# Is a “split plot” ANOVA with two factors the same as two-way ANOVA with repeated measures in one factor?

Is a "split plot" ANOVA with two factors identical to two-way ANOVA with repeated measures in one factor? if not, what is the distinction?

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ANOVA with one repeated-measures factor and one between-groups factor is identical to ANOVA with 3 factors - the formerly repeated-measures factor, the between-groups factor, and the subjects (respondents' ID) factor nested in the previous one.

In SPSS, for instanse, three following commands are equivalent:

(RM-ANOVA):
GLM time1 time2 time3 /*3 RM-factor variables*/
BY group /*between-group factor*/
/WSFACTOR= time 3 /*name the RM-factor of 3 levels*/
/WSDESIGN= time /*within-subject design is it*/
/DESIGN= group /*between-subject design is group*/.

(Split-plot ANOVA):
GLM depvar /*dependent variable as concatenated of time1 time2 time3*/
BY time /*variable indicating RM-levels*/ group subject
/RANDOM= subject /*respondent is a random factor*/
/DESIGN= group subject(group) /*subject nested in group*/ time time*group /*interaction*/.

(Split-plot via mixed models):
MIXED depvar
BY time group subject
/RANDOM= subject(group) /*respondent is a random factor nestes in group*/
/FIXED= group time group*time.

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The case with one between factor, and one repeated-measures factor is one particular example that leads to a split-plot design. In this case, each observational unit (e.g., a participant in an experiment) is observed multiple times. One participant is one "whole plot" (or block). There are N different participants, representing N levels of the blocking factor ID. Now, one group of whole-plots is treated according to level 1 of an experimental factor A (say, a control group), another group of blocks is treated according to level 2 of A (say, is administered a drug).

Now, each whole block is split into multiple "sub-plots". Within each whole block, these sub-plots are treated according to the levels of a second experimental factor B. In your case, B is time, so each participant is observed under different levels of the influence of time, say before the treatment, shortly thereafter, and then again some time later.

There are three factors: The blocking factor ID, the (between) factor A, and the (within) factor B. ID is a random factor, meaning that its levels are not controlled by the experimenter, but are the result of a random sampling process. The levels themselves are not interesting per se, and one wants to generalize the results beyond these particular levels (note that "random factor" is not super well defined, I think there's a blog entry by Gelman that I can't find at the moment). A and B however are experimental (fixed) factors in the proper sense, their levels are interesting per se, intentionally chosen, and repeatably realized by the experimenter. So this is a 3-factorial design with 1 one observation per cell ID $\times$ A $\times$ B.

Importantly, there is a level of nesting, or confounding: Each level of the blocking factor is observed only in one condition of the between-factor A, so ID and A are not crossed. The confounding is that, conversely, each level of A only contains a subset of levels from the blocking factor, but not all of them. (B does, however).

In agricultural terms (the origin of the design name), one whole plot is actually one area of land that is then subdivided into split-plots. In that case, the between factor A is one that is hard to manipulate - the classical example is irrigation, which cannot easily be applied in a different manner to small plots. In the same vein, giving different drugs to the same person at different times is often not feasible (if the person is cured after drug 1, then drug 2 cannot be tested anymore). The second experimental factor B, on the other hand, can easily be manipulated within one whole plot, the classical example being different fertilizers.

As you can see, one whole plot need not be one person observed multiple times. It's just that each whole plot is a homogeneous entity that can be split into sub-plots that are equivalent in some respect. In the social sciences, it could also be one group of subjects that is roughly homogeneous with respect to a nuisance variable, say socio-economic-status, or severity of the illness. In this case, each person within such a homogeneous group is then a split-plot.

As a further read, split-plot designs are explained here, or here.

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+1, this is a really nice contribution. I wonder if this is the Gelman blog post you are referring to. – gung Nov 15 '12 at 15:52
Thanks @gung! That's the exactly the post I had in mind. – caracal Nov 16 '12 at 8:51