I have a crossover experiment design, detailed as follows.

1. there are 7 sites conducting the same experiment;
2. In the experiment of each sites, 5 different treatments are administered to $n_i, \, (i = 1, 2, \dots, 7)$ subjects during 5 different periods. The $n_i$'s are not necessarily equal;
3. In this experiment, only the fixed treatment effect and random subject effect are of interest. Other effects are ignored.

The variance of the random subject effect is thus the between-subject variability, and the remaining variability from experiment error is within-subject variability.

Three mixed-effects models in terms of different sites are investigated.

(1) Site 1 and 2 combined; (2) Sites other than 1 and 2; (3) All sites.

In each mixed-effect model, fixed treatment effect and random subject effect are specified. After running the three models, I got to know that the within-subject variability for the three are,

(1) 0.7904; (2) 0.8203; (3) 0.8526

respectively.

However, I was told that there results should NOT be like this. Instead, intuitively the within-subject variability from Model (3) should be less than that from Model (2). The argument is that since Model (1) has the smallest within-subject variability, then intuitively if we combine the sites in Model (1) with those in Model (2), which results to Model (3), we should expect the within-subject variability decrease.

I cannot be persuaded this the simple intuition. My questions are:

(1) Is the above argument against the results correct?

(2) Are there clearer intuitions that help explain this problem?

(3) Are there any proofs or arguments from theoretical aspect (I mean by derivations) to help understand this?

Any comments or ideas are welcome. Thanks in advance!