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.

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).

A field sectioned into two whole plots (fertilizer type) and eight subplots (seed type)

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. The subplots fall into the wells within each plate, within which experimenters can randomly apply the remaining factors.

Building the Design¶

1. To bypass the design build without having to enter the names of all the factors, go to Help, Tutorial Data and open PCR.dxpx. Rebuild via File, New Design and clicking Yes to “Use previous design info”. Then note the design specifications for this two-level-factorial split plot.

   • Total factors: 9. These includes both the whole-plot (hard-to-change) and subplot (easy-to-change).
   • Hard-to-change (HTC) factors: 3. These are the three thermocycler (whole-plot) factors.
   • HTC factors laid out as: Full factorial (the default).
   • Groups per replicate: 8.
   • Runs per Group: 32. This specifies a 29-1 factorial design for all 9 factors, which is resolution IX. Note that the box changes green for any design that is Res V or better, meaning you can fit main effects and two-factor interactions (2FI).

Starting design from scratch

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 easy-to-change (ETC) factors are uppercase—E, F, G, H, and J (skipping past letter I due it being reserved as label for model intercept).

Design factors

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 (whole plots), which restricts randomization.

Design power

4) Click Finish to exit the design-building wizard and 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 whole-plot groups. Then, via File, New Design, re-open PCR.dxpx to get the results back.

Now having rebuilt the design and collected the data, continue this feature tour to see Design-Expert’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 in Design-Expert, except for one key difference: The subplot and whole-plot effects are analyzed separately—each getting its own half-normal plot. To get started on analyzing the Amplification response, click the R1-Amplification node under the Analysis branch at the left.

1. Click the Sub-plot Effects tab, bypassing the transform options (none needed unless diagnostics indicate so later in the analysis process).
2. Select the significant effects (those that stand out to the right) by clicking them 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 Whole-plot Effects tab and press Yes when asked the question about hierarchy. As shown below, select the significant whole-plot effects not already chosen for hierarchy.

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 whole-plot and sub-plot 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 (outside the 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 in Design-Expert. Explore the graphs for conditions that maximize the “Amplification” response.

Conclusion¶

This concludes a quick pass through the two-level factorial tools provided by Design-Expert software for a 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!