Stat-Ease Blog

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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Most people who have been exposed to design of experiment (DOE) concepts have probably heard of factorial designs—designs that target the discovery of factor and interaction effects on their process. But factorial designs are hardly the only tool in the shed. And oftentimes to properly optimize our system a more advanced response surface design (RSM) will prove to be beneficial, or even essential.

Let’s start by introducing the Stat-Ease® software menu options for RSM designs. Once we understand the alternatives we can better understand when which might be most useful for any given situation and why optimal designs are great—when needed.

• First on the list is the central composite design (our software default)
• Next is the Box-Behnken design
• And third is something called optimal design

The natural question that often pops up is this. Since optimal designs are third on our list, are we defaulting to suboptimal designs? Let’s dig in a bit deeper.

The central composite design (“CCD”) has traditionally been the workhorse of response surface methods. It has a predictable structure (5 levels for each factor). It is robust to some variations in the actual factor settings, meaning that you will still get decent quadratic model fits even if the axial runs have to be tweaked to achieve some practical values, including the extreme case when the axial points are placed at the face of the factorial “cube” making the design a 3-level study. A CCD is the design of choice when it fits the problem and generally creates predictive models that are effective throughout the design space--the factorial region of the design. Note that the quadratic predictive models generally improve when the axial points reside outside the face of the factorial cube.

When a 5-level study is not practical, for example, if we are looking at catalyst levels and the lower axial point would be zero or a negative number, we may be forced to bring the axial points to the face of the factorial cube. When this happens, Box-Behnken designs would be another standard design to consider. It is a 3-level design that is laid out slightly differently than a CCD. In general, the Box-Behnken results in a design with marginally fewer runs and is generally capable of creating very useful quadratic predictive models.

These standard designs are very effective when our experiments can be performed precisely as scripted by the design template. But this is not always the case, and when it is not we will need to apply a more novel approach to create a customized DOE.

Optimal designs are “custom” creations that come in a variety of alphabet-soup flavors—I, D, A, G, etc. The idea with optimal designs is that given your design needs and run-budget, the optimization algorithm will seek out the best choice of runs to provide you with a useful predictive model that is as effective as possible. Use of the system defaults when creating optimal designs is highly advised. Custom optimal designs often have fewer runs than the central composite option. Because they are generated by a computer algorithm, the number of levels per factor and the positioning of the points in the design space may be unique each time the design is built. This may make newcomers to optimal designs a bit uneasy. But, optimal designs fill the gap when:

• The design space is not “cuboidal”— there are constraints on the operating region that make the design space lopsided or truncated.
• There are categoric or discrete numeric factors to deal with.
• The expected polynomial model is something other than a full quadratic.
• You are trying to augment an existing design to expand the design space or to upgrade to a higher order model.

The classic designs provide simple and robust solutions and should always be considered first when planning an experiment. However, when these designs don’t work well because of budget or practical design space constraints, don’t be afraid to go “outside the box” and explore your other options. The goal is to choose a design that fits the problem!

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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

Quantification of Menthol by Chromatography Applying Analytical Quality by Design
Chromatographia, Published: 17 January 2026
Authors: Santiago Puentes, Julian Puentes, Ronald Andrés Jiménez & Emerson León Ávila

Mark's comments: Well done deploying the Ishikawa (fishbone) diagram to consider all the factors that might affect the analytical method.  This is a great application of a multifactor approach to achieve quality by design.

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

More new publications from January

1. UHPLC Method Development for the Trace Level Quantification of Anastrozole and Its Process-Related Impurities Using Design of Experiments
   Biomedical Chromatography, Volume 40, Issue 2 February 2026 e70279
   Authors: Narasimha Rao Tadisetty, Ramesh Yanamandra, Subhalaxmi Pradhan, Gitanjali Pradhan
2. Combinatorial Anti-Mitotic Activity of Loratadine/5-Fluorouracil Loaded Zein Tannic Acid Nanoparticles in Breast Cancer Therapy: In silico, in vitro and Cell Studies
   International Journal of Nanomedicine, 2026; 21:1-27
   Authors: Mohamed Hamdi, Moawia M Al-Tabakha, Isra H Ali, Islam A Khalil
3. Synergistic effects and optimization of palm oil fuel ash and jute fiber in sustainable concrete
   Scientific Reports volume 16, Article number: 259 (2026)
   Authors: Muhammad Umer, Nawab Sameer Zada, Paul O. Awoyera, Muhammad Basit Khan, Wisal Ahmed, Olaolu George Fadugba
4. Harnessing sawdust for clean energy: production and optimization of bioethanol in Nigeria
   MOJ Ecology & Environmental Sciences, 2026; 11(1):5‒11.
   Authors: Ifeoma Juliet Opara, Eno-obong Sunday Nicholas, Ezekiel Adi Siman
5. Antibiotic-phytochemical combinations against Enterococcus faecalis: a therapeutic strategy optimized using response surface methodology
   Antonie van Leeuwenhoek, 119, 41 (2026)
   Authors: Monikankana Dasgupta, Alakesh Maity, Ranojit Kumar Sarker, Payel Paul, Poulomi Chakraborty, Sarita Sarkar, Ritwik Roy, Moumita Malik, Sharmistha Das & Prosun Tribedi
6. Breakthrough RP-HPLC strategy for synchronous analysis of pyridine and its degradation products in powder for injection using quality metrics
   Scientific Reports volume 16, Article number: 1344 (2026)
   Authors: Asma S. Al‐Wasidi, Noha S. katamesh, Fahad M. Alminderej, Sayed M. Saleh, Hoda A. Ahmed, Mahmoud A. Mohamed

