DoE Continuous Mixture Factors with Continuous and categorical response

[#1]

What should be the modeling for this kind of DoE? Any tips or steps are much appreciated.

Thank you guys. Much love

[#2 · Accepted solution]

Hi @Alainmd02 ,

That is a big question. In fact, it is really more than one question. You will probably get more useful answers if you can break it down a bit and add some more information.

There are at least 2 parts to this:

1. Mixture Factors

2. Categorial Response

Are you uncertain about mixture factors, categorical responses, or both?

It would be helpful to say more about the nature of the categorical response. Is it a binary, nominal response (pass/fail, good/bad)? Or is it an ordered categorical, like rating 1 to 5?

You could also tell the community more about the mixture factors. How did you design the experiment? Did you use Custom Design or a classical design approach? Presumably you designed this with JMP so you should have a model script in the design table.

I am sure you will get some great advice.

Phil

[#3]

Hello Phil, 

My experiment involves four different factors. I used custom design for mixture and selected Scheffe cubic. My responses are three continuous and 1 categorical (pass or fail).

May I know if there is a threshold value for Prediction Variance Profile and Fraction of Design Space plot. I notice they become bigger as the range of factor increases. 

May I also know how to set-up constraint such that the 2 of my factors have always the ratio of 1:2.25

Thank you Phil.

[#4 · Accepted solution]

First of all, if 2 of the 4 mixture factors should be at a constant ratio, you really only have 3 mixture factors. You should treat those 2 ingredients as 1 mixture factor for the design. Then calculate the amount of the individual ingredients from the completed design (very easy to do with formula columns in JMP).

I have never come across any thresholds for Prediction Variance Profile or the Fraction of Design Space plot. For one thing, these show prediction variance over the factor space, so you don't get a single value. Also, like most design diagnostics, the absolute values are not very useful. They are best used for comparison of possible different designs. I am surprised that you see variance increase with increasing factor range. I would expect to see the opposite.

For modelling, you will have the script for the Scheffe Cubic model in the design table. You really just need to run that. You will probably not want to reduce the model to take out insignificant effects as the factors in a mixture design are not completely independent. So there is not much that you need to do other than run the model.

For the binary response ensure that it is nominal modelling type and then JMP will default to fitting a Nominal Logistic model. You should know that nominal responses contain much less information than continuous responses, therefore the power to detect effects is much lower.

I hope this all helps,

Phil

[#5]

Hi Phil thank you for your answer, actually I have a similar question than Alain's (i have 4 mixture factors and 1 continuous (a mixture-process design)).

I hope you won't mind me jumping in on this answer... would it be ok to adjust only the Scheffe polynomial + the quadratic term for the continuous variable?

I have run the simulation with the Scheffe's cubic model, but unfortunately for my case, the number of runs required becomes way too large.

Thanks in advance and thank you Alain for sharing your question!

Mariana

[#6]

Hi @Mariana_Aguilar ,

Classic statistician answer: it depends.

How many runs can you afford? If you can only afford a small number of runs, then you will be constrained to looking at a simpler model of the system.

Do you think that the process factors might interact with the mixture factors. In some examples, scientists design experiments to test for how the process factor affects the behaviour of the mixture system with interaction effects between process and mixture factors in the model. This obviously increases the number of required runs.

The minimum sized design for a model with all mixture main effects and all second-order effects (mixture-mixture interactions, process-mixture interactions, and process quadratic effect) is 10. 12 is the recommended default.

I hope this helps.

Phil

DOE(
	Custom Design,
	{Add Response( Maximize, "Y", ., ., . ), Add Factor( Mixture, 0, 1, "Mix1", 0 ),
	Add Factor( Mixture, 0, 1, "Mix2", 0 ), Add Factor( Mixture, 0, 1, "Mix3", 0 ),
	Add Factor( Continuous, -1, 1, "Process1", 0 ), Set Random Seed( 4216334 ),
	Add Term( {1, 1} ), Add Term( {2, 1} ), Add Term( {3, 1} ),
	Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),
	Add Term( {1, 1}, {4, 1} ), Add Term( {2, 1}, {3, 1} ),
	Add Term( {2, 1}, {4, 1} ), Add Term( {3, 1}, {4, 1} ), Add Term( {4, 2} ),
	Set Sample Size( 12 ), Simulate Responses( 0 ), Save X Matrix( 0 )}
);

[#7]

Sorry, @Mariana_Aguilar . I didnt read your comment properly. I was looking at 3 mixture factors. With 4 mixture factors the run numbers obviously get bigger: Min = 15, Recommended = 20.

DOE(
	Custom Design,
	{Add Response( Maximize, "Y", ., ., . ), Add Factor( Mixture, 0, 1, "Mix1", 0 ),
	Add Factor( Mixture, 0, 1, "Mix2", 0 ), Add Factor( Mixture, 0, 1, "Mix3", 0 ),
	Add Factor( Mixture, 0, 1, "Mix4", 0 ),
	Add Factor( Continuous, -1, 1, "Process1", 0 ), Set Random Seed( 4216334 ),
	Add Term( {1, 1} ), Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ),
	Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),
	Add Term( {1, 1}, {4, 1} ), Add Term( {1, 1}, {5, 1} ),
	Add Term( {2, 1}, {3, 1} ), Add Term( {2, 1}, {4, 1} ),
	Add Term( {2, 1}, {5, 1} ), Add Term( {3, 1}, {4, 1} ),
	Add Term( {3, 1}, {5, 1} ), Add Term( {4, 1}, {5, 1} ), Add Term( {5, 2} ),
	Set Sample Size( 20 ), Simulate Responses( 0 ), Save X Matrix( 0 )}
);

[#8]

Thank you Phil, yes that's exactly what I have (20 runs).

For future experiments (perhaps with smaller number of factors) with mixture and process variables, you would then recommend to do Scheffe Cubic +  mixture and continuous factors main effects + process-mixture interactions + process quadratic effect??

I have also seen that sometimes people use the Scheffe special cubic. Any advantages/disadvantages on using Scheffe special cubic vs Scheffe ("regular") cubic?

Thank you!

[#9]

Hi @Mariana_Aguilar,

Yes. What you have seen is that it is a question of balancing the complexity of the behaviours that you need to model with the number of runs that you can afford. You can run an experiment for a simple model with a small number of runs. Then, if you find it is not an adequate model, you can augment with more runs to test higher order effects (e.g. cubic model).

The special cubic model is a reduced form of the cubic scheffe model. It is simpler, with fewer parameters. Therefore, an experiment to estimate the special cubic model can have fewer runs, but it will not capture the same complexity of behaviours.

This Mastering JMP resource should be useful.

I hope this helps,

Phil

[#10]

thank you Phil!

appreciate your help