Interpreting a Mixture of Mixtures Study

[#1]

Can I get some help interpreting the results of my mixture of mixtures study? I'm studying the friction of asphalt, specifically the effects of aggregate type, aggregate blend ratios at coarse and fine fractions, and asphalt gradation type.

Aggregate Types, 6 total:
- Class A (high quality): A1, A2, A3
- Class B (low quality): B1, B2, B3

Aggregate Blending (Coarse and Fine portions)
- Blend percent of Coarse Class A and B aggregate in total mix: 0 to 50%
- Blend percent of Fine Class A to B aggregate in total mix: 0 to 50%
- Total of Coarse and Fine = 100%

Gradation Type, 2 types
- Type C
- Type D

Attached is my study .jmp file. I feel comfortable with multivariate regression analysis, but this mixture stuff is tripping me up. Any direction to get started is welcome. Thanks! 

[#2]

There is just not enough information provided about the experimental situation to give sound advice (e.g., Do you know measurement system error?  How much of a change in friction matters?  What is an experimental unit?  How was it measured?...). I'm not sure I understand why you are calling this a mixture design?  Two factors (it doesn't appear Aggregate and gradation type are mixture factors) and since you did ratios rather than the independent amounts of fine and coarse.  You actually have an alternate to a mixture design (not good or bad).  Analyze as typical fractional factorial design starting with the model you used to design the experiment and use fit model.

OK, I had a look at the data table, I'm still confused.  It seems you  have the aggregate types included in your 4 component mixture design?  It appears you have confounded those Aggregate types with the mixture components?

[#3]

Thanks for your comments. My study has both factorial and mixture components.

There are three categorical factors:

• Agg Type-Class A (Levels: A1, A2, A3)
• Agg Type-Class B (Levels: B1, B2, B3)
• Gradation Type (Levels: Type C, Type D)

Then there are two mixtures, where each is a two-component mixture.

• AggBlend, Coarse (A_Coarse + B_Course = 0.5)
• AggBlend, Fine (A_Fine + B_Fine = 0.5)

Can you clarify what you mean by "you did ratios rather than the independent amounts of coarse and fine?" I think I did not do ratios. 

• Coarse (A=0.2, B=0.3) + Fine (A=0, B=0.5) = 1.0

Because it's just two components, are you saying I could have changed the 4 mixture variables to 2 continuous ratio variables? (Ratio of A/B_Coarse, and Ratio of A/B_Fine). When I've done this in the past, I ran into problems with the analysis of AggType * AggRatio interactions. (I can expound on those problems if needed).

Here's what the DOE dialogue looked like.

I don't know how I got here with the model, but I think there are problems. (1) I am missing the AggType main effects. (2) Some interaction terms don't make logical sense.

• The first interaction, AggType, Class A * AggBlend, Coarse_Class A, would show how the Class A Agg type varies by it's amount in the mixture. And for test runs where Class A is not in the mixture (Blend Class A=0), it's effect wouldn't be considered. But the next interaction, AggType, Class A * AggBlend, Coarse_Class B, seems to be looking at the exact same thing, but the inverse. The mixture constraints mean that B Blend = 0.5 - A Blend. 
• The interaction, AggBlend, Coarse_Class A * AggBlend, Coarse_Class B, doesn't seem helpful at all. This is just the interaction of A Blend and 0.5-A Blend.

If I build the model intuitively, I end up with the following. Does this seem valid?

[#4]

I'm sorry, I am not an SME on the experimental factors you are considering.  My thought process would question whether you should try to do all of this in one experiment or whether a sequential approach is better?  If you want to embed a mixture design in a factorial design, I suggest you read:

Cornell, John A., 1990 "Embedding Mixture Experiments Inside Factorial Experiments", Journal of Quality Technology, Vol. 22, No./ 4, October 1990.

[#5]

As probably happens too often, the time has past for experimental design. The data is collected, and decently covers the design space. But I need to make some sense of it. I'm struggling to set up the model variables/interactions in JMP properly. for my scenario.
Thanks for your help. I've requesting the article from our librarian.

[#6]

I can't and won't try to comment on the design and the adequacy of it. I just considered this as a dataset that needs to be analyzed. I may have made some assumptions that are not appropriate, if so, feel free to ignore this post.

Because you have two different mixtures, each with two components, you can simplify the problem. Rather than analyzing with the total of four mixture components (two for each mixture), I ignored the AggBlend, Course_Class B and AggBlend, Fine_Class B factors. By knowing the Coarse_Class A and Fine_Class A, we can determine the others. This will greatly simplify the analysis because now we can treat all of the factors as continuous. I then modified your data table to take that into account. That table is attached here.

Now I tried to do the analysis. I think you wanted to have a model with interactions, so I specified the 5 factors with all 2-way interactions and used Forward Stepwise regression (using minimum AICc as the selection criteria) to find a model. The data did not support a model with all 2-way interactions, that is why I used Forward Stepwise. In my explorations, I saw some possible non-constant variance for friction, so I created and used the log(friction) as the response (try fitting these models with friction as the response and you should see the possible non-constant variance). See the Starting Model ... script in the data table to see the starting point.

With all of that said, I found a model that SEEMS decent. See the Possible Final Model script. Is it really good? I don't know. I do not know any of the background on this data collection or the science behind this situation. However, the key thing that I think that I have provided is a different way to think about this problem and possibly get to a result that is easily interpretable as this is just a typical regression analysis (nothing special with a Scheffe mixture model).

[#7]

Great workaround @Dan_Obermiller !

Reading through the messages here, I do wonder if a mixture design was really needed here, since the problem seems to involve only two formulation parts, each with two raw materials. So the use of continuous factors and ratio should be enough to define the design.

The analysis methodology you propose is very similar to:
Coding with Continuous and Mixture Variables to Explore More of the Input Space (2022-US-45MP-1103): https://community.jmp.com/t5/Abstracts/Coding-with-Continuous-and-Mixture-Variables-to-Explore-More-... : This talk shows how switching from categorical to continuous factors the definition of the design space enable to have a broader inference space where it is possible to find solutions.

I think your solution has a similar mindset, as it enables to simplify the model by considering factors ratios instead of using mixture factors and a (complex) mixture model. If the model can be validated by @bryantw, from a domain expert point of view and with the realization of validation runs, that would further increase the confidence and reliability of the model found.

[#8]

This might be helpful a Case Study in the JMP Library by Marie Guadard

https://community.jmp.com/t5/JMP-Sample-Data/Design-of-Experiments-Example-A-Mixture-of-Mixtures-Des...

[#9]

Thanks. A good read. The example is strictly mixtures, whereas my situation is both categorical and mixture variables. This is where I'm tripping up.