FAQ
Can I extrapolate a mixture model to a new formulation space?
Original question from a Biological Chemistry Master's Student:
“Thank you very much for your help setting up my catalyst-blending experiment. The results from the optimal mixture design produced a very strong predictive model. I see now that further experimentation outside of my original component ranges may be productive. But before doing so, I would like to see what the current model predicts. Can Stat-Ease® software extrapolate outside of my current mixture space?”
Answer :
Disclaimer: Based on my experience as a chemical engineer, I discourage extrapolating any empirical model —process and/or mixture—outside of the current experimental region, other than to guess at what may be out there before setting up a new DOE.
Process models can easily be extrapolated by widening the factor ranges in the numerical optimization tools provided by Design-Expert® or Stat-Ease 360 software. However, due to the way components get coded, doing so for a mixture model (Scheffé polynomial) requires a ‘workaround’ developed by my colleague Joe Carriere (Research Programmer/Statistician):
Save your mixture design file under a new name.
Click on the Constraints node and then the Edit constraints button.
Adjust one or more limits to make the corresponding range constraint less restrictive.
Refit the same models as you fit in the original file. It helps to have the two designs side-by-side on the same screen.
Predict in the extrapolation region by using the Point Prediction node using this new file (or examine the graphs).
The user asking about this reported that Joe’s workaround worked well for her. Furthermore, I verified the procedure on the detergent case provided in our mixture tutorial, changing the limits as follows:
A. Water from 3–5 to 3–8%
B. Alcohol from 2–4 to 0.5–4%
C. Urea from 2–4 to 0.5–4%
As explained in the tutorial, the chemist allowed for as much as 8% water. However, this had to be reduced to 5% to accommodate the required minimum of 2% for each of the other two ingredients and still fit the 9% total constraint (5+2+2=9). This new region lowers the lower limits to 0.5% each to make room for the higher level of water originally specified (8+0.5+0.5=9).
To illustrate the dangers of extrapolating like this (per my disclaimer at the outset of this answer), I looked at the outcome for the second response—turbidity, originally modeled by a special cubic. As you can see in the 3D plot, beyond the original design space (triangular region outlined by the red dots) the response either plummets to impossibly negative levels or skyrockets to ridiculously high outcomes.
