In short
An accord's character depends on the proportions of its components, which must sum to one. That constraint means you cannot vary each component independently, so ordinary factorial designs do not apply. Mixture designs place trials at systematic points of the simplex — pure components, binary blends, the centroid — and a Scheffé polynomial fitted to them shows where components blend synergistically and where they fight. The centroid then tests the model.
What you will learn
- Explain why proportions summing to one need mixture designs rather than ordinary factorial designs
- Lay out a simplex-lattice and a simplex-centroid design for three components
- Fit a quadratic Scheffé model and interpret its blending coefficients
- Use a checkpoint to test the model rather than trusting its fit
Before this lesson: Optimization Under Constraints: Cost and Composition, Ratio Ladders: Finding an Accord's Balance Point, Designing a Fragrance Experiment
Before you start
You should have built ratio ladders, designed a fragrance experiment, and met optimization in the previous lesson. A ratio ladder explores one edge of the simplex; this lesson explores its whole surface.
The geometry
Three components with x1 + x2 + x3 = 1 live on a triangle: the vertices are pure components, the edges binary blends, the interior three-way blends. Each "component" can itself be a sub-blend at a fixed dilution, so long as its internal ratio stays fixed throughout the experiment.
| Design | Points | Runs |
|---|---|---|
| Simplex-lattice {3,2} | (1,0,0), (0,1,0), (0,0,1), (½,½,0), (½,0,½), (0,½,½) | 6 |
| Simplex-centroid | the six above plus (⅓,⅓,⅓) | 7 |
The Scheffé model
Because the proportions sum to one, an intercept is not separately identifiable; Scheffé's canonical quadratic is
y = b1x1 + b2x2 + b3x3 + b12x1x2 + b13x1x3 + b23x2x3
On the {3,2} lattice, the six coefficients follow directly from the six responses. At a vertex only one term survives, so bi = yi. At a 50:50 binary, yij = ½bi + ½bj + ¼bij, so
bij = 4yij − 2yi − 2yj
Worked example
Panel mean ratings, invented for the arithmetic, of "how well it reads as the intended accord" on a 0–9 scale: pure components 5, 3 and 4; binaries 1–2 at 6, 1–3 at 4.5, 2–3 at 3.
- b12 = 4(6) − 2(5) − 2(3) = 8: strong synergy.
- b13 = 4(4.5) − 2(5) − 2(4) = 0: they simply average.
- b23 = 4(3) − 2(3) − 2(4) = −2: mild antagonism.
The model predicts the centroid at (5 + 3 + 4)/3 + (8 + 0 − 2)/9 = 4 + 0.67 = 4.67. If the panel scores the centroid at about 4.7, the quadratic model is adequate there. If it scores 6.5, a three-way interaction (the special cubic term b123x1x2x3) is at work, and the model's interior predictions should not be trusted.
Where it stops
- Six points, six parameters. The lattice model fits its own points exactly, so its fit says nothing; only checkpoints test it.
- Panel noise. With a handful of assessors, a difference of one scale point between blends may be noise. The Thurstonian lesson and the PCA lesson address what a panel can resolve.
- Amount matters too. A mixture design holds total dose fixed. How strong the accord is at use level is a separate factor (a mixture-amount design).
- Time matters. An accord rated at five minutes may rank differently at four hours; rate at stated times.
In professional practice
Mixture designs are standard in food and beverage formulation and in materials science. In fragrance they suit exactly the decisions a ratio ladder makes by intuition: where in a three-way blend the effect peaks. They also feed optimization: a fitted model becomes the character constraint the previous lesson needed.
Practical exercise: Run a seven-point simplex-centroid on one accord
- Choose three components for an accord, each at a fixed dilution. Make the seven simplex-centroid blends at the same total weight, plus a second centroid as a replicate.
- Code all eight. Have at least five people rate each on one clearly defined scale at a stated time after dipping, in randomized order.
- Fit the quadratic model from the six lattice points. Predict the centroid, and compare it with both measured centroids.
- Sketch the predicted response across the triangle, and mark where you would place the next run.
You have done this well if the blends differ only in proportion, the samples were coded and randomized, the two centroid replicates give you an estimate of noise, and your conclusion about the model's adequacy is stated against that noise, not against a single number.
Common mistakes
- Varying the total dose between points. Then you are testing amount, not proportion.
- Judging the model by its fit. Use checkpoints.
- Undefined rating scales. "Better" means different things to different raters; define the question.
Safety
All blends are evaluated on strips. Read each material's SDS and follow the safe handling lesson; panelists are briefed as in the ethics lesson.
Key terms
- Mixture experiment
- An experiment in which the response depends on the proportions of the components, which sum to one, rather than on their absolute amounts.
- Simplex-lattice design
- A mixture design whose points are all combinations of proportions 0, 1/m, 2/m … 1 that sum to one. The {3,2} lattice has the three pure components and the three 50:50 binaries.
- Blending coefficient
- In a Scheffé model, the coefficient of a product term such as x1x2. Positive means the two components together score higher than their average (synergy); negative, lower.
Knowledge check
Written and reviewed by the RUŌOD Lab team. This is general education, not legal, regulatory, medical or safety advice. Example dilutions and exercises are for learning, are not validated products, and have not been safety-assessed. Completing a lesson confers no qualification. How we check these lessons.