Split-Plot Multilevel Categoric Factorial

Introduction

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 Stat-Ease^® software makes it easy.

To illustrate how Stat-Ease 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.

Background for Case Study

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:

• Whole plots for the three batches of pulp (hard-to-change factor)

• Subplots for the four samples cooked at four different temperatures (easy to change factor).

You will set up this design as a blocked (by day) split-plot general factorial.

Design the Experiment

Let’s build the design. This is done via the usual procedure for general factorials, with days accounted for as blocks. From the 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 (HTC), so leave that be (note that HTC factors are lower cased). 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.

Note the warning due to lack of power on the HTC factor. To remedy this, 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.

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.

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. The program 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.

Analyze the Results

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, the program creates two separate ANOVAs for:

• Whole-plot treatment (pulp prep), and

• Subplot treatment (temperature) and the interaction between whole plot and subplot individually.

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. Press Start Analysis and continue on to the Model tab displayed at the top. The program preselects terms for the designed-for two-factor interaction (2FI) model—these are labeled with a green . To check if any terms can be safely taken out of the model, click 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 log 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).

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.

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.

Conclusion

This is an ingenious application of designed experimentation using a split-plot structure to handle the hard-to-change pulp preparation. The program 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.