In my opinion your design more strongly resembles a nested design. Individuals
are nested within the sites
. Hence, I would advise to either treat site
simply as a fixed effect covariate (due to the small number of levels) or use some sort of hierarchical modeling (could be difficult, again due to the low number of site
levels).
The question of whether or not you have a split plot design is dependent on the smallest experimental unit. If this would indeed be site
you would have a split-plot (or repeated-measures) design, as your experimental units would undergo different treatments. However, from your description it sounds like you sample experimental units within each site
making those the smallest experimental unit. As long as experimental units are not assigned to more than one condition, it is not a split-plot design. Rather, you seem to have a two-level multilevel model (e.g., sampling students within classes).
As for the analysis, number of blocks and replicates are the same thing in a split-plot design (at the main plot level). Whether you call them blocks or replicates depends on how you arranged them in your experiment.
Let's illustrate this with an example. Suppose you have four pea varieties A, B, C, D and you want to know how their yield differs among three fertilizer brands (X, Y, Z). Such an experiment is often set up in a split-plot design. In this case, the fertilizer brands are the main plots and the pea varieties are the subplots:
-----Y----- -----X----- -----Z-----
___________ ___________ ___________
| C | B | | C | B | | A | C |
|_____|_____| |_____|_____| |_____|_____|
| D | A | | A | D | | D | B |
|_____|_____| |_____|_____| |_____|_____|
Now at this point, your experiment has only one replicate at the main plot level (X, Y, Z), meaning you need replication. Since your study area appears to be heterogeneous (e.g. topography), you decide to block your experiment accordingly. The one replicate (at the main plot level) is also your first block now. Then you take that first block and replicate it with as many blocks as you need across the landscape (let's say three). In this case, number of blocks or replication at the main plot level is the same:
--------------- Block 1 ---------------
-----Y----- -----X----- -----Z-----
___________ ___________ ___________
| C | B | | C | B | | A | C |
|_____|_____| |_____|_____| |_____|_____|
| D | A | | A | D | | D | B |
|_____|_____| |_____|_____| |_____|_____|
--------------- Block 2 ---------------
-----Z----- -----Y----- -----X-----
___________ ___________ ___________
| A | B | | D | B | | B | C |
|_____|_____| |_____|_____| |_____|_____|
| D | C | | A | C | | D | A |
|_____|_____| |_____|_____| |_____|_____|
--------------- Block 3 ---------------
-----X----- -----Z----- -----Y-----
___________ ___________ ___________
| D | C | | A | B | | C | D |
|_____|_____| |_____|_____| |_____|_____|
| B | A | | C | D | | B | A |
|_____|_____| |_____|_____| |_____|_____|
Now you could also arrange your experiment differently if you know that the area where you establish your experiment is homogeneous and no nuisance variables are present that need to be account for. One possibility would be in form of a completely randomized design (CRD). In this case, you replicate (and randomize) the main plot levels (X, Y, Z) first (lets's say three times), and then randomize the subplots (A, B, C, D) into it:
-----Y----- -----X----- -----Z-----
___________ ___________ ___________
| C | B | | C | B | | A | C |
|_____|_____| |_____|_____| |_____|_____|
| D | A | | A | D | | D | B |
|_____|_____| |_____|_____| |_____|_____|
-----Z----- -----Y----- -----X-----
___________ ___________ ___________
| A | B | | D | B | | B | C |
|_____|_____| |_____|_____| |_____|_____|
| D | C | | A | C | | D | A |
|_____|_____| |_____|_____| |_____|_____|
-----X----- -----Z----- -----Y-----
___________ ___________ ___________
| D | C | | A | B | | C | D |
|_____|_____| |_____|_____| |_____|_____|
| B | A | | C | D | | B | A |
|_____|_____| |_____|_____| |_____|_____|
In the CRD setting, since there are no blocks, you are only dealing with replicates.
Best Answer
I think you know this already, but you can calculate the F Statistic by the ratio of MS and the F Statistic follows $F_{df1, df2}$ where df1 and df2 are degrees of freedom of the numerator and denominator.
About your estimating the block effect question:
First, I don't understand what is your block effect? Normally in split plot design time is not the block, but sometimes it does introduce correlation (like block). Generally, the blocks are the subjects (but if they are blocks as in RBD or samples as in CRD depends on the design)
If I understand your question correctly:
your whole plot treatment is the fixed factor treatment and your split plot treatment is t and you have four observation per individual (lets assume a block). You have got a longitudinal mixed effects model (aka repeated measures). If you have access to SAS can easily handle that using the glimmix (or mixed procedure). All you need to do is impose a temporal correlation (generally AR (1)). If you want to use R, please refer to George Casella's website, an incredible resource for R related experimental design programs. The relevant example for you will be Hypertension.R His book's 5th chapter is my favorite piece on split plot design.
Please let me know if I misunderstood you.