Design-Expert^{®} software offers designs that combine mixture components
and process factors (numeric and/or categoric). In this tutorial, you will apply
Design-Expert’s unique features for mixture-process design and analysis by
investigating John Cornell’s famous fish-patty experiment from his textbook
*Experiments with Mixtures*, 3rd edition, published by John Wiley and Sons, New
York. Zip through it all by skipping the sidebars, or, if you like to explore,
take the time to study these.

Note

Before embarking on this tutorial, you should first complete the previous tutorial on mixture design.

P.S. Stat-Ease provides in-depth training on combined mixture-process in its Mixture and Combined Designs for Optimal Formulations workshop. Call for information on content and schedules, or better yet, visit our web site at www.statease.com.

The food formulators hope to make something delicious from a blend of three tasty-sounding (?) fish: mullet, sheepshead, and croaker. Yum! The seven blends in the design include one each of the three fishes, plus three binary blends, and a blend of one-third each of the individual types of fish. The fish patties can be cooked at varying deep-frying time, oven temperature, and oven time. Combining each of these process factors at two levels for all seven blends creates a total of 56 experimental runs. The diagram below shows the blends as points on triangles, repeated at each of the eight corners of a cube representing the process.

The response measure is patty texture. (Never mind about taste!)

Start the program by finding and double clicking the Design-Expert
icon. Take the quickest route to initiating a new design by clicking the **New
Design** button on our opening screen as shown below.

Click **Custom Designs** to expand its menu and select the **User Defined**
option. The user defined option allows you to reproduce Cornell’s design with all
points chosen. Normally, to save on runs, you want to use the first choice on the
list, Optimal, which selects an ideal subset of design points from a candidate
set for a specified model. We’ll explore this option later.

For **Mixture 1 Components**, select **3** from the droplist. (Notice that Design-Expert
offers the option for adding a second mixture, for example in a two-layer
cake, film, or coating. For the number of **Numeric Factors** choose **3**. Leave
Categoric Factors at zero, but keep in mind for future experimentation that you
can add discrete variables – such as who supplies a given component.

Press **Next** and enter the **Total** for mixture components (fish types) at
**100** and **Units** as **%**. Then enter fish names as shown below. Low limits
remain zero. Set all high limits to **100**.

Press **Next** to move on to the process design factors. Enter factor names,
units, and ranges (low – L[1] and high – L[2]) as shown below.

Press **Next** to define the model you want to design for the combination of
mixture and process variables in this experiment. The default model is quadratic
by quadratic. Click the **Edit model** button and change the default for **Process
Order** to **Linear**.

Click the **OK** button to implement the change. Now, consider your “candidate”
points. Normally only some of these points would actually make it to the design,
but by choosing the User Defined option, you get all of them. However, there’s a
problem—the **Mixture** candidate set of points (Mixture tab at upper-left screen
box) contains more points than those chosen by Cornell. Fix this by clicking off
**Axial check blends**, and **Interior check blends**. Only vertices, centers of
edges, and overall centroid should now be checked. Press **Calculate points** to
get a count — this subtotals to 7 for the mixture candidate points.

Now click the **Process** candidate points tab (right of Mixture tab). This
design also contains more points than those chosen by Cornell, so click off every
checkmark except **Vertices**.

You should now see a subtotal of 8 process combinations, which combined with the 7 mixture candidates, generates 56 total runs (points).

Press **Next** to see the response specification form. Enter **Ave Texture** with
units of **gm*10^-3**.

Press **Finish** to complete the design. Design-Expert now displays the 56
experiments in random run order.

To view response data, click on the **Help, Tutorial Data** menu and select
**Fish Patties**. Now click on the **Design** node on the left.

Look over all resulting response data: Results within the range of 2 to 3.5 are desirable.

To analyze the data, click the **Analysis** node labeled **Ave Texture**. Then
click the **Fit Summary** tab. You now see a unique matrix of probabilities that
helps you determine the best crossed model for mixture and process. As shown below,
the software provides suggestion(s) on which combination is best—in this case, a
quadratic (“Q”) mixture model crossed with a linear (“L”) process model.

