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Measurement plan (DoE) where factors can be present or absent. Interactions and concentrations are of interest.

Tina11

Hi everyone,

I am struggeling with setting up a measurement plan for the following situation. Maybe someone has had a similar situation and could give me some input on that:

Factor 1 can be chosen from 3 candidates. In addition, its optimal concentration is of interest.
Factor 2 can be present or absent. If it is present, its optimal concentration is of interest.
Factor 3 can be present or absent. If it is present, it can be chosen from 3 candidates, and its optimal concentration is of interest.

In a first step (design), it would be enough to get a rough estimation which combination of factors delivers the best results. As there might be interactions between the factors, we would like to take them into account.

I had some thoughts already. For example, I was thinking about modeling factor 2 as continuous factor only, but then it would include the 0 as minimum and the upper limit as maximum. However, I am aware that it is not best practice to do that, as a concentration of 0 would mean the factor is absent, and for the case that it is present, the lower limit would actually be at a higher concentration.

Also, I thought about using the disallowed combinations filter for factor 3 (and maybe 2), but that does not solve the "0 issue".

Your input is highly appreciated.

Kind regards

Accepted answers

Victor_G

Hi @Tina11,

Great, if the concentration ranges are the same for each candidate in each factor, this simplify a lot the situation.

Yes, you would have three combinations in your plan corresponding to no candidate for factor 3 like you mentioned (for each 3 candidates when concentration = 0).

It doesn't look like a problem for me, as these experiments could help you evaluate the repeatability and reproducibility of your experiments (experimental noise/error), since you're measuring different experimental units with the same configuration. You should have results for these 3 combinations showing similar values : you can think about the factors 3 (candidate and concentration) as a simple regression model with possible different slopes for the different candidates (different impact of the concentration depending on the candidate chosen) but the same intercept for concentration = 0 (since it corresponds to the same situation with no candidate for factor 3).

I would rather consider this repetition of conditions in the design than using disallowed combinations (which would reduce the experimental and inference space and could create disjoint experimental area) or creating a 4th level in factor 3 for "no candidate" choice. This last option would create more runs, and since your minimum concentration level would be different from 0 (or else you will be repeating the "no candidate" configuration with additional combinations), that means you're not investigating the concentration range between 0 and this min value.

Hope this response makes sense for you and will help you,

All comments

Victor_G

Hi @Tina11,

Welcome in the Community !

It appears that in this first stage, you are in a screening approach, to explore which factors are important and if any 2-factors interactions effects should be considered.

The list of the factors could be set as :

X1: Factor 1A : Categorical : Choice between 3 candidates
X2: Factor 1B : Continuous : Concentration for factor X1
X3: Factor 2 : Continuous : Concentration
X4: Factor 3A : Categorical : Choice between 3 candidates
X5: Factor 3B : Continuous : Concentration of factor X4

Are the concentration ranges the same for each candidate in each factor ? If no, nesting may be necessary. To build such design you might have to use coded levels for the concentrations, and use nested factors in the modeling : Nested DOE with continous factors?

If yes, I would probably start with a D-Optimal design with a model assuming 2-factors interactions and quadratic effects for continuous factors X2, X3 and X5, in order to avoid "Yes/No situations" for the responses (each continuous factor will have a min level = 0, an intermediate level, and a max level). JMP would recommend by default 36 runs (minimum 30 runs) with this setup :

DOE(
	Custom Design,
	{Add Response( Maximize, "Y", ., ., . ),
	Add Factor( Categorical, {"A", "B", "C"}, "X1", 0 ),
	Add Factor( Continuous, 0, 1, "X2", 0 ), Add Factor( Continuous, 0, 1, "X3", 0 ),
	Add Factor( Categorical, {"A", "B", "C"}, "X4", 0 ),
	Add Factor( Continuous, 0, 1, "X5", 0 ), Set Random Seed( 466420203 ),
	Number of Starts( 4670 ), Add Term( {1, 0} ), Add Term( {1, 1} ),
	Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 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( {2, 2} ),
	Add Term( {3, 2} ), Add Term( {5, 2} ), Set Sample Size( 36 ),
	Simulate Responses( 0 ), Save X Matrix( 0 ), Make Design}
);

Depending if this experimental budget is too high or not, here are some other suggestions :

Are you sure you want to start your investigation already with 3 levels for the categorical factor ? Can you choose the most dissimilar 2 levels first, to better understand the pattern and trends with the response data, before trying other candidates/categorical levels?
Having 2-levels categorical factors may also be beneficial, as you could have access to other type of designs, like the Mixed-Level Screening Designs (which are efficient and may save you some runs), or simply help lowering the runs number in your optimal design.

