Split plot with factorial design vs three way ANOVA

Question

I have three site temperature types: cold, medium, and hot. This is a fixed effect - I am specifically am testing if temperature affects my response variable. I believe temperature type is my 'block'.

I will have three sites for each temperature type. I think site is a random effect. Within each site, I will have a factorial design, where two other fixed factors, A and B, are crossed.

I am interested in the interaction of all three fixed effect (tempxAxB).

From reading my stats book, I think this is a split plot design, but my stats book does not have information about factorial design (the cross of A and B) within blocks. Is what I just described a split plot design or a three way ANOVA? And if it is a funky split plot, could someone point me to an ANOVA table (how to calculate sum of squares) with a similar set up?

Here is a schematic of the setup:

Answer 1

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).