This tutorial illustrates Design-Expert® software tools for applying split-plot design to experiments that combine both mixture and process factors.
Baking cake is the perfect example to test out the tools in Design-Expert (DX) for the combined split-plot design. It involves all of the elements of a good experiment, from mixing together various portions of ingredients, to treating the mixture at temperature, and even adding another mixture (frosting) to the mix (pun intended). One particularly popular dessert, especially in the southern United States, is the Lady Baltimore Cake—rich and delicious with a fluffy frosting full of nuts and raisins. Supposedly the cake was first baked by Alicia Rhett Mayberry of Charleston, South Carolina for novelist Owen Wister, best known for The Virginian, published in 1902. Wister described the cake in his next book—Lady Baltimore (1906).
Many recipes for the Lady Baltimore Cake can be found via internet, e.g., the one pictured here detailed by wikiHow at www.wikihow.com/Bake-a-Lady-BaltimoreCake. These recipes vary, of course. For example, the wikiHow instructions specify the use of all purpose (plain) flour whereas the recipe shown below demands specialized cake flour.
|Cake||Frosting and Filling|
|3 cups sifted cake flour||2 egg whites, unbeaten|
|3 teaspoons baking powder||1 ½ cups sugar|
|½ teaspoon salt||5 tablespoons water|
|½ cup butter||1 ½ teaspoons light cornsyrup|
|1 ½ cups sugar||½ teaspoon vanilla|
|1 ¼ cups milk||6 dried figs; CandiedCherries|
|1 teaspoon vanilla||½ cup raisings|
|4 egg whites||½ cup nuts, chopped|
Immediately an experimentally inquisitive mind must question whether it really is 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 a blend of these three ingredients—all-purpose flour, cake flour and sugar–to develop a better tasting cake. The amounts will be expressed in ounces, holding all other ingredients constant.
Cake is great but it really comes up short without a good frosting. To tailor this second mixture formulation, the key ingredients of water, corn syrup, and vanilla will be varied, again holding the other ingredients constant.
Last but not least, the amount of frosting and filling (F&F) will be varied (the Lady Baltimore Cake features the same mixture in and out). This will be measured in cups—a numerical factor.
As we just explained in the Introduction, the Lady Baltimore Cake experiment encompasses two mixture formulations (cake and frosting) and an amount (F&F). That makes this a “combined” design, which, as you can imagine, results in many runs to provide enough data for a response surface method (RSM) optimization. Luckily, our bakery has an oven that can bake 12 cakes, i.e., a dozen, at a time, in conjunction with a large mixer that can mix that much batter. Obviously, therefore, it will be most convenient to bake the cakes in batches of a dozen each. That’s where the split-plot design steps in. Normally, an experiment would be completely randomized and thus require a new cake recipe for every run, but, as you will soon see, the split plot sorts cake batters into convenient groups for hard-to-change factors (HTC) within which will be randomized the easy-to-change (ETC) factors. Let’s get going on this combined design so you can see how to go about making the best Lady Baltimore Cake in the land.
To set up the experiment, open the DX software. Then click the blank-sheet icon () on your toolbar. Under the Custom Designs section near the bottom choose the Optimal (Combined) design. Select 3 from the Mixture 1 components droplist. Then, choose 3 for Mixture 2 components, and 1 for the Numeric factors. 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. Only all purpose flour, cake flour, and sugar are being investigated. That portion of the recipe comes to 36 total ounces. Enter 36 in the Total box and ounces for Units.
