Stat-Ease Blog

Years ago, I attended Stat-Ease’s Modern DOE workshop in Minneapolis—a five day deep dive into factorial and response surface methods (RSM). I then completed a four day course on Mixture Design for Optimal Formulations. Since then, I’ve trained practitioners and coached users through hundreds of experiments. One pattern is consistent: most people—myself included—gravitate toward familiar factorial or RSM designs and hesitate to use mixture designs for formulation work.

The result is force-fitting RSM tools onto mixture problems. Like using a flathead screwdriver on a Phillips screw, it can work, but it’s rarely ideal. And, avoiding mixture designs can actually create real problems. So, what makes mixtures unique, and what goes wrong when we ignore that?

Why Mixtures Are Different

In mixtures, ratios drive responses, not absolute amounts. The flavor of a cookie depends on the ratio of flour, sugar, fat, and salt; not the grams of sugar alone. And because mixture components must sum to a total (often 100%), choosing levels for some ingredients automatically constrains the rest.

The Ratio Workaround—and Its Limits

A common workaround is to convert a q-component mixture into q-1 ratios and run a standard RSM design¹. For example, suppose we’re formulating a sweetener blend (A = sugar, B = corn syrup, C = honey) that always makes up 10% of a cookie recipe. If we express the system using ratios B:A and C:A, we can build a two factor RSM design with ratio levels like 1:1, 2:1, and 3:1.

But compared to a true three component mixture design, the difference is clear. The ratio based design samples only narrow rays of the mixture space, leaving large regions unexplored. Standard error plots show that a proper mixture design provides far better prediction capability across the full region.

In short: the ratio trick can work, but it never matches the statistical properties of a proper mixture design.

The Slack Component Argument

Another justification for using RSM is when one ingredient is believed to be inconsequential. Perhaps the component is believed to be inert or is simply a diluent that makes up the balance of a formulation. The idea is to treat this component as a slack variable and allow it to fill whatever space remains after setting the other ingredients. One slack approach is to simply use the upper and lower values as levels of the non slack components in a standard RSM. Below is a comparison of a three-component system analyzed as a true mixture design alongside a two-factor RSM that eliminates the diluent as a component.

In this case, both approaches found essentially the same optimal conditions. Ignoring the diluent really didn’t impact the story, but the RSM approach is not specifically assessing the interactive behavior between the reactants and the diluent. If we study the system as an RSM, we assume the interactions involving the omitted component were not consequential—which may not be true. Cornell² states that the factor effects we are seeing are actually the effects confounded with the opposite effect of the ignored component. Without using a mixture design, we would have no way of validating our assumptions about these interactions.

Cornell³ also describes an alternative slack approach where the slack component is included in the design but excluded from the predictive model. Some practitioners believe this approach makes sense when the diluent interacts weakly with the key ingredients, the omitted component is the one with the widest range of proportionate values, or if that component makes up the bulk of the formulation. But statistically, this presents some interesting complexities.

Using the above chemical reaction example, Figure 6 shows the model differences between the Scheffé approach and the resulting models when each component is considered the slack component.

Note that in this example, while some of the models are similar, the one involving the diluent as the slack variable differs most from the Scheffé standard. Had we assumed the diluent could have been used as the slack variable, we would have poorly modeled and optimized the system.

Because slack variable models exclude at least one component and its interactions, they’re best avoided when possible.

When Components Don’t Share a Scale

Mixture designs require all components to share a common basis (percent, ppm, etc.). This becomes awkward when ingredients span vastly different scales—for example, large amounts of reactants plus a catalyst at ppm levels. The phenomenon is often called the “sliver effect” because the design space becomes a very narrow region for the low-level component, as shown in Figure 7.

