In some experiment designs you must restrict the randomization. Otherwise it wouldn’t be practical to perform the runs. For example, temperature cannot be easily changed in many applications. The solution may be a “split-plot” design, which originated in the field of agriculture. Experimenters divided large areas of land, called “whole plots,” into smaller “subplots” that could be treated separately. For example, they planted several crop varieties—one each per whole plot, and then applied different fertilizers on each subplot.
The analysis of a split-plot design is tricky, even for statisticians. But Design Expert® version 11 software makes it easy.
To illustrate how Design-Expert does design and analysis of split plots, let’s follow an example from Montgomery’s Design and Analysis of Experiments. Feel free to skip the boxed bits—these are sidebars for those who want to know all the details and spend more time to explore things.
Details on this case study: Refer to Section 14.4 “The Split-Plot Design” in the 8th Edition of the book referenced above—published by Wiley in 2012—for more details on the experiment’s structure and this particular case. It’s also covered in Stat-Ease’s computer intensive workshop on Factorial Split-Plot Designs for Hard-to-Change Factors.
A manufacturer wants to determine the effect of three methods of pulp preparation and four levels of cooking temperature on the tensile strength of paper. These twelve combinations can be conducted in one day.
To increase statistical power, the paper chemist decides to perform three replicates of this twelve-run general factorial design over three days, which produces a total of 36 runs. However, it is not feasible to perform the experiment in a random fashion. Instead, on each of the three days, this experimenter produces a batch of pulp and divides it into four samples to be cooked at four different temperatures. The process is repeated for pulp prepared by the two other methods.
A flowchart of the experiment is shown below.
In statistical terms, the split plot experiment can be structured as:
You will set up this design as a blocked (by day) split-plot general factorial.
Let’s build the design. This is done via the usual procedure for general factorials, with days accounted for as blocks. From the Design-Expert welcome screen, click the New Design Button (or click the blank-sheet icon on your toolbar, or choose File, New Design).
Then, from the default Factorial tab under Split-Plot click Multilevel Categoric. Choose 2 as the number of factors. Click the Vertical radio button. Then enter Pulp Prep as your Factor A name. It’s already set up as a Hard-to-Change factor, so leave that be. Mouse down to the Levels column and type in 3. Then enter a1, a2, and a3 as your L(1), L(2), and L(3) treatment names.
Still on the same screen, name your second factor Temp, with deg F for units, 4 for the number of levels, and 200, 225, 250, and 275 as your treatments. This factor is Easy to Change as already set by the program. For Type select Ordinal, thus denoting its numeric nature. Your screen should now match that shown below.
For Replicates enter 3 and check on (√) Assign one block per replicate to produce 36 runs (36=3x3x4 levels from the three factors) in 9 groups.
Press Next. Now for Block Names enter Day 1, Day 2 and Day 3.
Press Next again and for the one response Name enter Tensile in Units of strength. Then to assess the power of this design, type in Delta of 6 and Sigma of 2. Also, change the variance ratio to 0.25.
Next click Edit Model… and change the Process Order to 2FI as shown below and press OK.
Design-Expert defaults to a variance ratio of 1 for group (whole plot) to residual variance. But in this case, the whole-plot factor, although hard to change, can be controlled with little variation relative to the sub-plot factor of temperature.
P.S. The changing of the order presumes that the experimenters must be interested in the interaction of the pulp prep with temperature, otherwise why bother doing it at all. Therefore they want to know if this model term (aB) can be estimated with a ‘decent’ (80% being generally acceptable) level of power.
Press Next and review the power for the design. Notice that, due to the restriction in randomization in the split plot, the power for seeing effects of the hard-to-change factor “a” (Pulp prep) is reduced, whereas the power for seeing the effects of the easy-to-change factor “B” (Temperature) and its interaction “aB” are increased, albeit only slightly.
Due to the structure of this design, interactions of hard to change (whole plot) factors with ones that are easy to change (sub-plot) must be tested against the sub-plot error.
Notice the loss in power for the hard-to-change factor “a” (Pulp Prep) still leaves it above the recommended minimum. Finish to complete the design-building wizard. Press OK on the warning to reset factors group-to-group. Design-Expert now displays your design in groups by level of the hard-to-change factor “a” (Pulp prep), within which the easy-to-change factor “B” (Temperature) is laid out in completely random order.
At this stage the design would normally be printed as a run-sheet for the actual experiment, with resulting responses of tensile strength recorded.
Load the design by clicking Help, Tutorial Data -> Paper Pulp.
The analysis of a split-plot design must be done differently than a completely randomized experiment. To get the correct statistical tests for each, Design-Expert creates two separate ANOVAs for:
Then it fits the full model in order to get meaningful diagnostics and model graphs.
Start by clicking the Analysis node labeled R1:Tensile which can be found in the tree structure along the left of the main window. Then press ahead to the Model tab displayed in the progressive toolbar. Design-Expert preselects terms for the designed-for two-factor interaction (2FI) model—these are labeled with a green . Enable the program to check if any terms can be safely taken out of the model by clicking the Auto Select… button. Then, change the criterion to p-values and the Selection to Backward as shown in the screen shot below.
Click the Start button and the selection algorithm is detailed with any terms that exceed the “Alpha out” criterion (p-value of 0.1 by default) being removed from the model. They will be re-designated to Error () after you click Accept. If after looking over the statistics on the ANOVA, you want to remove a term, simply go back and double-click it (or right-click for a menu of designation choices).
Heads-up on effects selection for general-factorial split plots: Due to the structure of these designs, the handy graphical half-normal plot of effects is not produced like it would be for an experiment with factors only at two levels. Instead, run one of the Auto Select… algorithms (we used backward in this case)—details for which can by clicking Help button.
Continue to the ANOVA (REML) tab and review the p-values for significance. In this case all the designed-for model terms survive the elimination process, that is, they are all statistically significant.
A note about terminology: As you probably know, ANOVA is the acronym for analysis of variance—the fundamental approach to assessing model significance for all DOE where runs can be completely randomized. However, in a split plot the components of variance are estimated by a different approach called REML, which stands for restricted maximum likelihood estimation. Sometimes, depending on the structure of the design, REML agrees with ordinary-least-squares ANOVA exactly, but often the results differ somewhat.
Details on statistics presented in ANOVA (REML) report: Search Help, Contents for much more information, including references, on everything you see presented on this screen. Since these tutorials only provide the highlights, we must move on.
Move on to the next phase of analysis—verifying the statistical assumptions—by pressing the Diagnostics tab. First up is the normal plot of the residuals— externally studentized to provide the most effective view of any obvious discrepancies. There are no obvious irregularities in this case—the points fall in line as expected from normal data.
Examine all the graphs available via the Diagnostics tool. You should see nothing abnormal.
Press ahead to Model Graphs. On the Factors Tool right-click the B: Temperature bar and change it to the X1 axis.
You now see an interaction plot of the continuous factor temperature at three discrete levels of pulp preparation. From this graph we can see pulp preparation a2 (symbolized by the green triangles) will maximize tensile strength. Evidently it will be most robust to temperature changes, especially in the temperature range of 225 to 275 degrees F.
This is an ingenious application of designed experimentation using a split-plot structure to handle the hard-to-change pulp preparation. Design-Expert provides valuable insights on how power is reduced by restricting randomization. In this case the actual effect of pulp prep (factor “a”) far exceeded that of minimal importance so the experiment succeeded.