Mixture designs apply in cases where the factors, or mixture components, must sum to a fixed value, typically denoted as 1. Thus with three components, a design might examine the response for the mixture 1,0,0 or 0.5,0.5,0, but not 1,0,-1 or 1,0,1. Mixture designs essentially follow the same pattern as other multi-factor designs, but the underlying regression model is amended to account for the constraint applied. For example, a two-factor linear model with no interaction components would typically be of the form:
but we now have the additional constrain that the factors must sum to 1, i.e.
but this is equivalent to:
thus can be modeled as a simple multiple linear regression model with no intercept. Discussion of such designs is beyond the scope of this book, and interested reader should refer to the definitive text on the subject by Cornell (2002, [COR1]).