Why is the power of my mixture design so low, even with an effect size of 20 standard deviations?

[#1]

Hi all,

I'd appreciate any advice.  I'm helping a scientist at my company design a mixture experiment with 5 ingredients.  We wish to measure main effects and 1st order interactions. 

I asked the scientist to provide mins and maxes for the ingredient concentrations, the test precision standard deviation, and to estimate the effect size of the dominant ingredient.  I'm relying on Power to decide how many runs are needed.  The effect size of the dominant ingredient is about 20 standard deviations.  I find for 30 runs, the power for this ingredient is only 0.168.  For 60 runs, it's 0.33. 

An effect size of 20 standard deviations is obviously very large, so I don't understand why the Power is so low for this case.

I investigated further by simulating results for 5 instances of the response variable with the appropriate coefficient and standard deviation, and the ingredient with the large effect size is statistically significant (p < 0.001) in all 5 cases. 

What can I do?  I'm not sure I trust the power calculation for this case.  Am I hitting up against a limitation of the power calculation?  How can I provide guidance on the number of runs required? 

The scientist's goals for the design are perfectly reasonable, and I feel frustrated that I can't give him the guidance that he needs. 

Many thanks!  I attached a screenshot of the power, the .jmp design, and the text output of the models with the simulated responses, if you are inclined to take a look. 

 

[#2]

You should think of a mixture experiment as a response surface experiment rather than a screening experiment. Because of the built-in correlations between main effects and main effects with two-factor interactions due to the mixture constraint, the coefficient estimates are virtually meaningless. Therefore it is very difficult to do model selection with mixture experiments. One possibility is to consider removing all the two-factor interactions as a group. If the regression sum of squares does not drop significantly, then you can remove all these effects. Generally removing individual effects is of questionable utility.

However, the prediction variances for new responses inside the region of experimentation are usually quite low especially when the error variance is not large. So, you can rely on predictions and choose a new operating setting based matching your desired response. 

[#3]

When sizing mixture designs you might want to do as Brad recommends and regard them in the same way as you would a response surface design for optimising your responses. In which case, consider using the fraction of design space (FDS) plot to check your experiment has been sized sufficiently to enable you to predict your mean response(s) precisely, as you would for an RSM design.

Interactions between the formulations components or ingredients in your mixture models are in fact blends: two components, a two-part blend of the ingredients, three components, a three-part blend, and so on. These interactions or blends between your components cause curvature in the response, which is why you don't see pure quadratic terms in a Scheffé model. When you include blending, you're often attempting to construct a model to look for the optimal formulation in the same way as you might look for an optimum resulting from a response surface experiment involving process factors. Therefore use the FDS plot to assess the capability of your mixture design as you would if you set up a response surface. For all the good reasons all the good folks have mentioned on this thread (e.g., multicollinearity and soaring VIFs), the power calculation is not just inappropriate for response surface objectives but also for mixture designs.