[R-sig-ME] Random effects in a nested factorial design - Problems understanding the effect of reference categories

[michael.zombok at googlemail.com]

Hello,

I have problems understanding the following different, but somewhat
related mixed models and would thankful for any comments or references.

Suposse you have repeated measurements from an experiment with two factors
- factor 1 treatment: notreatment/treatment
- factor 2 time: t1/t2/t3

Every subject is either in the notreatment or treatment group and is
clearly distinguished by an unique ID (named IDS) in the data. So the
data (given in the long format) looks like the following:

IDS treatment time resp
1 TR T1 2.3
1 TR T2 4.2
1 TR T3 8.2
2 TR T1 3.2
2 TR T2 3.1
.
.
10 NT T1 3.2
10 NT T2 3.5
10 NT T3 3.2
11 NT T1 1.2
11 NT T2 3.5
11 NT T3 2.2

Both factors are treated 'as.factor' in R and not as numeric. One
is interested in the effects of both, treatment and time.

As the intercept in this model is the level in the response of the
notreatment group at T1, I wonder what is modelled with the following model:

resp ~ treatment + time + (1|IDS)            #M1

? Only the variability of the subjects in the no treatment group at time
T1 or the overall variability of all subjects? 

What is the difference to the following two models: 

resp ~ treatment + time + (1 + treatment|IDS)       #M2
resp ~ treatment + time + (0 + treatment|IDS)       #M3

As every subject is in only one treatment group, the overall error
variance remains the same, but the random effect variance is somehow
divided - how? Do the individual estimates make sense? Can they be interpreted?

An additional question: How do I model the variability of subjects over
time in this setting with random effects?

resp ~ treatment + time + (1 + time|IDS) ?

If I am correct, the intercept in this model refers to the variability
of the subjects at T1? Again: Does the intercept refer to all subjects
or only the subject of the reference category?

I appreciate every comment or reference. Thank you very much!

Best,
Friedericksen