7.1 - Split-Plot in RCBD

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)
Rep   580 600 620 640
I 5 217 158 229 223
  10 233 138 186 227
  15 175 152 155 156
II 5 188 126 160 201
  10 201 130 170 181
  15 195 147 161 172
III 5 162 122 167 182
  10 170 185 181 201
  15 213 180 182 199

 Using SAS

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:

Source DF
  (Whole Plots)
rep 2
temp 3
rep*temp 6
  (Split Plots)
time 2
temp*time 6
rep*time 4
rep*temp*time 12
Residual 0

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;
Source DF Expected Mean Square
(Whole Plots)    
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)
Residual 16 Var(Residual)

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).

  Using Minitab

In Minitab, use

Stat > ANOVA > General Linear Model

and then fill in the dialogue boxes provided:

minitab GLM dialog box

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
rep 2 1962.7 1962.7 981.4 3.32 0.107
temp 3 12494.3 12494.3 4164.8 14.09 0.004
rep*temp 6 1773.0 1773.9 295.7 0.48 0.816
time 2 566.2 566.2 283.1 0.46 0.642
temp*time 6 2600.4 2600.4 433.4 0.70 0.655
Error 16 9933.3 9933.3 620.8    
Total 35 29331.0        

S = 24.9165  R-Sq = 66.13%  R-Sq(adj) = 25.92%

Note! Minitab is not able to generate mean comparisons for the whole-plot factor – we need the Tukey-Kramer Adjustment and Minitab does not currently have this capability. SAS should be used for the Split-plot designs