Augmenting Scheffé Linear Mixture Models with Squared and/or Crossproduct Terms

Abstract

In a mixture experiment, q ≥ 2 components are mixed in various proportions, and one or more responses are measured for each mixture. Scheffé quadratic models are often used to model responses as functions of the component proportions. A complete Scheffé quadratic model contains q linear terms βixi and Q = q(q – 1)/2 quadratic crossproduct terms βijxixj (i < j). Because Q increases rapidly as q increases, alternative models containing fewer quadratic terms than the complete Scheffé quadratic model are of interest. Traditionally, reduced Scheffé quadratic models, formed by augmenting linear terms with selected quadratic crossproduct terms, are used. We propose generating partial quadratic mixture (PQM) models by augmenting linear terms with selected quadratic crossproduct terms and/or squared terms βiixi2. The interpretations and potential advantages of PQM models compared to equivalent restricted Scheffé quadratic models and to reduced Scheffé quadratic models are discussed. The methods are illustrated using data from two constrained mixture experiments involving simulated waste glass.