Treating site and year as an integrated factor I’m analyzing an experiment where we collected data across 4 different treatments. These treatments were replicated over two years.
In each year, we set-up the experiment at 2 different sites.

*

*In year 1, sites A and B.

*In year 2, sites A and C. (We used site A twice but switched out site B for a new location in year 2).

Typically, I would include site and year as factors in the model, each with two-levels. However, since this is unbalanced (we have different sites in yrs 1 and 2) I was thinking about creating a new variable called site-year, which combines the two into a single variable with four-levels (1A,1B,2A,2C).
Are there any benefits or drawbacks to considering site and year separately vs. as an integrated single variable?
 A: With your design it is not possible to estimate an interaction between site and year, so if you use an additive model with factors site and year, or one combined factor siteyear, the results will be the same. One fast way to see this is to do some experiments with R:
N <- 5
year <- factor(rep(1:2, each=2*N))
site <- factor( c(rep(c("A", "B", "A", "C"), each=N))  )

siteyear <- interaction(site,  year)

and then calculating degrees of freedom this way, for three models:
 Matrix::rankMatrix( model.matrix(  ~ site + year)  )

Matrix::rankMatrix( model.matrix(  ~ site * year)  )

Matrix::rankMatrix( model.matrix(  ~ siteyear)  )


[1] 4
attr(,"method")
[1] "tolNorm2"
attr(,"useGrad")
[1] FALSE
attr(,"tol")
[1] 4.440892e-15
> > 
[1] 4
attr(,"method")
[1] "tolNorm2"
attr(,"useGrad")
[1] FALSE
attr(,"tol")
[1] 4.440892e-15
> > 
[1] 4
attr(,"method")
[1] "tolNorm2"
attr(,"useGrad")
[1] FALSE
attr(,"tol")
[1] 4.440892e-15

Note the rank is 4 in all three cases, so for the results it does not really matter which you use.  Your tags indicate that you want a random effects model, I'm not sure that's a good idea with so few levels, but the conclusion is the same.
