Design of Experiment - Optional Mixture Additives

[#1]

I'd like to make a custom design to develop a mixture with some optional additives. The basic design is:

Y (Performance)
X1 (Main component physical characteristic): Categorical - 2 level
X2 (Main component chemical characteristic): Categorical - 2 level
X3 (Additive type): Categorical - 2 level (A & B, only one or the other would be used, never in combination)
X4 (Additive concentration): Continuous or Discrete Numeric (0.1, 0.2, 0.3)

What is the best approach for making a design that will also evaluate performance when no additive is used?

• Can I simply add 0 as an option for X4? Or, will that weaken my ability to evaluate performance differences between additives (presumably Additive A and Additive B would both perform the same at 0)?
• Could I consider adding "none" as a level in X3, include 0 as the lower limit in X4, and use factor constraints to ensure that non-viable combinations (e.g., X3 = "none" & X4 = 10%, X3 = "A" & X4 = 0%) are excluded from the design?
• Is there a different approach that would be better?

Any guidance or suggestions would be appreciated.

[#2]

Hi @BW11,

Welcome in the Community !

There are many questions to better understand your experimental setup :

• What is your goal regarding the use of this design (screening, exploration, prediction, ...) ?
• What is your knowledge about the system ? What is the assumed model for this design ? Are there any other factors you may be interested to include and study ?
• Can X1 and X2 vary independently if they relate to the same main component (these factors look like covariates) ? Why are X1 and X2 categorical if they are chemical and physical characteristics/properties ? They could be set/transformed to continuous covariates, see the presentation  )?

From a practical point of view and with the limited information available, I would go with the first approach of extending the range of X4 (as a continuous factor) to 0 to have the possibility to study the effect of no additive in the mixture. This option seems preferable, as it is more simple, less redundant in the experimental choices (no need to add disallowed combinations to avoid using no additive with concentration > 0) and enables to have a broader and continuous experimental space regarding the additives. Moreover, the absence of additive is represented as any additive with 0 concentration, which makes more intuitive sense than creating a new categorical factor level to represent the absence of additive in a "binary" way (Presence/Absence).

From a statistical point of view, adding a level 0 in your categorical additive type factor (second option) may increase the minimum required number of runs, while also creating a discontinuous design space (evolution of response for each additive from 0 to max concentration is not seen), reducing power and increasing non-uniformly prediction variance. It also creates a more complex correlation/aliasing structure in your design, and reduce design optimality (D-, G-, A- and I-efficiencies).

As a side note, factor X4 "additive concentration" seems to be a continuous numerical factor. Depending on your goal and needs, you can add in the model panel the quadratic term X4.X4 if you want to have three levels for X4 in your design for example.

Hope this first answer may help you,

[#3]

@Victor_G,

Thank you for the response. It is very helpful. I have responses to your questions below and an additional question at the end.

What is your goal regarding the use of this design (screening, exploration, prediction, ...) ?

I would say we are in the exploration phase. We have done some past screening work and are interested in learning more about the impact of a couple of new additives at various concentrations.

What is your knowledge about the system ? What is the assumed model for this design ? Are there any other factors you may be interested to include and study ?

We have decent knowledge of the product, but are evaluating it for a different use with new additives. We have decided to hold other factors steady at this point.

Can X1 and X2 vary independently if they relate to the same main component (these factors look like covariates) ? Why are X1 and X2 categorical if they are chemical and physical characteristics/properties ? They could be set/transformed to continuous covariates, see the presentation Coding with Continuous and Mixture Variables to Explore More of the Input Space (2022-US-45MP-1103) )?

X1 and X2 can vary independently. They are categorical because in practice, that can't be controlled with the precision necessary to use continuous variables. Rather than getting a product with a specific value for chemistry, we are dealing with ranges that we then assign categorical values (e.g., low, med, high).

I'm inclined to add a 0 to X4 and move forward.

I am curious about the tradeoffs between using a discrete numeric vs continuous variable. X4 could be either. But, it seems to me that I could get more information about the effect of concentration by setting more discrete values (e.g., 0, 0.1, 0.2, 0.3) than it is possible to set with continuous variable and a mid-point (0, 0.15, 0.3). I'm expecting a linear response, but am not certain that it will be. Any thoughts on this?

[#4 · Accepted solution]

Hi @BW11,

The choice between discrete vs. continuous numeric variable can be made at two stages :

• During design creation : If you want to test specific levels/values of concentration (aka force certain level values in the design), setting this factor to discrete numeric will help you define an optimal design with these levels constraints. A continuous numeric variable will have 2 levels by default in the design (with assumed model including main effects, and interactions), so to increase the number of levels, you have to specify higher order terms like quadratic (3 levels), cubic (4 levels), ... terms in the model panel. The levels will be equally distant from each others.
  Note that the choice of the number of levels and their values depend on your objective ("pick a winner" or system understanding with model interpolation) and domain expertise : in screening phase, having multiple levels for the same factor won't help much in filtering active (main) effects from non-active ones. But if you have already knowledge about your system and specific values/points, then integrating these interesting values directly in the design may be helpful in a comparative study.
• During analysis : Again, the experimental setup and objective of the study can help defining the modeling type of the factor. If the concentration value can only take certain values, because of feasibility and/or weighing/measuring capacity, then a discrete numeric modeling type may be more appropriate. If your concentration value can take any value between 0 and 0,3 then a continuous numeric modeling type will help infering the response value in the accessible concentration range of the design.

Covariate factor type may also help in forcing a design to accept specific values for a factor.

Some related discussions :

force levels in DoE 

Continuous Vs Discrete Numeric Factor 

discrete numeric factor good choice? 

And explanation by @DonMcCormack about this discrete numeric type : Dear Dr. DOE, isn’t Discrete Numeric just Categorical? 

Hope this answer make sense for you and will be helpful,