Component Scaling in Mixture Designs

Three scales are used to transform the component levels to a convenient scale for analysis. The original scale used to input the component levels is the “Actual”. These are converted to the “Real” scale, with each component re-scaled so that the total is 1.0. In the real scale, the components are then fractions of each component in the mixture. If the total of the components is set to 80% then the “Real” values are the actual values divided by 0.80.

For analysis, the “Pseudo” scale is preferred. In the pseudo scale each component is rescaled to be a fraction of the “active” part of the mixture. L-Pseudo is the default scale, but U-Pseudo will be offered if collinearity is lower in this scale. (See mathematical description below.)

To illustrate, suppose that a three-component nut mixture will always contain at least 50% peanuts, at least 15% pecans and at least 5% cashews. Therefore, 70% of the nut mix is predetermined. We are going to do an experiment to determine the remaining 30%. In pseudo terms,

Peanut pseudo component = (% peanut – 50%) / 30%

Pecan pseudo component = (% pecan – 15%) / 30%

Cashew pseudo component = (% cashew – 5%) / 30%

If there are no lower limit constraints, the real and pseudo scales are identical.

After building the design you may change the scale via the Display Options menu.

L-Pseudo vs U-Pseudo

Lower bounded pseudo values: the minimum value of each component becomes zero; the maximum value approaches one. How close it comes to one depends on the constraints. For a simplex the component minimums are zero and the maximums are one.

Upper bounded pseudo values: the maximum value of each component becomes zero; the minimum value approaches one. How close it comes to one depends on the constraints. For an inverted simplex the component maximums are zero and the minimums are one.

Slack Variable Model Form

In some cases of constrained mixtures, it becomes difficult to get a good regression fit of the Scheffe model. This occurs when you introduce a wide disparity in the constraint ranges. For example, you might make a blend with one ingredient, such as a catalyst, that’s very powerful, so you constrain it to fractions of volume percent, while all other ingredients can be varied over wide volumes. The resulting space will be a narrow “sliver” with very poor design properties. If it’s really bad, then Stat-Ease will recommend you try the “slack variable” mixture model. This model resembles the usual RSM models except that one component must be removed, because it will be a linear combination of all the others. The best term to remove is usually the one with the widest range. You can change this if you like.

You cannot eliminate any of the other main effect terms, because they incorporate information on other terms. They will be forced into the model.

Because it excludes at least one component and all its interactions, we recommend that you not use the slack variable form if at all possible. Instead consider changing the metric of your mixture to a form that results in more even ranges. For example, you might convert from weight percent to mole percent in a chemical mixture. In any case, if it works, the design you get with the Scheffe model will likely be nearly as good as the slack form. You will still be able to analyze your data with the slack form should you feel it necessary.

We advise that you always evaluate your mixture design by using the Design Evaluation node before you run it. Any special problems with the design may then be discovered before it’s too late.

Mixture Coding for Model Fitting

Stat-Ease allows the user to specify the coding (pseudo, real or actual) to use for fitting the model.

• If you specify pseudo, the X-matrix uses pseudos for the coefficient

• calculations and the equation is back transformed to both reals and

• actuals.

• If you specify reals, the X-matrix uses reals for the coefficient

• calculations and the equation is back transformed to actuals, but not

• pseudos.

• If you specify actuals, the X-matrix uses actuals for the coefficient

• calculations and the equation is back transformed to reals, but not

• pseudos.

Scheffé Mix Models

Scheffé models were specifically developed to handle the natural constraints of mixture designs.

Mixture models are only readily interpretable when the mixture components all go from 0 to the total for the design. Most mixture designs cover a more constrained space. Use the Model Graphs to better understand the models.

The Scheffé model forms are as follows:

Linear

$\hat{y}= \sum_{i=1}^{q}\beta _{i} x_{i}$

Example

$12A + 8B + 4C$

$\hat{y}=\sum_{i=1}^{q} \beta_{i}x_{i}\,+\sum_{i<j}^{q-1}\sum_{j}^{q} \beta_{ij} x_{i} x_{j}$

Example

$12A + 8B +4C + 8AB - 8AC$

Special Cubic

$\hat{y} \sum_{i=1}^{q} \beta_{i}x_{i}+\sum_{i<j}^{q-1}\sum_{j}^{q} \beta_{ij}x_{i}x_{j}+\sum_{i<j}^{q-2}\sum_{j<k}^{q-1}\sum_{k}^{q}\beta_{ijk} x_{i}x_{j}x_{k}$

Example

$12A + 8B + 4C + 8AC - 8BC + 54ABC$

Full Cubic

$\hat{y}=\sum_{i=1}^{q}\beta_{i}x_{i}+\sum_{i<j}^{q-1} \sum_{j}^{q}\beta_{ij}x_{i}x_{j}+\sum_{i<j}^{q-1}\sum_{j}^{q}\delta_{ij}x_{i} x_{j}(x_{i}-x_{j})+\sum_{i<j}^{q-2}\sum_{j<k}^{q-1}\sum_{k}^{q}\beta_{ijk}x_{i} x_{j}x_{k}$

Example

$12A + 8B + 4C + 8AB - 8AC + 54ABC + 48AC(A - C)$

Standard Scheffé polynomials are available up to the fourth order. There are also partial quadratic mixture (PQM) models using a combination of linear, squared, and quadratic terms (see reference by Piepel, et al.).

References

• G. Piepel, J. Szychowski, and J. Loeppky. Augmenting scheffe linear mixture models with squared and/or crossproduct terms. Journal of Quality Technology, 2002.

KCV Model for Combined Mixture-Process Designs

Standard crossed models generate far more terms than needed to detect likely interactions between mixture and process variables. For example, see the tutorial on Combined Mixture-Process – Fish Patties . Kowalski, Cornell, and Vining (KCV) simplified the equation by only crossing the linear models; adding mixture and process terms beyond that. When specified for an optimal (custom) design, this streamlined KCV model saves many runs, yet it detects probable interactions between components and factors. Furthermore, it accommodates categorical factors.

In the case of fish patties with 3 mixture components and 3 process factors the KCV model reduces the model to 21 terms versus 56 for the crossed Special Cubic x 2FI equation. That produces a savings of over 50 percent (21/56) on required runs for an optimal (custom) design. The KCV model (process factors bolded and crossed terms underlined) is: A, B, C, AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF, D2, E2, F2. The optional crossed model (not detailed—it being too tedious) includes many high-order terms that will almost certainly be insignificant, e.g., AD2, ABCDE, ABCD2, etc. All these terms must be fitted by a unique run. This would be very wasteful.

Refer to the reference by Kowalski, Cornell and Vining for complete mathematical details on KCV model construction.

References

• S. Kowalski, J. Cornell, and G. Vining. A new model and class of designs for mixture experiments with process variables. Communications in Statistics - Theory and Methods, 29(9-10):2255–2280, 2000.