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

Sustainable protection for mild steel (S235): Anti-corrosion effectiveness of natural Paraberlinia bifoliolata barks extracts in HCl 1 M
Hybrid Advances, Volume 11, December 2025, 100526
Authors: Liliane Nga, Benoit Ndiwe, Jean Jalin Eyinga Biwolé, Armand Zébazé Ndongmo, Achille Desiré Omgba Béténé, Yvan Sandy Nké Ayinda, Joseph Zobo Mfomo, Cesar Segovia, Antonio Pizzi, Achille Bernard Biwolé

Mark's comments: I like this for its application of response surface methods to thoroughly explore three process factors for optimization of a 'green' corrosion inhibitor. They then provide a scientific explanation of its "remarkable effectiveness."

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

More new publications from December

1. Sustainable synthesis of Mangenese cobalt oxide nanocomposite on natural clay and optimization using response surface methodology for ciprofloxacin degradation via peroxymonosulfate activation
   Applied Surface Science, Volume 712, 7 December 2025, 164136
   Authors: Amira Hrichi, Nesrine Abderrahim, Hédi Ben Amor, Marta Pazos, Maria Angeles Sanromán
2. Optimization of fermentation conditions for lactic acid production by Lactiplantibacillus plantarum OM510300 using plantain peduncles as substrate
   Scientific African, Volume 30, December 2025, e02998
   Authors: Oluwafemi Adebayo Oyewole, Japhet Gaius Yakubu, Nofisat Olaide Oyedokun, Konjerimam Ishaku Chimbekujwo, Priscila Yetu Tsado, Abdullah Albaqami
3. QbD-Based HPLC Method Development for Trace-Level Detection of Genotoxic 3-Methylbenzyl Chloride in Meclizine HCl Formulations
   Biomedical Chromatography, Volume 39, Issue 12, December 2025, e70238
   Authors: Naga Kranthi Kumar Chintalapudi, Naresh Podila, Vijay Kumar Chollety
4. Multi-objective intelligent optimization design method for mix proportions of hydraulic asphalt concrete facings
   Ain Shams Engineering Journal, Volume 16, Issue 12, December 2025, 103781
   Hanye Xiong, Zhenzhong Shen, Yiqing Sun, Yaxin Feng, Hongwei Zhang
5. Sustainable pretreatment and adsorption of chemical oxygen demand from car wash wastewater using Noug sawdust activated carbon Scientific Reports volume 15, Article number: 42464 (2025)
   Authors: Getasew Yirdaw, Abraham Teym, Wolde Melese Ayele, Mengesha Genet, Ahmed Fentaw Ahmed, Assefa Andargie Kassa, Tilahun Degu Tsega, Chalachew Abiyu Ayalew, Getaneh Atikilt Yemata, Tesfaneh Shimels, Rahel Mulatie Anteneh, Abathun Temesgen, Gashaw Melkie Bayeh, Almaw Genet Yeshiwas, Habitamu Mekonen, Berhanu Abebaw Mekonnen, Meron Asmamaw Alemayehu, Sintayehu Simie Tsega, Zeamanuel Anteneh Yigzaw, Amare Genetu Ejigu, Wondimnew Desalegn Addis, Birhanemaskal Malkamu, Kalaab Esubalew Sharew, Daniel Adane, Chalachew Yenew
6. Ternary phase optimized indomethacin nanoemulsion hydrogel for sustained topical delivery and improved biological efficacy
   Scientific Reports volume 15, Article number: 43322 (2025)
   Authors: K. R. Nithin, G. M. Pallavi, K. S. Srikruthi, Kasim Sakran Abass, Nimbagal Raghavendra Naveen
7. Numerical and experimental investigation of CI engine parameters using palm biodiesel with diesel blends
   Scientific Reports volume 15, Article number: 43399 (2025)
   Authors: B. Musthafa, M. Prabhahar
8. Development a practical method for calculation of the block volume and block surface in a fractured rock mass
   Scientific Reports volume 15, Article number: 44068 (2025)
   Authors: Alireza Shahbazi, Ali Saeidi, Alain Rouleau, Romain Chesnaux
9. Biofabrication, statistical optimization, and characterization of collagen nanoparticles synthesized via Streptomyces cell-free system for cancer therapy
   Scientific Reports volume 15, Article number: 43835 (2025)
   Authors: Noura El-Ahmady El-Naggar, Shaimaa Elyamny, Ahmad G. Shitifa, Asmaa Atallah El-Sawah