One way to avoid a sliver is to change the metric: in this case, changing to molar percent may put the components on a comparable basis and all components could have been included in the mixture design. Or, if I’m still avoiding mixtures, a practical solution is a combined design: treat the main ingredients as a mixture and the catalyst as a process variable. Both the mixture and the catalyst should be modeled quadratically to capture interactions. However, the interactive nature of components is best resolved when all ingredients are included in the mixture design.

The Bottom Line

For formulations, and recipes, the best results come from designs built specifically for mixtures. They’re not gimmicks or magic; they’re the right tools for the job. Stat-Ease provides tutorials and webinars to help you get started:

• A Crash Course in Mixture Design of Experiments
• Optimal experiment designs that combine mixture, process and categorical inputs

Or, if you’d prefer a hands-on, instructor-led experience (maybe with me!), sign up for one of the following courses:

• Fundamentals of Mixture DOE
• Mixture Design for Optimal Formulations

Mixture models (also known as "Scheffé," after the inventor) differ from standard polynomials by their lack of intercept and squared terms. For example, most of us learned about quadratic models in high school and/or college math classes, such as this one for two factors:

These models are extremely useful for optimizing processes via response surface methods (RSM) such as central composite designs (CCDs).

Mixture models look different. For example, consider this non-linear blending model for the melting point (Y) of copper (X₁) and gold (X₂) derived from a statistically designed mixture experiment*:

As you can see, this equation, set up to work with components coded on a 0 to 1 scale, does not include an intercept (ß₀) or squared terms (X₁², X₂²). However, it works quite well for predicting the behavior of a two-component mixture. The first-order coefficients, 1043 and 1072, are quite simple to interpret—these fitted values quantify the measured** melting points in degrees C for copper and gold, respectively. The difference of 29 characterizes the main-component effect (copper 29 degrees higher than gold).

The second-order coefficient of 536 is a bit trickier to interpret. It being negative characterizes the counterintuitive (other than for metallurgists) nonlinear depression of the melting point at a 50/50 composition of the metals. But be careful when quantifying the reduction in the melting: It is far less than you might think. Figure 1 tells the story.

First off, notice that the left side—100% copper—is higher than the right side—100% gold. This is caused by the main-component effect. Then observe the big dip in the middle created by a significant, second-order impact from non-linear blending. Because of this, the melting point reaches a minimum of 923 degrees C at and just beyond the 50/50 blend point. This falls 134 degrees below the average melting point of 1057 degrees. Given the coefficient of -536 on the X₁X₂ term, you probably expected a much bigger reduction. It turns out 541 divided by 4 equals 134. This is not coincidental—at the 50/50 blend point the product of the coded values reaches a maximum of 0.25 (0.5 x 0.5), and thus the maximum deflection is one-fourth (1/4) of the coefficient.

If your head is spinning at this point, I advise you not to attempt to interpret coefficients of the mixture model beyond the main component effects and, if significant, only the sign of the second-order, non-linear blending term, that is, whether it is positive or negative. Then after validating your model via Stat-Ease software diagnostics, visualize the model performance via our program’s wonderful model graphics—trace plot, 2D contour, and 3D surface. Follow up by doing a numeric optimization to pinpoint an optimum blend that meets all your requirements.

However, if you would like to truly master mixture modeling, come to our next Fundamentals of Mixture DOE workshop.

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A few years ago, while evaluating our training facility in Minneapolis, I came up with a fun experiment that demonstrates a great application of RSM for process optimization. It involves how sound travels to our students as a function of where they sit. The inspiration for this experiment came from a presentation by Tom Burns of Starkey Labs to our 5th European DOE User Meeting. As I reported in our September 2014 Stat-Teaser, Tom put RSM to good use for optimizing hearing aids.

Background

Classroom acoustics affect speech intelligibility and thus the quality of education. The sound intensity from a point source decays rapidly by distance according to the inverse square law. However, reflections and reverberations create variations by location for each student—some good (e.g., the Whispering Gallery at Chicago Museum of Science and Industry—a very delightful place to visit, preferably with young people in tow), but for others bad (e.g., echoing). Furthermore, it can be expected to change quite a bit from being empty versus fully occupied. (Our then-IT guy Mike, who moonlights as a sound-system tech, called these—the audience, that is—“meat baffles”.)

