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
 















Pure mixtures

























Mixtures of 2















Mixtures of 3
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
































































































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.