Stat-Ease » v23.1 » Tutorials » Combined Split-Plot Design (2024)

Introduction

This tutorial illustrates Stat-Ease^® software tools for applying split-plotdesign to experiments that combine both mixture and process factors.

Baking cake is the perfect example to test out the tools in Stat-Easefor the combined split-plot design. It involves all of the elements of a goodexperiment, from mixing together various portions of ingredients, to treating themixture at temperature, and even adding another mixture (frosting) to the mix(pun intended). One particularly popular dessert, especially in the southernUnited States, is the Lady Baltimore Cake—rich and delicious with a fluffyfrosting full of nuts and raisins. Supposedly the cake was first baked by AliciaRhett Mayberry of Charleston, South Carolina for novelist Owen Wister, best knownfor The Virginian, published in 1902. Wister described the cake in his nextbook—Lady Baltimore (1906).

Many recipes for the Lady Baltimore Cake can be found via internet, e.g., theone pictured here detailed by wikiHow atwww.wikihow.com/Bake-a-Lady-BaltimoreCake. These recipes vary, of course. Forexample, the wikiHow instructions specify the use of all purpose (plain) flourwhereas the recipe shown below demands specialized cake flour.

Immediately an experimentally inquisitive mind must question whether it reallyis necessary to use the fancier flour. Perhaps a combination of flours might do.Also, it may help to adjust the ratio of flour to sugar. Let’s investigate ablend of these three ingredients—all-purpose flour, cake flour and sugar–todevelop a better tasting cake. The amounts will be expressed in ounces, holdingall other ingredients constant.

Cake is great but it really comes up short without a good frosting. To tailorthis second mixture formulation, the key ingredients of water, corn syrup, andvanilla will be varied, again holding the other ingredients constant.

Last but not least, the amount of frosting and filling (F&F) will be varied (theLady Baltimore Cake features the same mixture in and out). This will be measuredin cups—a numerical factor.

Design the Experiment

To set up the experiment, open the software. Then click the blank-sheet icon() on your toolbar. Under the Custom Designs section near the bottomchoose the Optimal (Combined) design. Select 3 from the Mixture 1 componentsdroplist. Then, choose 3 for Mixture 2 components, and 1 for the Numericfactors. Click Next in the bottom right corner to advance to the next page.

Here, you will enter the components for Mixture 1, the cake recipe. Onlyall purpose flour, cake flour, and sugar are being investigated. That portion ofthe recipe comes to 36 total ounces. Enter 36 in the Total box andounces for Units.

That constant total of 36 ounces will be added to the rest of the unchangedrecipe, so the proportions are the same every run. The cake mixture is the HTCportion of the design, so switch the change column in A from Easy to Hardusing the droplist. In a mixture, all of the components need to be either Easy orHard, so changing the first one changes them all to Hard.

Click back on the component A name to change it to all purp flour (all-purposeflour). Note that all the components went to lower-case. HTC factors arelowercase in the software, differentiating them from the usual uppercase labels for ETCfactors. This will really come in handy when manipulating the graphs and labels.

Tab over and enter 0 for the Low and 26 for the high. Then enter cakeflour as component b with a low of 0 and high of 26, and sugar forcomponent c with a low of 10 and high of 14. Varying the levels of thesecomponents in relation to the rest will determine if the cake flour is necessaryand what level of sugar is best. The design should now look like this.

There is one part of the recipe that shouldn’t be strayed from too much: thetotal amount of flour. To ensure there’s enough flour, add a constraint byclicking on the Edit Constraint button near the bottom of the page. The totalflour consists of a+b, so enter that into the middle constraints column. Tomaintain the right amount of flour, enter a Low Limit of 20 and a High Limitof 26.

Press OK and then Next to move on to Mixture 2.

Enter the total for the filling and frosting mixture of 17 and units ofteaspoons. We’re only experimenting on the small (potent) ingredients of therecipe. This mixture is ETC, so no need to adjust the change column. Change thecomponent D name to water, with a low of 13.5 and high of 16.Component B should be corn syrup with a low of 0.5 and high of 2.5,and component C is vanilla with a low of 0.5 and high of 1.

Click Next to move on to the numeric factor. Change the name to amountF&F (filling and frosting), with a low level, L(1), of 3 and a high level,L(2), of 4.

Click Next and the design options appear. The optimal design has manyoptions to choose from, and with the combined design there are even more. Looknear the “Edit Model…” button and you see “Quadratic x Quadratic x Quadratic”,so Mix 1 model is Quadratic, as is the Mix 2 model and the numerical model.

The combined model will multiply the terms from each of these models together,resulting in 108 coefficients, or 108 required model points (see the upperright). That combined model is very complex and allows a very intricate model thatmay be overkill. To save runs, click the Edit model… button. One way to saveruns would be to change the model order of one of the individual models, Mix 1,Mix 2, or the process model. Choosing a simpler model, say linear would result infewer terms, and runs. However, instead of simplifying the individual models,getting rid of some of the very high order interactions caused by themultiplication of these models is a better bet. To do that, change the Combinedorder limit: to quartic.