Sound is measured on a logarithmic scale called “decibels” (dB). The dBA adjusts for varying sensitivities of the human ear.

Frequency is another aspect of sound that must be taken into account for acoustics. According to Wikipedia, the typical adult male speaks at a fundamental frequency from 85 to 180 Hz. The range for a typical adult female is from 165 to 255 Hz.

Procedure

This experiment sampled sound on a 3x3 grid from left to right (L-R, coded -1 to +1) and front to back (F-B, -1 to +1)—see a picture of the training room above for location—according to a randomized RSM test plan. A quadratic model was fitted to the data, with its predictions then mapped to provide a picture of how sound travels in the classroom. The goal was to provide acoustics that deliver just enough loudness to those at the back without blasting the students sitting up front.

Using sticky notes as markers (labeled by coordinates), I laid out the grid in the Stat-Ease training room across the first 3 double-wide-table rows (4th row excluded) in two blocks:

1. 2² factorial (square perimeter points) with 2 center points (CPs).
2. Remainder of the 32 design (mid-points of edges) with 2 additional CPs.

I generated sound from the Online Tone Generator at 170 hertz—a frequency chosen to simulate voice at the overlap of male (lower) vs female ranges. Other settings were left at their defaults: mid-volume, sine wave. The sound was amplified by twin Dell 6-watt Harman-Kardon multimedia speakers, circa 1990s. They do not build them like this anymore 😉 These speakers reside on a counter up front—spaced about a foot apart. I measured sound intensity on the dBA scale with a GoerTek Digital Mini Sound Pressure Level Meter (~$18 via Amazon).

Results

I generated my experiment via the Response Surface tab in Design-Expert® software (this 3³ design shows up under "Miscellaneous" as Type "3-level factorial"). Via various manipulations of the layout (not too difficult), I divided the runs into the two blocks, within which I re-randomized the order. See the results tabulated below.

Notice that the readings at the center are consistently lower than around the edge of the three-table space. So, not surprisingly, the factorial model based on block 1 exhibits significant curvature (p<0.0001). That leads to making use of the second block of runs to fill out the RSM design in order to fit the quadratic model. I was hoping things would play out like this to provide a teaching point in our DOESH class—the value of an iterative strategy of experimentation.

The 3D surface graph shown below illustrates the unexpected dampening (cancelling?) of sound at the middle of our Stat-Ease training room.

Perhaps this sound ‘map’ is typical of most classrooms. I suppose that it could be counteracted by putting acoustic reflectors overhead. However, the minimum loudness of 57.4 (found via numeric optimization and flagged over the surface pictured) is very audible by my reckoning (having sat in that position when measuring the dBA). It falls within the green zone for OSHA’s decibel scale, as does the maximum of 73.6 dBA, so all is good.

What next

The results documented here came from an empty classroom. I would like to do it again with students (aka meat baffles) present. I wonder how that will affect the sound map. Of course, many other factors could be tested. For example, Rachel from our Front Office team suggested I try elevating the speakers. Another issue is the frequency of sound emitted. Furthermore, the oscillation can be varied—sine, square, triangle and sawtooth waves could be tried. Other types of speakers would surely make a big difference.

What else can you think of to experiment on for sound measurement? Let me know.

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Here's the latest Publication Roundup! In these monthly posts, we'll feature recent papers that cited Design-Expert® or Stat-Ease® 360 software. Please submit your paper to us if you haven't seen it featured yet!

Featured Article

Green extraction of poplar type propolis: ultrasonic extraction parameters and optimization via response surface methodology
BMC Chemistry, 19, Article number: 266 (2025)
Authors: Milena Popova, Boryana Trusheva, Ralitsa Chimshirova, Hristo Petkov, Vassya Bankova

Mark's comments: A worthy application of response surface methods for optimizing an environmentally friendly process producing valuable bioactive compounds. I see they used Box-Behnken designs appropriately - good work!

