Evaluating Mixture Designs
Use Fraction of Design Space
The Fraction of Design Space (FDS) plot provides a precision based metric to evaluate response surface designs including mixture designs. Mixture designs are response surface designs where there is a constraint such that all the component proportions must add up to 1. Click on the graphs tab to see the FDS plot.
Power
Low power is inherent for mixture designs, particularly when there is a further imposition of tight constraints on one or more of the components. Two features combine to make the power low. First, the mixture nature of the design and the tight constraints combine to make the standard deviations of the estimated effects large. (The fixed sum constraint, and the additional constraints on the other components, make the model predictors correlated, which in turn make the estimated coefficients for those predictors have higher standard deviations.) Second, the 2 standard deviation effect is measured across the entire simplex, but we only get to see a small part of that for highly constrained components. For example, C ranges between 0 and 0.01, so for even a 2 SD-sized C effect, the observed range of the C effect in the design is only 0.02 SD. These two features combine to make the signal to noise ratio very low, which in turn makes our power (our probability to detect an effect) low as well. (Note: the floor for power is whatever threshold risk level (alpha value) established (5% by default)).
Leverage
Mixture designs also have problems being reasonable with regards to leverage. Especially for full order modes, the leverages can be very high. As the analysis progresses, it is likely that some effects will be found to be statistically insignificant and can be reduced out of the model. This will improve the leverages associated with the design points.
VIF
As mentioned above mixture design are highly correlated structure. VIF (Variance Inflation Factor) is a measure of correlation. As a result, VIF’s will be high for mixture designs and should not be used to determine appropriateness of a design. Properly reducing the model during analysis will also improve the VIF’s associated with the remaining effects.