Design experiments that combine mixture components with categorial factors - Case Help

[#1]

Hello everyone, 

I have recently attended several courses on Mixture Design. I am now ready to try the principles out in real-life scenarios. 

I have the following case. There will be the following things: 

3 mixture factors 

3 Categorical factors 

Most of the recipe is fixed at 83,275% which I will call the BASE - this part will not be changed.

So I have used the Mixture Sum function and set that to the remaining 16,725% (which will be the space for the 3-4 mixture factors) 

However, this starting base will have 2 different enzymes added (1 and 2) - a categorical factor 1 that will be made in big batches - meaning this will be a hard-to-change factor. 

Those things I know how to do in JMP. 

My issue or question relates more to how to combine and select the right terms for my model when combining the categorical ingredient type factors with the corresponding mixture factors. 

Categorial factor 2: Stabilizer type (3 levels)

Categorial factor 3: Fiber type (2 levels)

I then have 3 mixture factors: 

Mixture 1: Stabilizer conc. (expected to affect the responses)

Mixture 2: Fiber conc. (expected to affect the responses)

Mixture 3: Water (do not expect this to do anything - merely a filler)

I would have all the main terms included. 

But is it necessary to have all the second-order terms between the categorical factors and mixture factors?? 

For example, if we don't expect water (more like a filler) to do anything, should that even be included?

Also, we don't expect Fibers and Stabilizers to affect each other. 

This would be my best guess for a model: see the attached screenshot from JMP. 

Should other terms be included or??

I would like some input on this. My overall goal is to make some predictions. So my thought was to use an I-optimal design to get as low a prediction variance as possible. However, as a first step, it would also be cool/beneficial if the design could somehow "screen" which of the stabilizers or fibers work best - so they could be excluded going forward. Would a D-optimal design be better for this???

For the first trial my maximum amount of runs i 14. But more could be added later (augmenting). 

The JMP file is also attached below. 

Looking forward to hearing your thoughts. I am not capable of using JSL or scripting in general. 

Thank you in advance. 

Best regards 
Kristian Bertelsen 

[#2]

@Victor_G  - Hoped that you might know who to ask. 

[#3]

Some thoughts:  There is no one way to go about this situation.  Here are some options:

1. A sequential approach. First study the categorical factors.  Do they have an effect?  What is rank order of those effects? Is there a level that is better than the other?  Then you can do your mixture design.  You actually have options here as well because you have a filler (water) you may not have the same restrictions that a mixture design typically has (collinearity for example).

2. A split-plot type design.  You could put the 3 categorical factors in the whole plot and run a res. III experiment in the whole plot (aliasing the 3rd categorical factor with the interaction of the 1st 2) and then the mixture factors in the sub-plot.

3. Since you may not have the mixture restrictions due to the water filler, you might just run a factorial design (or some fractional...)

in any case, design multiple experiments.  Compare and contrast them (e.g., What can you learn from each vs. resources required).

You might find Cornell's paper Embedding Mixture Experiments Inside Factorial Experiments (JQT Vol. 22, No. 4, October 1990) useful.

In my mind, mixture designs are optimization designs.  That man we already know the significance of the factors in the mixture.  The mixture design is intended to create a "surface"  or contour map for where the results can be optimized.  Optimization designs should be used after you have determined the optimum design space (so you are mapping inside the design space.  This means you have discovered the significant effects and you understand the effects of noise.

[#4]

Hey statman, 

Thanks a lot for your suggestions. 

I think that the sequential approach sounds like a good idea. Since the people I am helping with this design would like to know which of the stabilizers and fiber types work best. I guess the split-plot is also needed for the base categorial factor. The the design would be the following: 

Then a follow-up question regarding setting up a future mixture design afterwards. 

Let's say we find that a specific stabilizer and fiber work best. Then you could also introduce a new ingredient (like protein). But I guess you can not augment this design into a mixture right? 
Another question is then, could you add the 3-5 runs from the first design to the mixture design as "extra" runs? 

[#5]

I'm not sure what you mean "introduce a new ingredient"?  Why don't you include this in the study to start?  Is this a new formulation?  Was this "protein" already present, just not manipulated?  Conclusions you draw from any experiment are contingent on the inference space.  While adding factors in subsequent iterations is always an option, realize that conclusions you drew from previous experiments may no longer be valid (e.g., new interaction effects may impact the conclusions).

When adding data from previous experiments to new experiments you must be aware of the "block" effect.  What factors/conditions changed between the first and second experiment? (e.g., raw materials, ambient conditions)

[#6]

You have gotten some good advice from @statman .  I have only two things to add. One thing is that if you have process variables (in your case, your categorical variables) and mixture variables, then you SHOULD cross the process variables with the mixture variables. This will allow you to see if any mixture effects change when you change those process variables -- a very real concern, otherwise why put the process variables into the design?

The other point that I will make is that you might be able to avoid the mixture scenario all together. Since one of your ingredients will be water, you could consider that inert. Ignore it, and that will allow you to treat the other mixture ingredients as "continuous" variables and you will be back into the realm of more traditional experimental design and modeling. You may not want to do that, but it is at least an option to point out.

[#7]

Hi Dan, 
Thanks a lot for commenting. 

Yes, I am aware that you should cross the process and mixture variables. However in this case for example with the two categorial factors: Fiber type and Stabilizer Type. We expect based on domain knowledge that they should not interfere with each other. Is it then never "okay" to remove certain terms to simplify the model and get fewer runs?? 

Yes, that is a very valid point. I guess I was just really eager to try the mixture design approach after attending several courses. 

[#8]

@Bertelsen92 , I am not saying that you must include all 2-way interactions. However, I think it would be important to put the Stabilizer Type*Stabilizer conc. and the FiberType*Fiber conc. interactions in the model. And to go one step further, the "interactions" in a mixture model are actually estimating curvature. That fact along with the fact that mixture designs are really all about developing predictive models, I would recommend ALL 2-way interactions (and the special cubic terms), if at all possible. Too bad resources are not free to allow that! 

[#9]

Hi again, 
The trials we run are rather expensive and the developer wants as few ingredients in the formulation as possible. 
So hopefully the current setup will solve the issues. However, if that is not the case we might need to test new ingredients like proteins.

But really good point that if the inference space changes in subsequent experiments by adding new factors or changing the levels of existing factors, the conclusions drawn from previous experiments may no longer be valid. But I guess a categorial factor called Protein type as well as a continuous factor with the concentration could be added with levels like: Protein A, Protein B or Non-protein. Then all previous experiments would be "non-protein" at 0%? 

How would you tackle the above regarding the "block effect"?