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Taguchi Arrays for Mixtures?

bwanders

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Sep 20, 2013


Some of my collaegues are using Taguchi Arrays for their mixture DOE's.  In your opinion, what are the biggest mistakes you can make by using an L9 design for 3 ingredients at 3 levels instead of using a proper mixture DOE?

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I am somewhat curious how people are analyzing their mixture models with all of the components in the model specified as continuous factors. The linear constraint of the components adding to one should give a Singularity Details warning in JMP, one of the model terms will be zeroed out and others may be listed as Biased. Maybe those things are being ignored, but they should not be. Anyhow, having things specified as mixture components is nice because then the profiler will enforce the blend to add to 1 and will not allow "impossible blends" to happen.

The answer to why the profiler does not show curvature will depend on the model that was fit.

Look at the terms in the model. Would any of the functional forms exhibit curvature? If not, the profiler won't show curvature. However, when working with mixture models, things are different. A quadratic Scheffe mixture model will not show any squared terms as people would EXPECT to see them. In a Scheffe model X1*X2 is actually a quadratic term. Similarly, X1*X2*X3 is a cubic term. So if the model has those terms, you should see curvature in the prediction profiler unless those binary blend terms are not significant.

Dan Obermiller
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bwanders

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Sep 20, 2013

Correction- colleagues

Dan_Obermiller

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Apr 3, 2013

For your scenario, I am assuming that the formulation actually has FOUR components, but they use three of them when creating the Taguchi L9 design (this is the only way it could be done). There is nothing wrong with this approach as long as that fourth component (which is not included in the design) is inert. You can perform a "standard" regression analysis (Scheffe mixture models will not be needed) and the interpretations are fairly straight forward. However, the design space may have a different shape than if you had created a straight mixture design.

In most mixture experimentation, the goal is to get a good prediction for the response (cause and effect conclusions don't really make sense: was it component A going up or component B going down??). A Taguchi L9 design is not intended to estimate any two-way interactions. That severely limits predictive abilities and two-way interactions are EXTREMELY common in mixture scenarios (that is the entire idea behind blending things together).

In my opinion, if a more traditional design were desired, I would recommend a response surface design for three factors and let the inert factor make up the rest of the formulation. This should give good predictions. However, for these situations, an I-optimal mixture design for all four components from the custom designer typically gives better mathematical properties for prediction and often uses fewer runs.

Dan Obermiller
bwanders

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Sep 20, 2013

Dan,

I definitely like your explanations in all three paragraphs.

In most cases where I help colleagues analyze their data, the designs were not setup in JMP as a Taguchi Array design.  So when I receive the JMP file, all mixtures are designated as continuous variables in which the sum totals 100%.

I have compared the regression analysis when all mixture factors are continous to when they are properly set to the mixture role.  This way I can show the difference in predictions using the profiler.

Now what would you say to the people that are confused to why the prediction profiler is not showing curvature?

Solution

I am somewhat curious how people are analyzing their mixture models with all of the components in the model specified as continuous factors. The linear constraint of the components adding to one should give a Singularity Details warning in JMP, one of the model terms will be zeroed out and others may be listed as Biased. Maybe those things are being ignored, but they should not be. Anyhow, having things specified as mixture components is nice because then the profiler will enforce the blend to add to 1 and will not allow "impossible blends" to happen.

The answer to why the profiler does not show curvature will depend on the model that was fit.

Look at the terms in the model. Would any of the functional forms exhibit curvature? If not, the profiler won't show curvature. However, when working with mixture models, things are different. A quadratic Scheffe mixture model will not show any squared terms as people would EXPECT to see them. In a Scheffe model X1*X2 is actually a quadratic term. Similarly, X1*X2*X3 is a cubic term. So if the model has those terms, you should see curvature in the prediction profiler unless those binary blend terms are not significant.

Dan Obermiller
bwanders

Community Trekker

Joined:

Sep 20, 2013

Thank you, Dan.  I now have enough information to back up what I am trying to teach.  It would be so much easier to start out with the right design in the first place.

joe_n_hockman

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Joined:

Dec 3, 2014

For what it may be worth, I will offer my opinion on this one.  For the most part I agree with DanO.  Use of a Taguchi L9 makes no sense to me in a mixture context with 3 components.  As DanO rightly pointed out, the overall mixture constraint that component proportions must sum to a constant (usually 1 unless a portion of the formulation is "fixed") is not taken into account if 3 (independent) continuous factors are defined.  Even if there is a 4th component not explicitly defined (usually refered to as the slack variable) the L9 may include runs with nonfeasible blends once slack variable or 4th component is calculated.

Under the scenario that the L9 does translate into a well defined and space-filling 4 component space, then sure it is possible to analyze the data with a slack variable model which would look like an ordinary 2nd order model for 3 factors including the overall constant, linear effects and quadratic terms but no 2-way interaction terms.  I differ a bit with DanO in that the excluded 4th component need not be inert in order to fit such a model.  Perhaps if it is inert then one can claim that some interpretability of normal regression coefficients or effects is preserved.  However slack variable models have limited or no useful interpretability of effects.  One also has to consider what does it mean for a component to be "inert" in a mixture context.  I will not go into that here.  In general focus on the plots or prediction profiler and not on interpretability of regression coefficient output with a slack variable approach and you should be OK.

Perhaps most troublesome is that nonlinear blending effects in a Scheffe modelling context (what JMP supports) are the terms that look like ordinary 2-way interactions.  It was correctly pointed out that terms that look like quadratic terms (what the L9 could support) are not really the same thing as nonlinear blending effects.  So in my opinion use of a Taguchi L9 in a mixture problem is really a misapplication of DOE or at the very least a poor fit of DOE principles in a mixture context.  In almost all cases, mixture problems should be explicitly dealt with as a mixture DOE even when one has either no component constraints or only nonzero lower bound constraints were a regular simplex geometry applies.  Hope this input is helpful.

bwanders

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Sep 20, 2013

"use of a Taguchi L9 in a mixture problem is really a misapplication of DOE or at the very least a poor fit of DOE principles in a mixture context"- joe.n.hockman

I completely agree with your statement above, Joe.