# Understanding the split plot

Can someone explain me the intuition behind the split plot?

From what I understand, it is essentially restricted randomization. But I still do not quite understand it. Is there a resource or example anyone can give me to make it clearer?

Split plots are often used out of necessity, but there can be statistical advantages in term of precision of your contrasts (or also disadvantages). Here is my rudimentary understanding of intuition for using split plot:

First, let me establish that two common terms in split plot design are "whole plot factor" and the "sub-plot factor." In an agricultural study the whole plot factor are at a larger spatial scale, say entire fields, which represent different levels of some treatment such as drainage efficiency. The sub-plot factors are spatially nested within the whole plot factor. Subplot factors are often something that can be applied at a smaller spatial scale, such as crop type. Aside from reasons of practicality (which may be the case in the example I wrote above), split power may be efficient (or inefficient!). Federer and King 2007 suggest that one reason to use Split plot is that in comparison to a 2-way ANOVA you generally have increased precision to detect contrasts between the sub-plot factors. Also, interaction effects may be easier to detect. In contrast, precision to detect contrasts between the whole plot factor generally decreases.

These differences are explained by the fact that two separate residual error terms are used for hypothesis testing. The whole plot error term is calculated by first averaging the subplots within each whole plot.

Spit plot is also sometimes used as a split plot in time, which as I understand is similar to a repeated measures, often used on subjects. I'm not sure what the advantage one way or another is on this. The terminology maps as follows:

 split-plot design = repeated-measures design
whole plot        = subject
whole plot factor = between-subject factor
split-plot factor = within-subject factor = repeated-measures factor


A very comprehensive reference on split plot theory and implementation is: Federer WT & King F (2007) Variations on split plot and split block experiment designs (John Wiley & Sons).

A good resource would be Mead's "The design of experiments" (1988), chapter 14. I think there is a new version here. But you don't really need the new version to understand split-plot, and I am assuming you have access to these books at your local library.

I can give you my 2-cent-worth. In the ideal world, if you have 2 treatments, you would want to do a factorial design. It is probably the most efficient design you can use. However, there is often practical limitation. Perhaps the 2 treatments have to be applied to different levels of the unit (1 larger, 1 smaller), then you will have to contend with split-plot. So my view of split-plot is that it arises out of practical limitation.

Linking to the notion of restricted randomization, yes, split plot is a type of restricted randomization. The treatment that is applied to the main unit ('larger' plot) is randomized in a restricted sense. But the restriction is posed by practical limitation rather than statistical ideal.