This will take out all of the terms in the model that involve fifth or sixthorder terms, things like ABEFG2, or ADFG2, with are sixth and fifth order,respectively. These aren’t absolutely necessary to get a good picture of thesystem, and eliminating them will save some runs. Only 4th order (quartic) termswill remain.

Press OK and the required model points goes down from 108 to 72, a nicesavings.

Now, look at the Groups column in the middle. These are the “whole plot groups”,where the HTC factor levels are held constant. In other words, these groupscontain runs that will be mixed together and baked in one big batch, many cakesat a time. By default, there are 9 groups, so with 81 total runs, that’s 9 cakesper group. Remember, 12 cakes can be baked at once, so reduce the Additionalgroups to 1 and press the tab key to update. Say no to the warning to keeponly seven groups. That leaves a total of 79 runs in 7 groups, or a little morethan 11 cakes per group, making much better use of the large oven and mixer.

Click Next. For the R1 Name, enter Rating, which in this case will bemeasured on a 100 point scale—the higher the better the taste.

Click Finish to build the design. It may take a few minutes. The program isgoing through many trials (20 by default) to pick the right set of runs to fit themodel terms as precisely and efficiently as possible. After the iterations, thebest design (judged by the statistical criteria chosen in the options) ispresented.

You will get a warning to reset the factor levels between groups. Just clickOK to bypass this warning, for now. The runs on your screen will most likelybe different due to the randomizing (where not restricted) of the design. Thefirst two groups are shown below. For group 1, cakes will be baked with 22 ouncesof all-purpose flour, 0 ounces of cake flour, and 14 ounces of sugar.Coincidentally, group 2 contains another set of runs with no cake flour, but alittle less sugar. The great thing about the split-plot design is that thesegroups contain 11 or 12 straight runs with the same cake recipe. That allows adozen or so cakes to be mixed and baked all in one big batch, an enormous timesavings.

After the big batch of cakes are baked, the frosting and filling recipespecified by the Mix 2 components (D, E, and F) can be whipped up and applied inthe proper amount, specified by Factor G.

Analyze the Results

Load in the results by clicking on the Help, Tutorial Data menu andselecting Lady Baltimore.

The design with data should look like the screenshot below. Note that the customdesign you built is replaced with the tutorial run-order for consistency.

To get started with the analysis, click the node labeled R1: Rating under theAnalysis branch. As with a normal RSM analysis, a new set of tabs appears at thetop of your screen and they are arranged from left to right in the order neededto complete the analysis.

There are a variety of Transforms that can be applied on this page. Not knowingif they will help at this point, click ahead to the Model tab. There arediagnostics that are checked later that can determine if a transform will help.

On the model tab, the combined model (reduced to only quartic terms) ispresented for consideration (denoted by the green “” next to the terms andthe “Design model” in the process order. Clicking ahead to the ANOVA (REML)screen at this point will evaluate that full designed for model. However, it’sbest to do some analysis to select the best model from among the possible terms,eliminating insignificant ones.

To allow the computer to do this automatically, click on the Auto Select…button. Accept the defaults for AICc criterion and forward selection and clickthe Start button to run the analysis. The software will go through the termsin the design model and select which ones improve the AICc criterion the mostand add them to the model one at a time until adding terms will no longer improvethe criterion.

The software shows you the terms added in selecting the model, showing AICc criterion foreach step. Click the Help button for more details on algorithmic model selectionand the criterion used. Otherwise, click Accept to continue and evaluate theresulting model. To get the results, click on ANOVA (REML) tab. You will geta warning that the model you have selected is not hierarchical. Be sure to clickYes to correct for hierarchy. This will give you a more statistically soundmodel, ensuring lower order terms are present to support higher order terms, evenif they are insignificant. This is good statistical practice. Click on the helpbutton in the warning box for more information. The model statistics are thenshown.

This is not your standard ANOVA analysis, which relies on randomization forvalidity. The analysis done for split-plot designs in the software is a form of maximumlikelihood estimation, more specifically, restricted maximum likelihood (REML),as noted at the top of the results table.

The big difference between the statistics on this table and a normal ANOVA isthe grouping of variance terms into a Whole-plot section for the HTC factors anda subplot section for ETC factors. However, for this design, there are noWhole-Plot terms selected. In other words, there are no terms consisting of justA, B, and C that are significant. There are terms involving A, B, and C, but theyare always interacting with the frosting mixture and frosting amount (G) termsand are thus part of the subplot. That’s somewhat expected, because in a split-plotdesign, the subplot terms (and their interactions) have more power and can bedetected more easily. In fact, these subplot interactions can often make up fora lack of power for the HTC factors.