You could also set the estimability of 2nd order terms in your model as "If Possible" which can lower the recommended number of runs (from 36 to 18 runs)

DOE(
	Custom Design,
	{Add Response( Maximize, "Y", ., ., . ),
	Add Factor( Categorical, {"A", "B", "C"}, "X1", 0 ),
	Add Factor( Continuous, 0, 1, "X2", 0 ), Add Factor( Continuous, 0, 1, "X3", 0 ),
	Add Factor( Categorical, {"A", "B", "C"}, "X4", 0 ),
	Add Factor( Continuous, 0, 1, "X5", 0 ), Set Random Seed( 799900606 ),
	Number of Starts( 11461 ), Add Term( {1, 0} ), Add Term( {1, 1} ),
	Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ),
	Add Potential Term( {1, 1}, {2, 1} ), Add Potential Term( {1, 1}, {3, 1} ),
	Add Potential Term( {1, 1}, {4, 1} ), Add Potential Term( {1, 1}, {5, 1} ),
	Add Potential Term( {2, 1}, {3, 1} ), Add Potential Term( {2, 1}, {4, 1} ),
	Add Potential Term( {2, 1}, {5, 1} ), Add Potential Term( {3, 1}, {4, 1} ),
	Add Potential Term( {3, 1}, {5, 1} ), Add Potential Term( {4, 1}, {5, 1} ),
	Add Potential Term( {2, 2} ), Add Potential Term( {3, 2} ),
	Add Potential Term( {5, 2} ), Set Sample Size( 18 ), Simulate Responses( 0 ),
	Save X Matrix( 0 ), Make Design}
);

If you're interested about nested designs, here are some relevant conversations similar to what you intend to do :

Disallowed Combinations not working

Design of Experiment - Optional Mixture Additives

Hope this first answer may help you,

Tina11

Hi @Victor_G ,

thank you very much for your comprehensive reply.

I like your proposal but have a question as well. The concentration ranges are the same for each candidate in each factor. However, for Factor 3A/3B I miss the option to have no candidate chosen. Would this be modeled through having conc. = 0 in Factor 3B? But then, I would have the three combinations Factor 3A = candidate 1 and Factor 3B = 0, Factor 3A = candidate 2 and Factor 3B = 0, Factor 3A = candidate 3 and Factor 3B = 0 which would all mean that no candidate is chosen. Correct? How would I interpret this when I fit the model? 36 or a few more runs would be acceptable.

Kind regards,

Tina

Victor_G

Hi @Tina11,

Great, if the concentration ranges are the same for each candidate in each factor, this simplify a lot the situation.

Yes, you would have three combinations in your plan corresponding to no candidate for factor 3 like you mentioned (for each 3 candidates when concentration = 0).

Hope this response makes sense for you and will help you,

P_Bartell

So far an interesting discussion...I'm going to come at this issue from a bit of a different perspective that hopefully you've thought through? From a first principles viewpoint, do you know what happens in the system under study when one of the factors is set to zero? Sometimes I've seen in the past that the system 'falls apart' and does not behave in the same manner if one of the factors is set to zero as opposed to some minimal level. At worst you end up introducing some nonlinearity in the response that ends up being modeled rather poorly by a linear in the parameters model. It sounds like maybe you are in factor 'screening' mode so maybe a poorly fitting model is no big deal?

When I step back and think about DOE...what are we trying to do? In large measure understand how a system behaves over some n dimensional factor space when those factors are set at some levels. But the system may behave very differently when one or more of those factors is absent so generalizing over that absent factor space is not the same as if all factors were present. Just wanted to ask the question.

Tina11

Thanks a lot for your thoughts! This is what we will try to do. What I am not yet 100% sure about is if the concentrations of factor 2 and 3 should be modeled as continuous factors or discrete numeric. Discrete numeric will add some runs and the design will be much larger, however, so probably we will go with the continuous modeling.

Tina11

@P_Bartell thank you very much for your thoughts! In our case, it will not totally blow up everything if the factors are not present. However, it might change the response significantly. We will try the design @Victor_G proposed and see what happens