That constant total of 36 ounces will be added to the rest of the unchanged recipe, so the proportions are the same every run. The cake mixture is the HTC portion of the design, so switch the change column in A from Easy to Hard using the droplist. In a mixture, all of the components need to be either Easy or Hard, 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-purpose flour). Note that all the components went to lower-case. HTC factors are lowercase in DX, differentiating them from the usual uppercase labels for ETC factors. 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 cake flour as component b with a low of 0 and high of 26, and sugar for component c with a low of 10 and high of 14. Varying the levels of these components in relation to the rest will determine if the cake flour is necessary and 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: the total amount of flour. To ensure there’s enough flour, add a constraint by clicking on the Edit Constraint button near the bottom of the page. The total flour consists of a+b, so enter that into the middle constraints column. To maintain the right amount of flour, enter a Low Limit of 20 and a High Limit of 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 of teaspoons. We’re only experimenting on the small (potent) ingredients of the recipe. This mixture is ETC, so no need to adjust the change column. Change the component 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 amount F&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 many options to choose from, and with the combined design there are even more. Look near 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 upper right). That combined model is very complex and allows a very intricate model that may be overkill. To save runs, click the Edit model… button. One way to save runs 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 in fewer terms, and runs. However, instead of simplifying the individual models, getting rid of some of the very high order interactions caused by the multiplication of these models is a better bet. To do that, change the Combined order limit: to quartic.
This will take out all of the terms in the model that involve fifth or sixth order terms, things like ABEFG2, or ADFG2, with are sixth and fifth order, respectively. These aren’t absolutely necessary to get a good picture of the system, and eliminating them will save some runs. Only 4th order (quartic) terms will remain.
Press OK and the required model points goes down from 108 to 72, a nice savings.
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 groups contain runs that will be mixed together and baked in one big batch, many cakes at a time. By default, there are 9 groups, so with 81 total runs, that’s 9 cakes per group. Remember, 12 cakes can be baked at once, so reduce the Additional groups to 1 and press the tab key to update. Say no to the warning to keep only seven groups. That leaves a total of 79 runs in 7 groups, or a little more than 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 be measured 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 is going through many trials (20 by default) to pick the right set of runs to fit the model terms as precisely and efficiently as possible. After the iterations, the best design (judged by the statistical criteria chosen in the options) is presented.
You will get a warning to reset the factor levels between groups. Just click OK to bypass this warning, for now. The runs on your screen will most likely be different due to the randomizing (where not restricted) of the design. The first two groups are shown below. For group 1, cakes will be baked with 22 ounces of 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 a little less sugar. The great thing about the split-plot design is that these groups contain 11 or 12 straight runs with the same cake recipe. That allows a dozen or so cakes to be mixed and baked all in one big batch, an enormous time savings.
After the big batch of cakes are baked, the frosting and filling recipe specified by the Mix 2 components (D, E, and F) can be whipped up and applied in the proper amount, specified by Factor G.
Load in the results by clicking on the Help, Tutorial Data menu and selecting Lady Baltimore.
The design with data should look like the screenshot below. Note that the custom design you built is replaced with the tutorial run-order for consistency.
To get started with the analysis, click the node labeled R1: Rating under the Analysis branch. As with a normal RSM analysis, a new set of tabs appears at the top of your screen and they are arranged from left to right in the order needed to complete the analysis.
There are a variety of Transforms that can be applied on this page. Not knowing if they will help at this point, click ahead to the Model tab. There are diagnostics that are checked later that can determine if a transform will help.
On the model tab, the combined model (reduced to only quartic terms) is presented for consideration (denoted by the green “” next to the terms and the “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’s best 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 click the Start button to run the analysis. DX will go through the terms in the design model and select which ones improve the AICc criterion the most and add them to the model one at a time until adding terms will no longer improve the criterion.
DX shows you the terms added in selecting the model, showing AICc criterion for each step. Click the Help button for more details on algorithmic model selection and the criterion used. Otherwise, click Accept to continue and evaluate the resulting model. To get the results, click on ANOVA (REML) tab. You will get a warning that the model you have selected is not hierarchical. Be sure to click Yes to correct for hierarchy. This will give you a more statistically sound model, ensuring lower order terms are present to support higher order terms, even if they are insignificant. This is good statistical practice. Click on the help button in the warning box for more information. The model statistics are then shown.
This is not your standard ANOVA analysis, which relies on randomization for validity. The analysis done for split-plot designs in DX is a form of maximum likelihood estimation, more specifically, restricted maximum likelihood (REML), as noted at the top of the results table.
