When the ingredients must add up to 100%
Concrete, an alloy, a paint, a feed mix: you are choosing proportions, and proportions sum to 1. This one constraint breaks the ordinary machinery of experimental design, and doing it anyway is a quiet, convincing kind of wrong.
Why a factorial cannot work here. A factorial varies one factor while holding the others fixed. In a formulation you cannot do that: adding more cement means less sand. The question "what is the effect of cement?" has no answer unless you say what it replaces. So the classical main effect is not merely hard to estimate — it is undefined.
Why there is no intercept. If x₁ + x₂ + x₃ = 1 always, then a column of ones is exactly the sum of the component columns. The design matrix is singular; the intercept and the components cannot be told apart. Scheffé's canonical polynomials are written without an intercept for precisely this reason. Fit a formulation with ordinary regression and the software will still hand you coefficients and a healthy-looking R² — which is what makes the mistake dangerous.
How to read the coefficients. A single-component coefficient is the predicted response of that component on its own. That is a directly meaningful number: it is what you would measure if you ran the pure material. A cross term is the blending effect. Positive means the pair does better together than their proportions alone would predict — synergy, and it is usually the thing worth finding. Negative means they interfere.
Designs. A simplex lattice spaces the points evenly over the triangle (or tetrahedron, for four components). A simplex centroid uses every equal-share subset: each pure component, every 50/50 pair, every third, and the overall blend. Either way you need points in the interior, not just the corners: pure components alone can only ever fit a straight plane, and the blending effects — the whole reason you are doing this — live in the middle.
One practical detail. If you record ingredients as percentages or as kilograms per batch, they sum to 100 or to the batch weight rather than to 1. SenSight rescales them to proportions and tells you it did. The fit is unchanged, but you should know which units the coefficients are in.
Why a factorial cannot work here. A factorial varies one factor while holding the others fixed. In a formulation you cannot do that: adding more cement means less sand. The question "what is the effect of cement?" has no answer unless you say what it replaces. So the classical main effect is not merely hard to estimate — it is undefined.
Why there is no intercept. If x₁ + x₂ + x₃ = 1 always, then a column of ones is exactly the sum of the component columns. The design matrix is singular; the intercept and the components cannot be told apart. Scheffé's canonical polynomials are written without an intercept for precisely this reason. Fit a formulation with ordinary regression and the software will still hand you coefficients and a healthy-looking R² — which is what makes the mistake dangerous.
How to read the coefficients. A single-component coefficient is the predicted response of that component on its own. That is a directly meaningful number: it is what you would measure if you ran the pure material. A cross term is the blending effect. Positive means the pair does better together than their proportions alone would predict — synergy, and it is usually the thing worth finding. Negative means they interfere.
Designs. A simplex lattice spaces the points evenly over the triangle (or tetrahedron, for four components). A simplex centroid uses every equal-share subset: each pure component, every 50/50 pair, every third, and the overall blend. Either way you need points in the interior, not just the corners: pure components alone can only ever fit a straight plane, and the blending effects — the whole reason you are doing this — live in the middle.
One practical detail. If you record ingredients as percentages or as kilograms per batch, they sum to 100 or to the batch weight rather than to 1. SenSight rescales them to proportions and tells you it did. The fit is unchanged, but you should know which units the coefficients are in.
Try it in the app
Try: DOE tab, Mixture Design. Generate a 3-component simplex lattice, note that every row sums to 1, and look at whether the cross terms come out positive (synergy) or negative.