Note

For a detailed explanation about how to interpret
sequential p-values in this Fit Summary table for combined designs, see
Stat-Ease’s *Handbook for Experimenters* (provided free of charge to all
registered users of Design-Expert) section titled “Combined
Mixture/Process Analysis Guide.” This section is available upon request to trial
users.

Now click the **Model** tab atop your screen. Design-Expert uses the suggested
model as its default. Accept this for the moment by clicking ahead to the
**ANOVA**. Notice that many of the model terms, particularly ones of third-order
(for example, ABF) exhibit high Prob>F values. This is a byproduct of crossing
the two mixture and process models, which creates many superfluous higher-order
terms. When analyzing designs like this, you will often find it beneficial to
reduce your model to only the significant terms, subject to requirements for
maintaining hierarchy (no interactions without their parent terms).

To reduce the model, go back to **Model**. You could manually deselect the
insignificant terms observed from the ANOVA, but it is quicker to let
Design-Expert do this for you. To do this, click on the **Auto Select…** button.
Change the criterion to **p-values**. The statistical criterion is like the
scoreboard for comparing the adequacy of alternative designs, here we’re comparing
p-values. Next, change the selection mode to **Backward**.

This enables automatic model reduction by a backward (stepwise) algorithm. The
criterion for elimination is a probability value of 0.1000, as specified in the
Alpha field. Click **Start**. You will be presented with a summary of the terms
that were eliminated via the backward selection method. Click **Accept** to
continue with the selected model. Then, click on the **ANOVA** to evaluate it
(shown below). Note that all of the terms in the selected model have significant
(at the <0.05 alpha level) p-values.

The ANOVA table also shows that the reduced model produces a highly-significant
F Value (p-value < 0.0001). Move over to the **Fit Statistics** pane (location
will vary based on pane layout) — they look really good.

Click the **Diagnostics** tab and examine the graphs of residuals.

The normal plot of residuals looks good, so move on to the **Model Graphs** to
view the response in mixture (triangular) space (use blue layout icons to match
layout in screenshot).

Click or grab various process factors on the Graphs Tool and adjust them to see how response changes. For example, push all three bars to the right as shown below.

Notice by the hotter colors that texture increases at the 100 percent mullet (top of triangle).

Now right-click the **Factors Tool** where it shows **D:oven temp** and select
**X2 axis**. As shown below, the view changes to process (rectangular) space.

Now shift back to mixture coordinates by right-clicking the **Factors Tool** where
it lists **A:mullet**. Select **X1 axis**.

On the **Graphs Toolbar** click **3D Surface** and notice the strong tilt in the
A-B axis.

Press ahead to the **3D Surface Mix-Process** display. It’s very likely you have
never seen a plot like this!

Now you see the impact of shifting component A (mullet) in direct substitution for B (sheepshead) on one axis versus the process factor D (oven temp) on the other axis. Isn’t that something!

Continue if you like and explore different combinations of the mixture components and process factors in 2D contour view and 3D plots. Keep in mind that any textures outside of 2 to 3.5 are unacceptable.

The goal of the experimental program is to learn how to produce fish patties
with a texture in the range of 2.00 to 3.50 x 10^{-3} grams. The target is
2.75. To find optimal combinations of formulas and processing, click the
optimization node labeled **Numerical**. Then select **Ave Texture**. Select a
**Goal** of **target->** and enter **2.75**. For **Limits** enter a **Lower** of
**2** and **Upper** of **3.5**. Leave the weights and importance settings at their
default levels. Your screen should now look like that below.

Click the **Solutions** tab. If not already there by default, mouse to the
**Solutions Toolbar** and select **Ramps**. Notice the many solutions listed in
the solutions drop-down menu. (Due to the random nature of the optimization
algorithm, your screen may show fewer or more solutions.) The software defaults
to the most desirable solution.

