OPTIMAL DESIGNS UNDER LOG MODELS FOR CONSTRAINED MIXTURE COMPONENTS

What the paper says

Mixture experiments arise in a wide range of application areas, including food processing, agriculture, and chemical and industrial research. The statistical theory underlying such experiments was first developed in the mid-1950s. In mixture experiments, the response variable depends on the relative proportions of the mixture components rather than on their absolute amounts, which, whether expressed as volume or weight is usually fixed, and the component proportions therefore sum to one (or equivalently, 100%). Consequently, changes in component proportions directly influence the characteristics of the resulting product. The primary objective of mixture experiments is thus to model the blending surface, which represents the relationship between the response variable and the proportions of the mixture components. Several statistical models have been fitted to data of experiments involving mixture. However, these models have a difficulty in some way to capture the data of such experiments. Thus, the present work focuses on the modelling and design of mixture experiments subject to lower and upper bounds on the component proportions. Model fitting in mixture experiments is challenging due to the inherent constraints imposed by the mixture structure, particularly the requirement that component proportions sum to one and remain within specified limits. To address these challenges, we propose a class of logarithmic models motivated by connections with compositional data analysis. These models demonstrate superior performance compared to existing mixture models under standard model comparison criteria. Once an appropriate model has been selected, it is necessary to represent the response surface accurately over the experimental region of interest. This, in turn, requires the construction of suitable experimental designs. Accordingly, we develop and investigate both exact and continuous optimal designs under commonly used optimality criteria. These designs provide efficient and informative sampling schemes for estimating model parameters and exploring the response surface within constrained mixture regions.