Looking at the subplot as a whole, the model has a quite significant F value(p-value of <0.0001). Most of the terms are also significant (at the 0.05 alphalevel) or needed for hierarchy. For example, the insignificant term ABE is neededfor the significant ABEG term.

Next, press the Variance Components tab and the software presents various statistics toaugment the REML analysis.

Here, you will see more details on the variance components. Move to the ModelComparison Statistics tab to view likelihood ratios for the selected model,including the information criterion (AIC, BIC, and AICc). More can be learned onthose in the help menus.

One important number to look at is the Adj. R-squared (adjusted R-squared),found under the Fit Statistics tab . This number goes from 0 to 1, with 1being the best. In this case, the 0.75 Adj. R-Squared shows that the selectedmodel captures most of the variation in the data (~75%).

To learn more about the various model criterion, this would be a great place toexercise the software’s context sensitive help. Just click on a number you are interestedin to highlight it and then press the F1 key (or right-click and select Help).For example, look at the information obtained about the Adj R-Squared criterion.

That’s enough on the model statistics. It seems we have quite a strong model.Click the Diagnostics tab and examine the graphs of residuals.

The residuals graphs found via the floating diagnostics tool are important tocheck, but these have been covered extensively in other tutorials. For instance,see the Response Surface tutorial. In this case, the diagnosticsall look good, so press on to the Model Graphs to view the response in mixture(triangular) space.

Remember that this mixture interacts with both Mixture 2 (Frosting) and theProcess factor (F&F amount), represented on the floating Factors Tool. Click anddrag the bars for those factors and the response graph will change. For example,drag the amount of F&F from the Process portion of the factors tool to theleft, low level. Reducing the amount of frosting seems to reduce the tasterating across the board, which makes sense. Who wants cake without much frosting?

Let’s see what this process factor looks like on its own. Right-click on theamount F&F process factor in the Factors Tool and select X1 axis. Notethat “One Factor” is now highlighted in the graphs tool. Clicking on that buttonis the other way to pull up this graph.

This will put the amount F&F on the x-axis. As seen in the prior graph, raisingthe amount of F&F improves the taste, but only to a point. If the amount is raisedto the highest level, the taste rating goes down. This is only part of the story,as you may note from the red Warning atop the graph! The amount of F&F interactswith the two mixtures. To see how changing the frosting mixture affects the graph,drag the red bars for Mixture 2 from the floating Factors tool. For example,drag water to its high level and see how the amount F&F graph responds.

Note that as more water is added, the corn syrup and vanilla must be removed tomaintain the constant total for the frosting mixture. Adding all that waterdoesn’t change the overall rating much and there is still a peak (slightlysharper) for the optimum amount of F&F.

In a combined design like this, there are many other interesting graphs toinvestigate. For instance, click on the Mix-Process button on the graphstoolbar. This reveals how substituting all-purpose flour (from left toright) for cake flour affects the rating along with the amount F&F (from bottomto top).

Find the Optimal Solution

The goal of the experimental program is to learn how to customize the LadyBaltimore cake recipe to get the highest overall rating. To find optimalcombinations of formulas and processing, click the optimization node labeledNumerical. Then select Rating. Choose maximize from the Goal droplistand leave everything else at the default settings. Your screen should now looklike that below.

Click the Solutions tab. The solutions are presented in the ramps view, bydefault.

The ramps view makes it easy to see the levels of each component/factor and theresulting rating (73). Unfortunately, all-purpose flour is set to the low level.The recipe makers knew what they were doing when they called for cake flour. Theamount of F&F (G), is set to the upper middle level seen previously wheninvestigating that factor on the one factor plot.

That’s solution number one, with the highest rating. To investigate a few otheroptions, look at some other solutions in the dropdown menu on the Factors Tool.

Once you click in the solutions dropdown, you can easily toggle through the othersolutions using the up and down arrow keys. Even going to solutions with lowerratings, it seems the all-purpose flour must be set to a low level to get goodratings.

Click on the Graphs tab to investigate the graphs at the optimal solution (besure to select solution number 1 on the solutions bar). There will be a flagplanted at the optimum (at low levels of all-purpose flour). By default, allresponses are shown side by side, including the desirability plot used to searchfor the optimum.

To concentrate on just the rating, select Rating from the Response dropdown.

Having explored many graphs already, the recipes have been pretty welloptimized. We’ll leave you on your own if you’d like to investigate the graphssome more.

Final Comments

A split-plot design can be applied to save experimental effort. This can even beapplied in the case of a complex combined design like this, involving two mixturesand a process factor. Here, it allowed the bakers to make a dozen or so cakes at atime in big batches, instead of changing the batter every run and baking one cakeat a time. That’s a big savings of time and budget, allowing the cake recipe to befully optimized. With split-plot designs, you can have your cake and eat it, too!