General principles

In a similar way to experimental designs for unconstrained mixtures, the main requirement of a good design for a constrained mixture is that the design points should cover as much as possible of the design space. More specific requirements for fitting a quadratic surface are:

Also,

Designs for mixtures of 3 ingredients

With mixtures of only 3 ingredients, it is possible to sketch the design space and pick points 'by eye'. An example is given below.

Designs for constrained mixtures

Generating good mixture designs subject to constraints on the maximum values of proportions is harder when there are four or more ingredients in the mixture.

Most statisticians use computer software to generate mixture designs when there are constraints on the maximum values of some proportions.


Fat content of biscuits

The constraints on the mixture for this experiment are shown in the table below.

Ingredient  Minimum   Maximum 
Butter 5.5% 8%
Margarine 0% 0.7%
Canola    0% 1.8%

Of these, it is the constraints on Margarine and Canola that are most important since the limits for Butter do not further restrict the mixture. It is therefore possible to use a 32 factorial design for the levels of Margarine and Canola to cover the design space, with the proportion of Butter following as one minus the sum of the other levels.

Ingredient  Low level   Medium level   High level 
Margarine 0% 0.35% 0.7%
Canola    0% 0.9% 1.8%

This results in the following design points.

Margarine  Canola   Butter 
0% 0% 8%
0% 0.9% 7.1%
0% 1.8% 6.2%
0.35% 0% 7.65%
0.35% 0.9% 6.75%
0.35% 1.8% 5.85%
0.7% 0% 6.3%
0.7% 0.9% 5.4%
0.7% 1.8% 5.5%

The diagram below shows the design.

Click on points to read off the proportions.

Analysis of experimental results

The design of mixture experiments subject to constraints is harder than the corresponding problem for unconstrained mixtures. However after the experiment is conducted, the analysis of the resulting data proceeds in exactly the same way.

Again, we usually fit linear or quadratic models to the data (though we could only be confident about their fit within our design space). And analysis of variance can again test for curvature of the response surface and for lack of fit of the quadratic model.