「三角図」を用いて配合実験の等高線をグラデーション表示する方法

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In JMP, there is a platform in the Graph menu that allows you to create a graph called a " Territorial Plot ." This is a useful graph for examining distribution and variability by graphing data with three components, but there are also cases where it is effective to use it in combination experiments in experimental design.

The Ternary Plot allows you to draw graphs like the one shown below. The color gradation on the graph represents the predicted values ​​of the model created using the experimental design method as contour lines.

What steps do you take to draw a diagram like this? I will explain using a concrete example of a compounding experiment.

What is a blending experiment?

A mixture experiment is an experiment in which the components (ratios) of the mixture are factors, and each factor can take a value between 0 (0%) and 1 (100%). In addition, the sum of the ratios of all factors is restricted to 1 (100%). Therefore, the experimental space is a (number of factors - 1)-dimensional region.

For example, when there are three blending factors X1, X2, and X3, the constraint X1+X2+X3 = 1 exists, so the experimental space is a two-dimensional region formed by cutting a triangular plane out of a three-dimensional cube.

Example of a blending experiment

In tablet design for pharmaceutical development, suppose you want to investigate the relationship between the following three excipients (additives added to solid dosage forms) and tablet hardness:

Excipients	The upper and lower limits of the blending ratio
X1: Alpha Lactose Monohydrate	20%～70%
X2: Potato Starch	20%～80%
X3: Anhydrous Alpha-Lactose	0%～60%

Furthermore, from a physical property standpoint, the total of X2 and X3 must be kept within 70%.

Visualize your experimental points with a ternary plot

Using JMP's "Custom Design," we set the conditions for the above mixture factors. In this example, we set a model that includes the main effects of the mixture factors and two-way interactions, and run 12 experiments.

The created mixture experiment design can be visualized using the "Terriangle Plot" platform. The gray area in the terriangle plot is outside the range of possible values ​​for each factor, and the light blue area is the area where X2 + X3 ≦ 0.7 is not satisfied. In other words, the white area in the terriangle plot is the experimental space in this example.

The black points plotted in the experimental space are the experimental points. The vertices, edges, and areas near the center of the white area are the experimental points.

Based on the experimental points obtained from the custom design, measure the hardness (N) of the tablets manufactured with different excipient ratios and enter this in the Y (Response) cell of the table.

Once the experiment is complete, we fit the mixture model using the Fit Model platform. In Fit Model, there is an option called Mixture Profiler , which also allows us to plot the equivalent of a ternary plot.

Here you can plot contours of predicted values ​​and explore the predicted tablet hardness for different blend ratios, but you cannot create color gradations according to the contour values.

In academic papers, it is common to see contour lines displayed with a gradient, so to do this you will need to use the "Terriangle Plot" platform in the Graph menu, as shown at the beginning.

Ternary plot reflecting prediction formula

In the "Terriangle Plot" you can include the contour lines of the prediction formula in the plot, and you can add gradation to the contour lines. The specific operation method is as follows.

Creating a ternary plot with contour lines (operation)

1. In the Fit Model report, click the red triangle and select Save Columns > Prediction Formula to save a column containing the prediction formula to your data table.
2. Select Graph > Ternary Plot and specify the three columns of mixture factors in X, Plot , and the column containing the prediction formula you saved in Contour Formula .
3. From the red triangle button in the upper left of the "Territorial Plot" report, select [Fill Between Contour Lines] > [Fill Above] .
4. If necessary, change the contour line color theme in the Color Theme options, change the numbers in the legend in the upper right, and change the contour line thresholds.

With these operations, we were able to include contour lines with gradation in the ternary plot. If you want to increase tablet hardness, you should consider the mixture ratios in the darker areas. For example, the lower right of the experimental area, X1: 30%, X2: 20%, X3: 50%, is where the predicted tablet hardness value is the highest, so you could create tablets with these mixture ratios, measure their hardness, and see if you can reproduce a high hardness.

When expressing the relationship between the response and the mixture factors in a mixture experiment as a contour line, it is easier to understand if the contour line is displayed with a color gradient. Therefore, it is good to know that JMP can display a contour gradient in the "Ternilinear Plot".

by Naohiro Masukawa (JMP Japan)

Naohiro Masukawa - JMP User Community