Split-Plot Two-Level Factorial

Introduction

Very often, experimenters set up two-level factorial designs with the best intentions of running them in random order, but they find that a given factor, such as temperature, cannot be easily changed. In this case, the analysis should be done by the split-plot method, which conveniently divide the experimental runs into groups.

Split-plot designs originated in the field of agriculture where experimenters applied one treatment to a large area of land, called a “whole-plot,” and other treatments to smaller areas of land within the whole-plot—called “subplots”. For example, the whole-plot treatment might be fertilizer 1 vs. fertilizer 2, with the subplot treatment being seed type 1 through 8 (see picture below).

Case study: Polymerase Chain Reaction (PCR)

This example is based on a polymerase chain reaction—a biochemical technology that amplifies DNA for diagnosing hereditary diseases and other purposes. Due to equipment limitations, it is not convenient to fully randomize the treatments, so the biochemists chose a split-plot design. In this case the whole-plots are actually plates that are subjected to varying conditions of time and temperature by a “thermocycler”. The subplots fall into the wells within each plate (96 wells available as shown below), within which experimenters can randomly apply the remaining factors. As you will see, the split-plot design conveniently groups runs plate-by-plate.

Building the Design

To save on typing, you will rebuild an existing design, thus demonstrating how it is created. Then you will re-open the file to recover the results before moving on to the analysis. Follow these steps to learn how Stat-Ease software builds two-level factorial split-plots:

1. To bypass the design build without having to enter the names of all the factors, go to Help, Tutorial Data and open PCR. Rebuild via File, New Design and clicking Yes to “Use previous design info” and No to “Do you want to save changes…?”. Then note these design specifications for this two-level-factorial split-plot:

   • Total factors: 9. These include both the hard-to-change and easy-to-change factors.

   • Hard-to-change (HTC) factors: 3. These are the three thermocycler factors.

   • HTC factors laid out as: Full factorial (the default) with 8 combinations of factors (2³).

   • Groups per replicate: 8 (to accommodate the full factorial on the HTC thermocyler factors).

   • Runs per Group: 32. This number supports a 2^6-1 half fraction on the 6 easy-to-change (ETC) factors. For all 9 factors, this specification and the others produces a design with resolution IX.

   Note

   The Resolution box changes green for any design that is Res V or better, meaning you can fit main effects and two-factor interactions (2FI). Check this out by reducing the number of Runs per group to 16 (still green) and then 8 (turns yellow due to lower resolution—OK for screening main effects only) and then 2 (a red design—not recommended).

2. Click Next to see aliases (there are none of any consequence) and Next again to see the entries for factors. Note here that the HTC factors are labeled lowercase—a, b, and c; while the ETC factors are uppercase—E, F, G, H, and J (skipping past letter I due to it being reserved as the label for the model intercept).

3. Click Next to view the entry for the response and then, accepting the defaults for Signal/Noise ratio, Next again to view the Split-Plot Design Power. Note that lowered power for the HTC factors (a, b and c). This occurs due them being put into 8 groups, which restricts randomization.

4. Click Finish produce the experimental plan (recipe sheet), pressing OK on the warning to reset factor levels. Scrolling down you will see how this 256-run design is split up into 8 groups going into 8 separate plates.

5. When you have finished examining the design, restore the results to the response colum by selecting Help, Tutorial Data and PCR from the menu. Choose No when prompted to save the existing file.

Now having rebuilt the design and collected the data, continue this feature tour to see Stat-Ease’s specialized tools for selecting effects from a two-level-factorial split-plot, as well as the programs statistical analysis, diagnostics and informative displays for assessing the final outcome.

Analyzing the Design

Analysis is the same as for a factorial design, except for one key difference: The Run and Group effects are analyzed separately—each getting its own half-normal plot. To get started on analyzing the Amplification response, click the Amplification node under the Analysis branch at the left.

1. Click the Start Analysis button to bring up the Run Effects tab.

2. Select the significant effects (those that stand out to the right) by clicking them (or “lassoing” them with a mouse drag) as shown below.

Notice how the Pareto chart, at the right in this default side-by-side view [|], updates as you modify the half-normal effect selection.

3. Click the Group Effects tab and press Yes when asked the question about hierarchy. As shown below, select the significant Group effects not already chosen for hierarchy—in this case b-Denature_Temp.

4. Click on the ANOVA (REML) tab. The restricted maximum likelihood (REML) analysis is necessary to properly identify the significant effects and calculate p-values for split-plot designs. In this case, all the Group and Run terms are significant at the p < 0.05 level.

5. Click on the Diagnostics tab and analyze the diagnostics as you normally would. (For more details refer to the Two-Level Factorial tutorial, which goes into far more depth than this high-level feature tour.) There is one possible outlier (the point near the upper outside red lines), but in a design with 256 runs, that is not unexpected, so leave that run in.

6. The model graphs, numerical optimization and other post-analysis options work as they do for all two-level factorial designs. Explore the graphs for conditions that maximize the “Amplification” response.

Conclusion

This concludes a quick pass through the two-level factorial tools provided by Stat-Ease software for split-plot experiments. Consider saving your results and then seeing via Numerical Optimization what the program recommends for achieving maximum optimization for the polymerase chain reaction (PCR). It is quite amazing what DOE can do with the proper tools!