Split-plot analysis with four factors in SAS

Question

I'm trying to correctly model a split-plot design with more than two factors (specifically, four) in SAS. I'm interested in an example from Design and Analysis of Experiments (Montgomery , 8th Ed.), and would preferably use PROC MIXED (PROC GLM could be okay too, but PROC MIXED seems to do a better job with these types of designs in general from what I've read).

I'm first going to consider a more simple split-plot design - that with only two factors. Ultimately, I'm interested if anyone could tell me how to augment this code to reflect the change from two to four factors (as outlined below).

First, starting with the more simple case: The split-plot design with two factors is also given in Montgomery starting on pg. 621:

Variables:

• Replicate: Replicates/blocking factor (three replicates)
• Method: Pulp preparation method (three levels)
• Temp: Temperature (degrees F; four levels)
• Strength: Paper tensile strength (outcome)

Description of data collection: A batch of pulp is produced by one of the three methods, then the batch is split-up into four parts, and each is treated with one of the four temperature settings (i.e., temperature is nested within method). Finally, the paper tensile strength is measured. Then, another batch of pulp is made and the process repeated until all data has been collected. This equates to nine total batches (three pulp preparation methods, each replicated three times).

Data for importing into SAS:

DATA Mont;
INPUT Replicate Method Temp Strength;
DATALINES;
1   1   200 30
1   2   200 34
1   3   200 29
2   1   200 28
2   2   200 31
2   3   200 31
3   1   200 31
3   2   200 35
3   3   200 32
1   1   225 35
1   2   225 41
1   3   225 26
2   1   225 32
2   2   225 36
2   3   225 30
3   1   225 37
3   2   225 40
3   3   225 34
1   1   250 37
1   2   250 38
1   3   250 33
2   1   250 40
2   2   250 42
2   3   250 32
3   1   250 41
3   2   250 39
3   3   250 39
1   1   275 36
1   2   275 42
1   3   275 36
2   1   275 41
2   2   275 40
2   3   275 40
3   1   275 40
3   2   275 44
3   3   275 45
;
RUN;

PROC MIXED code:

PROC MIXED DATA=Mont;
   CLASS Replicate Method Temp;
   MODEL Strength = Method | Temp / DDFM=SATTERTH; 
   RANDOM Replicate Replicate*Method;
RUN;

PROC MIXED results:

                             Type 3 Tests of Fixed Effects

                                    Num     Den
                    Effect           DF      DF    F Value    Pr > F

                    Method            2       4       7.08    0.0485
                    Temp              3      18      36.43    <.0001
                    Method*Temp       6      18       3.15    0.0271

Now, for the four-factor split-plot design, Montgomery gives the following scenario (pg. 627): An experiment is conducted in a furnace to grow an oxide on a silicon wafer. The response variable is oxide layer thickness. There are four factors: temperature (A), gas flow (B), time (C) and wafer position on the furnace (D). These all have two levels. He then goes on to say, "Now factors A and B are difficult to change, whereas C and D are easy to change... Notice both replicates of the experiment are split into four whole plots, each containing one combination of the settings of temperature and gas flow. Once these levels are chosen, each whole plot is split into four subplots and a $2^2$ factorial in the factors time and wafer position is conducted, where the treatment combinations in the subplot are tested in random order. Only four changes in temperature and gas flow are made in each replicate, whereas the levels of time and wafer position are completely randomized." This is depicted as:

I'm specifically interested in this scenario - where the whole-plot level has two factors, and so does the sub-plot. Can anyone indicate how the PROC MIXED code above would need to change to account for this design? Also, may inference (F statistic and p-value) from the output be used as is, or do p-values need to be calculated manually (as is the case with split-split plot designs)?

Answer 1

I am pressed for time but I believe

 PROC MIXED DATA=TextBook ;
    CLASS Block A B C D ;
    MODEL Thickness = A|B|C|D / DDFM=SATTERTH; 
    RANDOM Block Block*A Block*B Block*A*B ;
 RUN;

You should only consider this a half-baked answer but this seems likely to work.