Analysis of Quantal Response Data From Mixture Experiments

Abstract

SUMMARYThis paper presents a logistic regression procedure for the analysis of mixture data. The design of the mixture experiment is based on the simplex-lattice designs where the components represent the proportions of a mixture. i.e. each component is expressed as a fraction of the mixture with the sum equal to one. The Scheffe (1958) and Cox (1971) polynomial andBecker (1968) non-polynomial forms are used to model relationships between the expected response and individual components of the mixture. In a three components mixture it was shown that if one component is completely linear with the remaining two components, then the proper representation of such a system is the Becker non-polynomial model. An experimental data set designed to study the interrelating effects of calories from dietary fat, carbohydrate and fibre on tumour incidence is used to illustrate the analysis and the additive effect of the fibre component. Each mixture (diet) of fat. carbohydrate and fibre was administered to provide the same total caloric consumption.

Dual‐objective Optimal Mixture Designs

“…Such a criterion is no longer differentiable but equivalence theorems are still possible, see Wong (1995), King & Wong (1998), Brown & Wong (2000), Chen, Wong & Li (2008). More interesting design issues arise and hitherto un‐researched is the case when the response is binary, as discussed in Chen, Li & Jackson (1996) where their primary concern was in data analysis for such experiments.…”

Nonlinear logistic regression mixture experiment modeling for binary data using dimensionally reduced components

“…Then, the DRC proportions g k satisfy the constraints 0 ≤ g k ≤ 1 and ∑ G k¼1 g k ¼ 1; (2) so that the DRCs themselves are components of a mixture. Note that the non-negativity constraint on the α k,j parameters also ensures that the mixture constraint in Equation (2) holds, since one or more negative α k,j values could cause the corresponding g k to be negative.…”

“…[1][2][3][4][5][6] Despite the limited literature, it is straightforward to model binary response data from mixture experiments using logistic or probit regression 7,8 where the right side of the model is any mixture experiment model linear in the parameters (eg, Scheffé polynomial, Becker model homogeneous of degree one, and Scheffé polynomial with inverse terms). [1][2][3][4][5][6] Despite the limited literature, it is straightforward to model binary response data from mixture experiments using logistic or probit regression 7,8 where the right side of the model is any mixture experiment model linear in the parameters (eg, Scheffé polynomial, Becker model homogeneous of degree one, and Scheffé polynomial with inverse terms).…”

“…The vast majority of examples in the mixture experiment literature involve continuous response variables, with only a few examples that address modeling for binary or multinomial response variables. [1][2][3][4][5][6] Despite the limited literature, it is straightforward to model binary response data from mixture experiments using logistic or probit regression 7,8 where the right side of the model is any mixture experiment model linear in the parameters (eg, Scheffé polynomial, Becker model homogeneous of degree one, and Scheffé polynomial with inverse terms). Cornell 9 discusses each of these classes of mixture experiment models for continuous response variables.…”

General Blending Models for Data From Mixture Experiments

“…Becker ( 1968 ) introduced models that allow for describing linear blending: and Reports of applications of these models include those of Becker ( 1968 ), Snee ( 1973 ), Johnson and Zabik ( 1981 ), Chen, Li, and Jackson ( 1996 ), and Cornell ( 2002 ), among others, most of whom demonstrate them to be advantageously used in comparison to Scheffé polynomials.…”