General Blending Models for Mixture Experiments: Design and Analysis

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Abstract

It is felt the position of the Scheffé polynomials as the primary, or sometimes sole recourse for practitioners of mixture experiments leads to a lack of enquiry regarding the type of blending behaviour that is used to describe the response and that this could be detrimental to achieving experimental objectives. Consequently, a new class of models and new experimental designs are proposed allowing a more thorough exploration of the experimental region with respect to different blending behaviours, especially those not associated with established models for mixtures, in particular the Scheffé polynomials. The proposed General Blending Models for Mixtures (GBMM) are a powerful tool allowing a broad range of blending behaviour to be described. These include those of the Scheffé polynomials (and its reparameterisations) and Becker's models. The potential benefits to be gained from their application include greater model parsimony and increased interpretability. Through this class of models it is possible for a practitioner to reject the assumptions inherent in choosing to model with the Scheffé polynomials and instead adopt a more open approach, flexible to many different types of behaviour. These models are presented alongside a fitting procedure, implementing a stepwise regression approach to the estimation of partially linear models with multiple nonlinear terms. The new class of models has been used to develop designs which allow the response surface to be explored fully with respect to the range of blending behaviours the GBMM may describe. These designs may additionally be targeted at exploring deviation from the behaviour described by the established models. As such, these designs may be thought to possess an enhanced optimality with respect to these models. They both possess good properties with respect to optimality criterion, but are also designed to be robust against model uncertainty.

Layman's Abstract

It is felt the position of the Scheffé polynomials as the primary, or sometimes sole recourse for practitioners of mixture experiments leads to a lack of enquiry regarding the type of blending behaviour that is used to describe the response and that this could be detrimental to achieving experimental objectives. Consequently, a new class of models and new experimental designs are proposed allowing a more thorough exploration of the experimental region with respect to different blending behaviours, especially those not associated with established models for mixtures, in particular the Scheffé polynomials. The proposed General Blending Models for Mixtures (GBMM) are a powerful tool allowing a broad range of blending behaviour to be described. These include those of the Scheffé polynomials (and its reparameterisations) and Becker's models. The potential benefits to be gained from their application include greater model parsimony and increased interpretability. Through this class of models it is possible for a practitioner to reject the assumptions inherent in choosing to model with the Scheffé polynomials and instead adopt a more open approach, flexible to many different types of behaviour. These models are presented alongside a fitting procedure, implementing a stepwise regression approach to the estimation of partially linear models with multiple nonlinear terms. The new class of models has been used to develop designs which allow the response surface to be explored fully with respect to the range of blending behaviours the GBMM may describe. These designs may additionally be targeted at exploring deviation from the behaviour described by the established models. As such, these designs may be thought to possess an enhanced optimality with respect to these models. They both possess good properties with respect to optimality criterion, but are also designed to be robust against model uncertainty.

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Three data setsCode for four model fitting procedures.

Keyword(s)

Design of Experiments; Mixture Experiments; Nonlinear Models

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