequently removed each term from the random structure, and  compared the AIC score.
From this procedure it turned out that the best random structure is:  (1 + var2 | season).

My concern here is, if subject is no longer included in the random structure, i'm no longer accounting for the pseudorplication, am I?

Is the fact that the term (1 + V2 | treatment/subject) does not improve the model fit telling that in fact there is no significant correlation between measures on the same subject?


To further explore the issue, i also analysed the following random structure:

(1 + V2 | subject) + (1 + V2 | season)

and in this case the model selection procedure identified this as the best random structure.

So here is where i got confused.
My question is how do i properly specify the random structure?

Searching for an answer to my doubt in the literature, in Crawley 2007, i came across a further way to account  for temporal pseudoreplication (in lme: random = ~ season | subject) translated in lmer would it be

(season | subject) ? 

at this stage I had to give up and ask for help.
Could anyone give me some advice? is my strategy somehow wrong?

Cheers 
matteo