Factorial DOE with severe constraints

[#1]

I recently planned and executed a DOE that appeared to function effectively; however, I am seeking feedback on the design and suggestions for potentially better alternative strategies.  Apologies for the lengthy post, but I wanted to include as much relevant detail as I could.

The DOE involved four additives.  The level of each additive could span 0 – 6.5% of the formulation but the sum of all four additives should be between 4.3 and 6.5%.  I considered two alternative approaches.

The first approach utilized four mixture variables which define the percentage of each additive in the total blend of additives.  The total percent additives on formulation is represented by a separate independent variable.   By employing the Custom Design platform and incorporating second-order interactions I generated the following.

The alternative design utilized four continuous factorial variables with constraints to limit the total additive level to between 4.3 and 6.5%.

The second approach seemed to be a good option since fewer runs were required.  Although I had some concerns regarding the results of the Power Analysis during the design evaluation, I proceeded with the design using simulated responses that reflected the expected signal strength and noise levels, and the outcomes were satisfactory.

Ultimately, I conducted a 13-run DOE based on the second approach.

Attached is a disguised version of the design (though the data itself is real) that includes one of the responses and a reduced model for that response.

Some observations about the design and analysis.  As a result of the constraints there is some correlation between the DOE factors.

Presumably, because of this, the Variance Inflation Factors are larger that I usually see for a factorial DOE.

It was also noteworthy that the profiler operates in a manner akin to that for a mixture DOE. When the level of one factor is adjusted, the levels of the other factors also change to ensure that the total additive remains within the range of 4.3% to 6.5%.

If you have read this far, I appreciate your attention and would welcome any comments you may have regarding this DOE. Is the design valid? Are there any special considerations needed due to the high Variance Inflation Factors (VIFs)? Was there a more effective option?

[#2]

I did not read your post thoroughly, but have some quick comments:

Mixture designs are not orthogonal, so you will likely get VIF's >1.  There is plenty of argument about how large the VIF should be before being concerned (e.g., Marquardt says >10 whereas Gorman suggests >100).  Just don't fall in love with the estimates.

IMHO, spend less time looking at factor significance and more time looking at the mixture response surface when analyzing Mixture designs.

see also here:

https://community.jmp.com/t5/Discussions/How-to-deal-with-multicollinearity-when-doing-multiple-mixt...

[#3]

Thanks for your quick reply.  Your comments make a lot of sense to me.

Just one point of clarification.  The design I ran was not a mixture design, but rather a factorial design with some major constraints (A, B, C and D vary between 0 and 6.5%, but A+B+C+D has to be between 4.3 and 6.5%).  It surprised me that with these constraints,.the DOE took on some characteristics that I usually associate only with mixtures (higher VIFs, not being able to move the factors in the profiler independently).

And I accept full responsibility for that not being evident from a quick read of the post - I did get a bit lengthy!

[#4]

I looked at your design...What is the objective of the experiment?  Is it to pick a winner or understand causal structure?

The design you attached will allow for estimation the following model:

Additive 2

Additive 1

Additive 3

Additive 4

Additive 2*Additive 2

Additive 2*Additive 1

Additive 1*Additive 1

Additive 2*Additive 3

Additive 1*Additive 3

Additive 3*Additive 3

Additive 2*Additive 4

Additive 1*Additive 4

Notice there are some quadratic effects in the model.  The VIFs are exceptional. I would not have run this design.

I added a fit model with a saturated model (Fit model all).  Always start saturated and remove un-interesting terms (the models for DOE are built with the subtractive method). There is a surface you can look at, but I can't analyze it without SME.

[#5]

Thanks for your advice and for taking the time to add the saturated model.  Much appreciated.

[#6 · Accepted solution]

Hi @drdrf,

Some comments and discussion besides @statman excellent guidance :

• Higher VIFs are normal in a situation involving constraints : VIFs measure the amount of collinearity between terms in the model (see Parameter Estimates for Original Predictors). When adding a constraint like A+B+C+D = 4,5 you also add collinearity between terms, as the amount of one factor can be deduced from the other thanks to the constraint, hence the higher VIFs. Factors are not independant anymore (hence the correlations seen), and so are the estimations of terms (hence the higher VIFs).
• In this scenario, I would have favored a Mixture design, since the 4 additives can vary on the same "scale", and you seem more interested in the ratio of the individual factors than in their individual quantities. If you had one solvent among the 4 factors with highly different concentrations range, maybe a factorial design only on additives (and using solvent as QS, completing the total quantity to a fixed sum by adding or removing some solvant) could have been done.
  What would have been possible for design creation is to use 4 mixture factors and 1 continuous "process" factor for the total quantity, like in your first approach.
• Even if you are not sure about mixture or constrained factorial design, you can do the analysis on both cases ; you already did it for the constrained factorial case, but if you change your original factors into ratio with a new factor for total quantity, you can try to see what happends in the modeling of a mixture scenario.
  I tried this approach, and it seems to provide a useful model (attached please find the modified datatable with the Mixture model tested, you might have to delete the "constraint" script on the table to avoid bug in the prediction profiler from the "Mixture model"). You can try to compare both models and see how they seem to be in accordance in terms of conclusion and optimal points (you may need some transformations from continuous to mixture factors to compare the outcomes of the models). I would recommend testing the optimum by doing validation runs, and possibly augment your original design if you need more precision in your analysis.

Hope this answer will help you,

[#7]

Thanks @Victor_G for your comprehensive and practical reply.

You are right that the constraints need to be removed in order to analyze the data as a mixture DOE.  I had tried this before without removing the constraints and wondered why the profiler was not making any sense.  Once I removed the constraints, everything fell into place. 

I compared mixture and constrained factorial models for all of the responses of interest.  In most cases, both models lead to the same predicted optimum.

Your advice to augment the design to enhance precision and to test the optimum is also well taken.   We have already started work to that effect.