Design space for mixture of four ingredients
The design space for mixtures of 3 ingredients can be understood most easily as an equilateral triangle in three dimensions. The corresponding diagram for 4 ingredients would involve four dimensions and therefore cannot be drawn.
However it is possible to extend our 2-dimensional view of the equilateral triangle representing the design space into a third dimension to visualise the design space for mixtures of 4 ingredients. The allowable mixtures of 4 ingredients correspond to points in a regular tetrahedron — a 3-dimensional pyramid with 4 faces that are each equilateral triangles.
Each vertex of this tetrahedron corresponds to 100% of one ingredient and the opposite face corresponds to none of that ingredient.
The diagram at the end of this page illustrates.
Simplex-centroid design
For k ingredients, this design consists of:
With k = 4 ingredients, the simplex-centroid design consists of the following 15 design points:
Run | Propn of A x |
Propn of B z |
Propn of C u |
Propn of D v |
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Pure mixtures |
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Mixtures of 2 |
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Mixtures of 3 |
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Mixture of 4 |
Simplex-lattice designs
This type of design is again defined in exactly the same way as for mixtures of 3 ingredients. The design depends on a constant m and the design points are:
For four ingredients and m = 3, the design points are:
Run | Propn of A x |
Propn of B z |
Propn of C u |
Propn of D v |
---|---|---|---|---|
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These designs are shown graphically in the diagram below.
Mixtures of four fruit juices
The tetrahedron below corresponds to all possible mixtures of four proportions. Each vertex represents 100% of one juice and the opposite face corresponds to mixtures that have none of this juice.
The diagram initially shows the design points for the simplex-centroid design. Click on any circle design point to read off the corresponding proportions.
Select Simplex-lattice design from the pop-up menu and alter the constant m to investigate the design points for these types of design.