Details on split-plot analysis: The aim of maximum likelihood estimation is to find the parameter value(s) that makes the observed data most likely. Restricted maximum likelihood estimation, which is generally used unless you click on the Analysis menu available on the Model screen to change the method, is another way to estimate variances. In the split plot case, REML estimates the Group variance for the whole plot factors and the residual variance for the subplot factors. Once the variances are estimated, Generalized Least Squares (GLS) is used to estimate the factor effects. The Kenward-Roger’s method is then used to produce F-tests and the corresponding p-values. You can learn even more by clicking on the lightbulb icon for screen tips and following the links.
The big difference between the statistics on this table and a normal ANOVA is the grouping of variance terms into a Whole-plot section for the HTC factors and a subplot section for ETC factors. However, for this design, there are no Whole-Plot terms selected. In other words, there are no terms consisting of just A, B, and C that are significant. There are terms involving A, B, and C, but they are always interacting with the frosting mixture and frosting amount (G) terms and are thus part of the subplot. That’s somewhat expected, because in a split-plot design, the subplot terms (and their interactions) have more power and can be detected more easily. In fact, these subplot interactions can often make up for a 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 alpha level) or needed for hierarchy. For example, the insignificant term ABE is needed for the significant ABEG term.
Next, press the Variance Components tab and DX presents various statistics to augment the REML analysis.
Here, you will see more details on the variance components. Move to the Model Comparison Statistics tab to view likelihood ratios for the selected model, including the information criterion (AIC, BIC, and AICc). More can be learned on those 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 1 being the best. In this case, the 0.75 Adj. R-Squared shows that the selected model captures most of the variation in the data (~75%).
To learn more about the various model criterion, this would be a great place to exercise DX’s context sensitive help. Just click on a number you are interested in 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 to check, but these have been covered extensively in other tutorials. For instance, see the Response Surface tutorial. In this case, the diagnostics all 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 the Process factor (F&F amount), represented on the floating Factors Tool. Click and drag 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 the left, low level. Reducing the amount of frosting seems to reduce the taste rating 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 the amount F&F process factor in the Factors Tool and select X1 axis. Note that “One Factor” is now highlighted in the graphs tool. Clicking on that button is 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, raising the amount of F&F improves the taste, but only to a point. If the amount is raised to 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 interacts with 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 to maintain the constant total for the frosting mixture. Adding all that water doesn’t change the overall rating much and there is still a peak (slightly sharper) for the optimum amount of F&F.
In a combined design like this, there are many other interesting graphs to investigate. For instance, click on the Mix-Process button on the graphs toolbar. This reveals how substituting all-purpose flour (from left to right) for cake flour affects the rating along with the amount F&F (from bottom to top).
The goal of the experimental program is to learn how to customize the Lady Baltimore cake recipe to get the highest overall rating. To find optimal combinations of formulas and processing, click the optimization node labeled Numerical. Then select Rating. Choose maximize from the Goal droplist and leave everything else at the default settings. Your screen should now look like that below.
Click the Solutions tab. The solutions are presented in the ramps view, by default.
The ramps view makes it easy to see the levels of each component/factor and the resulting 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. The amount of F&F (G), is set to the upper middle level seen previously when investigating that factor on the one factor plot.
That’s solution number one, with the highest rating. To investigate a few other options, 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 other solutions using the up and down arrow keys. Even going to solutions with lower ratings, it seems the all-purpose flour must be set to a low level to get good ratings.
Click on the Graphs tab to investigate the graphs at the optimal solution (be sure to select solution number 1 on the solutions bar). There will be a flag planted at the optimum (at low levels of all-purpose flour). By default, all responses are shown side by side, including the desirability plot used to search for the optimum.
To concentrate on just the rating, select Rating from the Response dropdown.
Having explored many graphs already, the recipes have been pretty well optimized. We’ll leave you on your own if you’d like to investigate the graphs some more.
A split-plot design can be applied to save experimental effort. This can even be applied in the case of a complex combined design like this, involving two mixtures and a process factor. Here, it allowed the bakers to make a dozen or so cakes at a time in big batches, instead of changing the batter every run and baking one cake at a time. That’s a big savings of time and budget, allowing the cake recipe to be fully optimized. With split-plot designs, you can have your cake and eat it, too!