The split plot design is often employed in a randomized complete block design, where one factor is applied to whole plots forming a complete block, and then the second factor is applied to sub-plots within the whole plots within each block.
As an example (adapted from Hicks, 1964), consider an experiment where an electrical component is subjected to different temperatures for different amounts of time. They wanted to have 3 replications and used a RCBD design (replication = block). However, it was impractical to run individual treatment combinations, and instead they opted to set the oven a temperature and then take out randomly selected components at different times. This process was repeated for each temperature until a replication was complete.
The data (Bake Time Data) were:
|Baking Time (min)||Over Temperature (degrees F)|
In SAS, we could specify the full model with the following statements:
model resp=temp time temp*time; random rep rep*temp rep*time rep*temp*time;
but there is a problem:
We can’t retrieve the residual (there are 0 d.f. for error). If repeat observations are made within the split plots, then a separate error term can be estimated. However, it is important to keep in mind that tests of replication effects are not of interest, but are being isolated in the ANOVA to reduce the error variance. As a result, the model that is usually run in this design drops out the rep × split plot factor and the rep*whole-plot factor*split-plot factor terms. This has the effect of combining these interactions with the true error variance, for a working error term.
So the Split-Plot RCBD ANOVA can be obtained as:
proc mixed data=baketime method=type3; class rep temp time; model resp=temp time temp*time; random rep rep*temp; run;
|Source||DF||Expected Mean Square|
|rep||2||Var(Residual) + 3 Var(rep*temp) + 12 Var(rep)|
|temp||3||Var(Residual) + 3 Var(rep*temp) + Q(temp, temp*time)|
|rep*temp||6||Var(Residual) + 3 Var(rep*temp)|
|time||2||Var(Residual) + Q(time, temp*time)|
|temp*time||6||Var(Residual) + Q (temp*time)|
In this model the rep*time and rep*temp*time are included in the residual error. Notice that the correct error term for the F test of the treatment applied to whole plots is the block × treatment interaction (assuming blocks are a random effect).
In Minitab, use
Stat > ANOVA > General Linear Model
and then fill in the dialogue boxes provided:
Notice that in Minitab all effects have to appear in the Model box, and that we are not specifying all the random effects in the Random factors box. Unlike SAS, where we do specify all the random effects in the random statement, in Minitab we only specify the simple factor name in the Random factors box. Minitab automatically treats interactions with the random effect as random effects and doesn’t ask you to include these in the Random factor dialogue box. It is a common mistake when using this program to include all the random factor interactions in this dialogue box and find that Minitab won’t run the model.
Analysis of Variance for resp, using Adjusted SS for Tests
|Source||DF||Seq SS||Adj SS||Adj MS||F||P|
S = 24.9165 R-Sq = 66.13% R-Sq(adj) = 25.92%