PNNL

Abstract

In a mixture experiment, the response variable depends on the proportions of the components, which must sum to one. Because of this constraint, standard polynomial models cannot be used to analyze mixture experiment data. To get around this, some researchers ignore one of the components and use standard polynomial models in the remaining components. Because the component proportions must sum to one, the ignored component (referred to as the slack variable (SV)) makes up the remaining proportion of the mixture. In the literature, there have been many examples of researchers using the SV approach instead of a mixture approach. We have analyzed data from several of these examples using both approaches. For examples whose goal was to screen the mixture components (screening examples), we fit full linear models and identified which components were important using both approaches. In the screening examples, the mixture approach revealed that the SV had a significant effect on the response. For examples that had sufficient data to fit quadratic models to the data (quadratic examples), we used stepwise regression to develop reduced quadratic models for the SV approach, and partial quadratic mixture (PQM) models for the mixture approach. In three quadratic examples, the PQM models identified the SV and/or one of its quadratic blending terms as having a significant effect on the response variable. Hence, by completely ignoring a component’s effect on the response, SV analysis carries an inherent risk of wrong conclusions. There are fewer possible reduced quadratic SV models than possible PQM models because the reduced quadratic SV models are a subset of the class of PQM models. As a result, the PQM models will always fit the data as well as, or better than, the best reduced quadratic SV model. Our research concludes that it is better to analyze mixture experiments using methods specifically developed for them instead of using standard methods with the SV approach.