Nominal labeling structure impacting results for random effect in mixed model

[#1]

Background: I had 6 individuals each assigned to either the treatment or control groups (3 per group). I wanted to compare the effect of the treatment and have included the "individual" as a random effect in my mixed model and "treatment" (T or C) as a fixed effects.

Issue: I am finding that I get different results based on how I label my replicates (individuals). In the experiment I had the individuals organised as 1,2,3, in the treated group and 4,5,6, in the control group. This was how I was including "individual" in my mixed model originally but I have been since advised that this is incorrect and confounds treatment effects because individuals weren't repeated across treatment and each individual was only assigned one treatment. Instead, I needed to have the nominal values in a repeated structure across treatments (1,2,3 in C and 1,2,3, in T). I get different results depending on how I have the "individual" random replicate effects structured:  (T-1,2,3, and C-1,2,3) vs (T-1,2,3 and C-4,5,6,)

Question: I dont fully understand why a repeated structure across treatments is correct and I am looking for further clarification. 

[#2 · Accepted solution]

Sounds like you got some bad advice.  Unless you nest Subject within Treatment, JMP treats the data as if you have 3 subjects that were measured in both the control and treatment group.  In reality, you have 6 subjects assigned to just 1 group.  That's why you're getting a mismatch, and as far as I can tell you had it right the first time.

If you use the second labeling method, you should get the same result if you nest subject ID in treatment.  I attached an example to illustrate where I generated some data for a fake study with 4 subjects in each treatment level measured at 0 and 1 weeks.  I created 2 subject ID columns to illustrate I could do it either way and get the same results if I nest it properly.  The first script uses unique IDs for each subject.  The second uses nominal IDs that are repeated for each treatment group, but I nest Subject within Treatment.  I get the exact same numbers.  This is because the nesting tells JMP that Subject 1 in the T group is a different person than Subject 1 in the C group.

The third script shows the incorrect way without the nesting.  Now the results are different because JMP treats Subject 1 in both groups as the same person and thinks I have 4 measurements on this person.  This is essentially what I think you have done with your data.

[#3]

Thank you very much for the informative response! Nesting subject within treatment gave me the same result as when I ran each subject individually. It is now clear that I need to run it with each individual having a unique nominal identifier and not in a repeated structure across treatment (b.c. jmp will run it as a repeated measure across trts).

[#4]

I also meant to ask if you have more than 1 data point on each person?  For example, in the data set I uploaded I measured them at 2 time periods.  If you only have 1 data point, then Subject is confounded with the residual (not with the treatment as you were advised).  If that's the case, that's not really a problem.  JMP just drops "Residual" from the REML variance component table and replaces it with "Subject."  Your treatment effect is still totally estimable.