topic Re: DoE Continuous Mixture Factors with Continuous and categorical response in Discussions

DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Thu, 08 Jun 2023 21:12:44 GMT

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

Thank you guys. Much love

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Tue, 01 Nov 2022 07:28:54 GMT

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

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Tue, 01 Nov 2022 09:51:05 GMT

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.

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Tue, 01 Nov 2022 10:51:23 GMT

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

Re: DoE Continuous Mixture Factors with Continuous and categorical response

statman — Tue, 01 Nov 2022 14:44:54 GMT

I have the following thoughts:

1. Did you look at the correlation of the response variables?

2. I don't understand why you would have a nominal response for mixture design? The analysis should give you contour plots for each response variable which, of course, is non-sensical for a nominal response. What is passing or failing? Can that be converted into a continuous response?

3. Phil already commented on your constant factor ratio.

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Mariana_Aguilar — Thu, 03 Nov 2022 01:03:51 GMT

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

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Thu, 03 Nov 2022 09:53:27 GMT

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 )}
);

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Thu, 03 Nov 2022 09:58:10 GMT

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 )}
);

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Mariana_Aguilar — Thu, 03 Nov 2022 18:00:25 GMT

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!

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Sun, 06 Nov 2022 07:11:53 GMT

Hello @Phil

Very well said. Thank you again

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Sun, 06 Nov 2022 07:15:02 GMT

Hi @stats

1. What do you mean by looking at the correlation of the response variables?

2. The binary nominal response. is Clear solution and Hazy solution . I want my product to be a clear solution

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Sun, 06 Nov 2022 07:41:29 GMT

Hello @Phil_Kay and @statman

How to set-up the following mixture design:

- Four Ingredients
- Water is 65 to 90%

- RM1 and RM2 Ratio should be greater or equal to 1:3.85
- RM 3 is 0 to 0.5%

Thank you

Re: DoE Continuous Mixture Factors with Continuous and categorical response

statman — Sun, 06 Nov 2022 14:48:43 GMT

1. Whenever you have multiple response variables, it is prudent to assess the relationship between the response variables. Particularly when some of the response variables have potential measurement system issues (e.g., lack discrimination, resolution, or precision, are unstable). Plotting the response variables on scatter plots, checking for outliers with Mahalanobis are easy ways to look for those associations (Analyze>Multivariate Methods>Multivariate).

2. I believe, perhaps with a little work, this response could be made more continuous (e.g., measure opacity, shine a light source through the solution, why is the clarity changing? Is it due to solids? Measure solids in the solutions). If nothing else, you aught to be able to rank order the clarity of the solution and use an ordinal response. It all relates to efficiency. To see changes in nominal response variables you typically need larger sample sizes. As the response becomes more continuous you can get the information you need with less samples. In addition, many nominal response variables are aggregate. They change as a result of multiple failure mechanisms. This creates a real challenge when trying to understand causal structure.

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Mon, 07 Nov 2022 08:48:19 GMT

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

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Mon, 07 Nov 2022 09:12:13 GMT

Hi @Alainmd02 ,

You need to set this up with a linear constraint to define that the ratio of RM1:RM2 >= 0.26.

I find the best way to get the equation for the constraint is to define 2 points that obey the constraint (e.g. 0.1, 0.385 and 1,3.85) and fit a line using Fit Y by X in JMP.

This gives you the equation, RM1 = 0 + 0.2597403*RM2, and you just need to rearrange into a form that fits the constraints interface.

Here is the script for the Custom Design.


DOE(
	Custom Design,
	{Add Response( Maximize, "Y", ., ., . ),
	Add Factor( Mixture, 0.65, 0.9, "Water", 0 ),
	Add Factor( Mixture, 0, 0.35, "RM1", 0 ),
	Add Factor( Mixture, 0, 0.35, "RM2", 0 ),
	Add Factor( Mixture, 0, 0.005, "RM3", 0 ), Set Random Seed( 2095116882 ),
	Number of Starts( 29816 ), Add Constraint( [0 -1 0.26 0 0] ), Add Term( {1, 1} ),
	Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ),
	Add Alias Term( {1, 1}, {2, 1} ), Add Alias Term( {1, 1}, {3, 1} ),
	Add Alias Term( {1, 1}, {4, 1} ), Add Alias Term( {2, 1}, {3, 1} ),
	Add Alias Term( {2, 1}, {4, 1} ), Add Alias Term( {3, 1}, {4, 1} ),
	Set Sample Size( 10 ), Simulate Responses( 0 ), Save X Matrix( 0 ),
	Set Run Order( Randomize ), Make Table}
);

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Phil_Kay — Mon, 07 Nov 2022 13:31:44 GMT

@statman and @Alainmd02 .

Regarding 2.: I did something like this in a project when I was working in the chemicals industry. We had the same aim, which was to understand what parts of the formulation space gave us clear, homogeneous solutions. We rated clarity on a scale of 1-5. And we also heated up the samples to 70C and rated them as they cooled to 60, 50, 40, 30 and 20. I struggled to build useful statistical models of the data. The visuals of this data were very useful though. Here is a plot of the (anonymised) data for 20C. You can clearly see the good, green regions. That was all we needed to know for our objectives.

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Mariana_Aguilar — Mon, 07 Nov 2022 16:54:33 GMT

thank you Phil!

appreciate your help

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Fri, 25 Nov 2022 02:47:03 GMT

I think this solved our problem. It is easier if we have continuous responses. That's why we found a way to quantify the clarity levels of our solutions.

Kudos to you @statman

Much Appreciated.

Re: DoE Continuous Mixture Factors with Continuous and categorical response

Alainmd02 — Fri, 25 Nov 2022 02:49:30 GMT

This is interesting. Its almost the same as my experiments.

Much appreciated Phil.