Be sure to check out this important study, and the other research listed below!

More new publications from October

1. Development and evaluation of a battery powered harvester for sustainable leafy vegetable cultivation
   Scientific Reports, volume 15, Article number: 33812 (2025)
   Authors: Kalluri Praveen, Yenikapalli Anil Kumar, Atul Kumar Shrivastava
2. Rutin/ZnO/mesoporous Silica-based Nano-hydrogel accelerated topical wound healing in albino mice via potential synergistic bioactive response
   European Journal of Pharmaceutics and Biopharmaceutics, Volume 216, November 2025, 114875
   Authors: Huma Butt, Haji Muhammad Shoaib Khan, Muhammad Sohail, Amina Izhar, Farhan Siddique, Maryam Bashir, Usman Aftab, Hasnain Shaukat
3. Production of improved Ethiopian Tej using mixed lactic acid bacteria and yeast starter cultures
   Scientific Reports, volume 15, Article number: 33460 (2025)
   Authors: Ketemaw Denekew, Fitsum Tigu, Dagim Jirata Birri, Mogessie Ashenafi, Feng-Yan Bai, Asnake Desalegn
4. Design and Optimization of Trastuzumab-Functionalized Nanolipid Carriers for Targeted Capecitabine Delivery: Anti-Cancer Effectiveness Evaluation in MCF-7 and SKBR3 Cells
   International Journal of Nanomedicine, Volume 2025:20 Pages 12075—12102, 3 October 2025
   Authors: Shubhashree Das, Bhabani Sankar Satapathy, Gurudutta Pattnaik, Sovan Pattanaik, Yahya Alhamhoom, Mohamed Rahamathulla, Mohammed Muqtader Ahmed, Ismail Pasha
5. **Design and test analysis of a rotary cutter device for root cutting of golden needle mushroom
   Scientific Reports, volume 15, Article number: 37219 (2025)
   Authors: Limin Xie, Yuxuan Gao, Zhiqiang Lin, Feifan He, Wenxin Duan, Dapeng Ye
6. Central composite design optimized fluorescent method using dual doped graphene quantum dots for lacosamide determination in biological samples
   Scientific Reports, volume 15, Article number: 36507 (2025)
   Authors: Ahmed Serag, Rami M. Alzhrani, Reem M. Alnemari, Maram H. Abduljabbar, Atiah H. Almalki
7. Innovation in functional bakery products: formulation and analysis of moringa-fortified millet cookies
   Journal of Food Measurement and Characterization, Published: 17 October 2025
   Authors: Anshu, Neeru & Ashwani Kumar
8. Improving the efficacy and targeting of letrozole for the control of breast cancer: in vitro and in vivo studies
   Naunyn-Schmiedeberg's Archives of Pharmacology, Published: 13 October 2025
   Authors: Shahira F. El Menshawe, Seif E. Ahmed, Amr Gamal Fouad, Amira H. Hassan
9. Optimization and Evaluation of Functionally Engineered Paliperidone Nanoemulsions for Improved Brain Delivery via Nasal Route
   Molecular Pharmaceutics, Published October 7, 2025
   Authors: Niserga D. Sawant, Pratima A. Tatke, Namita D. Desai
10. Innovative inhalable dry powder: nanoparticles loaded with Crizotinib for targeted lung cancer therapy
    BMC Cancer, volume 25, Article number: 1526 (2025)
    Authors: Faiza Naureen, Yasar Shah, Maqsood Ur Rehman, Fazli Nasir Fazli Nasir, Abdul Saboor Pirzada, Jamelah Saleh Al-Otaibi, Maria Daglia, Haroon Khan

Here's the latest Publication Roundup! In these monthly posts, we'll feature recent papers that cited Design-Expert® or Stat-Ease® 360 software. Please submit your paper to us if you haven't seen it featured yet!

