How to have a mixture of mixtures design

[#1]

Hi, I am developing a cell culture medium based on 4 prototypes. These 4 prototypes are basically contain same components but at different level, Prototype 1 is lean, prototype 2 is rich in amino acid, prototype 3 is rich in vitamin and prototype 4 is rich in trace element. I will mixture different level of the 4 prototypes to have a new mixture media.

I understand this is a mixture of mixtures design. There is an example of mixture-of-mixtures design from JMP. However, in that design the mixtures are different component, such in the cake factory experiment, mixture 1 is dry including cocoa, sugar and flour, and mixture 2 is wet including butter, milk and egg. 

In my case, the mixtures all contain same component, but at different level.

I am wondering how may I design this experiment ? Can the 4 mixtures be treated as 4 separate factors? What mixture design should I apply and Should I consider 2-way or 3-way interaction of the 4 factors. Is Sheffe cubic model is appropriate to be applied for analysis? 

Thanks, any suggestion will be appreciated. 

[#2]

Hi @loganshawn,

From your description, there may be (at least) two options to consider :

• Either you work with the individual components/raw materials: you find the components of each mixture and do a mixture design with the individual components, and define appropriate range for each factor based on your domain expertise and experience with the mixtures, and build a mixture design. Depending on the number of individual components, this option may require a high number of experiments.
• Or you work with the mixture preparations : you can then build your mixture design based on your 4 mixtures, and define appropriate ranges for each based on your experience. Once you find the optimum mixture (ratio) of the 4 mixtures, you can calculate the optimal amount for each constituants if you have the composition information for every mixture preparations.

There are indeed a lot of options for mixture design, from classical ones (Scheffe Cubic for example), to more specific ones (Custom designs and Mixture Space Filling). You can see an overview of available mixture designs here : Overview of Mixture Designs (jmp.com) and here : Examples of Mixture Design Types (jmp.com)
The choice will depend on several criteria, such as : experimental budget, expected prediction precision/variance in your experimental space, "continuous" or "discontinuous" response surface, model-based vs. model-agnostic approach ...

Some differences between models and terms used in different models :

• In Scheffé cubic model (done in Custom Design), you'll be able to create a model with main effects, 2 mixture factors interactions and 3 mixture factors interaction, for a total of 30 runs recommended by JMP (see screenshot). You can add manually the 4 factors interaction for not extra cost (JMP doesn't seem to add runs in the "recommended runs number" when you add this term in the model, but it does change the minimal amount of runs needed from 20 to 21 because of this extra term added).
• In Simplex Centroïd design (done in Classical -> Mixture Design), you are doing mixtures with up to 4 components for a total of 15 experiments recommended by JMP (see screenshot). You have fewer runs than the previous design, but at the expense of a larger relative variance prediction in your experimental space.
• A Space filling Mixture design might also be considered, in this case for 4 factors JMP recommends 40 runs. The advantage of this design is the homogeneous repartition of points/runs in your experimental space, without pre-defined model (it's a model-agnostic approach). That means you should be able to use other modeling approach for analysis, like Gaussian Process, or techniques from Machine Learning such as Support Vector Machines, Tree-based methods, Neural Networks, etc... The validation of such models could be then done with train/validation/test sets or SVEM method (Using Design of Experiments Methods for Efficient Modeling & Simulation - JMP User Community). But it comes at the expense of more aliasing (correlations) between factors, so it might be more difficult to understand which mixture factors or interactions is driving the response(s) (see screenshot Matrix_correlations_designs).

So the choice is up to you, if you prefer a "causal" model (Classical Mixture designs and Custom designs with pre-defined model) or only a model with good predictive performance but with limited knowledge. Based on the previous post you made, a Space-filling approach was considered for one of your similar topics, but you may change your mind depending on your objective and experimental budget.

There are so many things we can tell about Mixture designs, you can check on the community similar topics and the webinars dedicated to this topic (Designing Mixture Experiments - Part 1 - JMP User Community, Designing Mixture Experiments - Part 2 - JMP User Community, Accelerating Innovation with Space Filling Mixture Designs, Neural Networks and ... - JMP User Commu...).

I hope this first (long) answer will help you in your reflexion,

[#3]

Thank you so much, Vector. I really appreciate the long answer. I can't image how much time you spend on typing the answer. I am glad that I asked the question so that not only me but others also can learn from your answer. 

Best wishes

[#4]

I recommend that you keep this mixture experiment simple. I would use a mixture design with the essential components of the medium and explore a wide space of proportions of each instead of pre-conceiving "prototypes." I would use the Scheffe cubic model with Custom Design so that you can predict well over the space. It doesn't make sense to screen components in a mixture experiment, though you might discover that some higher-order terms are insignificant and remove them from the model. It might very well be that the optimal mixture is close to one of the conceptual prototypes that you had in mind.