Independently adjustable factors
The experiments that were considered in earlier sections all involved factors that could be independently controlled — any combination of factor values within the experimental region of interest was possible. For example, in an experiment to assess the effects of temperature and addition of an enzyme to the strength of a plastic, these two factors can be independently adjusted to give a run of the experiment at any desired combination of factor levels.
Mixtures
In contrast, some experiments are conducted to compare different mixtures of ingredients in a product. If there are k ingredients whose proportions in the mixture are denoted by x, z, ..., then the proportions are constrained to sum to exactly 1.0 (i.e. 100% of the product) so they cannot be independently adjusted.
Ingredient | Proportion |
---|---|
A | x |
B | z |
... | ... |
Total | 1.0 |
As an extreme example, if there is 100% of ingredient A in a run of the experiment, then the only allowable proportion for the other ingredients is zero.
Experimental designs for mixtures
When designing experiments for mixtures, we usually treat all k proportions symmetrically. For k = 3 ingredients whose proportions are x, z and u, the experimental region consists of values such that
0 ≤ x ≤ 1
0 ≤ z ≤ 1
0 ≤ u ≤ 1
x + z + u = 1
These constraints define an equilateral triangle in 3 dimensions, and the experimental design should choose design points within this triangle.
Mixtures of three fruit juices
Consider an experiment to compare the perceived sweetness of a drink that consists of a mixture of apple, orange and pineapple juices. The factors that can be controlled in the experiment are the proportions of these three juices.
The triangle in the diagram contains all possible proportions of the juices that sum to 1.0.
The 3 vertices of the triangle are mixtures containing only a single ingredient. Points on the edges are mixtures of 2 of the 3 ingredients.