Generating a mixture design and analysing the results with XLSTAT

This tutorial explains how to generate and interpret a mixture design with Excel using XLSTAT.

Dataset for generating a mixture design

The data is described by Phan Tan Luu. We want to generate a mixture design based on a Scheffé simplex of degree 2.

The design {3,2} has 6 experiments:

• Pure constituents (1; 0; 0)
• Binary mixtures (0.5; 0.5; 0)

The total quantity of mixture is defined as a proportion, equal to 1 (100%).

In order to have enough degrees of freedom to implement a t-test and an analysis of variance, we opt for 2 test replications.

Goal of this tutorial

Mixture designs are used to model the results of experiments aiming to optimize formulations. The output model is called "mixture distribution".

In this tutorial, we want to measure the effectiveness of a thinner by studying the hardness of a material. The mixture (diluent) has three components: lactose, CACO3 and cellulose.

Setting up a mixture design

Once XLSTAT is open, click on the DOE button on the XLSTAT ribbon and select Mixture Design.

The dialog box Mixture Design pops up.

In the General tab, select the table of quantitative factors (this table must contain two rows, one with the minimum and one for with maximum values per factors), the number of responses (1), and choose 2 repetitions.

In the Options tab, select a simplex design of two degrees as the type of design, and set the total quantity of mixture equal to 1.

Click OK to start the computations.

Results of the generation of a mixture design

A summary table containing information about the factors is displayed.

The experimental design is then presented in the form of a table, in which you need to fill the Response column with the results per experiments. The response optimization table is automatically set up once the response column is filled.

Let’s suppose, that the 12 experiments have been carried out and the values have been entered in Response column.

We select Maximum in the Goal cell of the optimization table in order to search for the optimum values and obtain the most possible effective diluent.

Setting up the analysis of a mixture design

There are two options to start the design analysis.

• Click on the launch analysis button located below the design. This first solution makes it possible to automatically fill in the fields of the dialog box.
• Click on the DOE button on the XLSTAT ribbon and select analysis of a mixture design.

A dialog box pops up. If the box is not pre-filled, select in the general tab the Response column and the table of the design. Be careful to select all the columns of the design so that the analysis will work.

In the Options tab, check the responses optimization option and select the appropriate table, including the columns headings as well.

Once you've clicked OK, the calculations begin. A new dialog box pops up in order to modify, if you wish, the settings of the analysis.

Interpret the results of a mixture design analysis

The first table provides the results of the responses optimization. This shows us several effective solutions to achieve the maximum efficiency of the diluent. Here, the pure lactose product is the best.

The results of the analysis of variance are then displayed, starting with the goodness of fits coefficients. R² equals to 1, which indicates a very good fit of the model.

The model parameters and the equation are also provided in the report sheet.

The experimental domain is displayed in a ternary diagram.

The optimum can be read directly on the ternary diagram (Maximum Value): this is the configuration with the pure Lactose product (configuration 1,0,0). It confirms the result previously found based on the response optimization.