When choosing a featured article for each month, we try to make sure it's available for everyone to read. Unfortunately, none of this month's publications that met our standards are available to the public, so there's no featured article this month. We still recommend checking out the incredible research done by these teams, and congratulations to everyone for publishing!

New publications from September

1. Evaluating the efficacy of nintedanib-invasomes as a therapy for non-small cell lung cancer
   European Journal of Pharmaceutics and Biopharmaceutics, Volume 214, September 2025, 114810
   Authors: Tamer Mohamed Mahmoud, Mohamed AbdElrahman, Mary Eskander Attia, Marwa M. Nagib, Amr Gamal Fouad, Amany Belal, Mohamed A.M. Ali, Nisreen Khalid Aref Albezrah, Shatha Hallal Al-Ziyadi, Sherif Faysal Abdelfattah Khalil, Mary Girgis Shahataa, Dina M. Mahmoud
2. Optimization of fermentation conditions for bioethanol production from oil palm trunk sap
   Journal of the Indian Chemical Society, Volume 102, Issue 9, September 2025, 101943
   Authors: Abdul Halim Norhazimah, Teh Ubaidah Noh, Siti Fatimah Mohd Noor
3. Microwave-assisted modification of a solid epoxy resin with a Peruvian oil
   Progress in Organic Coatings, Volume 206, September 2025, 109333
   Authors: Daniel Obregón, Antonella Hadzich, Lunjakorn Amornkitbamrung, G. Alexander Groß, Santiago Flores
   Authors: Hüsniye Hande Aydın, Esra Karataş, Zeynep Şenyiğit, Hatice Yeşim Karasulu
4. Development of an innovative method of Salmonella Typhi biofilm quantification using tetrahydrofuran and response surface methodology
   Microbial Pathogenesis, Volume 208, November 2025, 107992
   Authors: Aditya Upadhyay, Dharm Pal, Awanish Kumar
5. Bauhinia monandra derived mesoporous activated carbon for the efficient adsorptive removal of phenol from wastewater
   Scientific Reports volume 15, Article number: 31790 (2025)
   Authors: Bhojaraja Mohan, Chikmagalur Raju Girish, Gautham Jeppu, Praveengouda Patil
6. Quality by Design-Driven Development, Greenness, and Whiteness Assessment of a Robust RP-HPLC Method for Simultaneous Quantification of Ellagic, Sinapic, and Syringic Acids
   Separation Science Plus, Volume 8, Issue 9, September 2025, e70126
   Authors: V. S. Mannur, Rahul Koli, Atith Muppayyanamath
7. Dissolution and separation of carbon dioxide in biohydrogen by monoethanolamine-based deep eutectic solvents
   Journal of Chemical Technology and Biotechnology, Early View, 12 September 2025
   Authors: Xiaokai Zhou, Yanyan Jing, Cunjie Li, Quanguo Zhang, Yameng Li, Tian Zhang, Kai Zhang
8. Strategic Implementation of Analytical Quality by Design in RP-HPLC Method Development for Andrographis paniculata and Chrysopogon zizanioides Extract-Loaded Phytosomes
   Separation Science Plus, Volume 8, Issue 9, September 2025, e70125
   Authors: Abisesh Muthusamy, Vinayak Mastiholimath, Darasaguppe R. Harish, Atith Muppayyanamath, Rahul Koli
9. Improving the bioavailability and therapeutic efficacy of valsartan for the control of cardiotoxicity-associated breast cancer
   Journal of Drug Targeting, Published online: 29 Sep 2025
   Authors: Mary Eskander Attia, Fatma I. Abo El-Ela, Saad M. Wali, Amr Gamal Fouad, Amany Belal, Fahad H. Baali, Nisreen Khalid Aref Albezrah, Mohammed S. Alharthi, Marwa M. Nagib