Select solution number 2 and beyond: You will see many combinations that satisfy
the texture requirements. Next press **Graphs** and you’re presented with the
multiresponse graph showing the Desirability and Ave Texture graphs side by
side. Select Desirability from the Response droplist on the solutions bar to
focus on that response. Then, select **3D Surface** from the **Graphs Toolbar**.
Move your mouse cursor over the graph and when it turns into a hand (or
crosshair) grab it by holding the left mouse button and rotate it to a vantage
point like that shown below.

Note

Check out the desirability surface for other solutions. Remember that by right-clicking any of the process factors on the Factors Tool, you can create a graph in process space.

To preserve your modeling and optimization work, select **File, Save** or press
the icon.

As noted in the introduction, we wanted to reproduce the fish patty case exactly
as reported by Cornell in his textbook *Experiments with Mixtures*. Now let’s try
an optimal design, which is much more efficient. We recommend you use this
approach when designing your own mixture-process experiment. Assuming you
are continuing from before, proceed by choosing **File, New Design**. The software
asks whether to “**Use previous design info?**” Click **Yes**.

You see the builder from the previous user-defined design session. Select
**Optimal (combined)**.

Press **Next** to see that all component names and Low/High values remain
filled in for you. Press **Next** again to see your process factors, again at
their previous ranges. Press **Next** one more time to bring up the Optimal Design
screen. Notice it’s already set up for the quadratic-linear base model that you
want for the fish-patties process-mixture design.

The objective of this exercise is to pick a subset of runs from Cornell’s actual
experiment so the search must be limited to specific combinations – not just any
coordinate within the mixture-process space. Therefore you must change the
**Search** droplist from its default of “Best” to **Point Exchange**, as shown
below.

Leave the optimality criterion at its default of “I” — a good choice in this case where continuous variables must be optimized. This criterion will dictate the selection of the minimal subset of model points from the candidate set, which in a moment you will specify as the original combinations run by Cornell.

Because the original data had no replicates, set **Replicate** points to **0**.
Hit the tab key and you see a total of 29 runs. Leave “Model points” and “To
estimate lack of fit” at their default levels.

Before continuing, you need to re-create the points originally selected by
Cornell. (Recall doing this earlier when you set up the User Defined design.)
Click **Edit candidate points**. In the **Mixture 1** tab, check off **Thirds of
edges, Triple blends, Axial check blends**, and **Interior check blends**. Press
**Calculate points**. You now see a subtotal of 7 mixture points comprised of
vertices (3), centers of edges (3) and the overall centroid (1). Go back to the
screen shot shown on this earlier to be sure you have it right!

Next, press the **Process** tab and check off all points except for vertices.
Click **Calculate points**. This produces a subtotal of 8 process points. You now
have 56 total candidate points, the same as earlier when you set up this case
study as a User Defined design.

Click **OK** and **Next** to move on to the response specification. Accept the
default response name and units (as in the user defined example) by pressing
**Finish** again.

Design-Expert now builds the optimal design. Due to random elements in the algorithm (configurable via options), the 29 runs chosen may differ each time you do an optimal design, but they will be essentially identical in their matrix attributes.

Remember that the point of this follow up to the first part of the tutorial is to show how you can get by with only a fraction of all possible mixture-process combinations by using an optimal design — in this case one that includes 29 combinations from within the original experiment, which contained 56 runs. Thus, this is a “what-if” exercise. If you were to build a completely new design for this case, it would be advisable to include the 5 replicates (34 total runs) that Design-Expert suggests by default. This would provide 4 degrees of freedom to estimate pure error, which would allow a test for lack of fit, and generally improve the quality of the design. This subset is still much more efficient at fitting the specified model than the original 56 runs used by Cornell.

In all but the simplest cases, a User-Defined design (all possible combinations of mixture and process points) is too large to be practical. Consider using an optimal design, which provides a reasonable subset of points to estimate the model you specify. For any combined design, but especially if done via optimal for a minimum-run experiment, it’s advisable to override any suggested model that falls short of what can be estimated without aliases. Then perform a backward elimination to rid the model of the large number of superfluous terms that are characteristic of these experiments that combine variables from the mixture and the process. If all goes well, you can blend your fish and bake it too. However, we have a suggestion: include taste as one of your sensory attributes (not just texture!).