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I-Optimal Design of Mixture Experiments

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  • Peter Goos
  • Bradley Jones
  • Utami Syafitri

Abstract

In mixture experiments, the factors under study are proportions of the ingredients of a mixture. The special nature of the factors necessitates specific types of regression models, and specific types of experimental designs. Although mixture experiments usually are intended to predict the response(s) for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients, little research has been done concerning their I-optimal design. This is surprising given that I-optimal designs minimize the average variance of prediction and, therefore, seem more appropriate for mixture experiments than the commonly used D-optimal designs, which focus on a precise model estimation rather than precise predictions. In this article, we provide the first detailed overview of the literature on the I-optimal design of mixture experiments and identify several contradictions. For the second-order and the special cubic model, we present continuous I-optimal designs and contrast them with the published results. We also study exact I-optimal designs, and compare them in detail to continuous I-optimal designs and to D-optimal designs. One striking result of our work is that the performance of D-optimal designs in terms of the I-optimality criterion very strongly depends on which of the D-optimal designs is considered. Supplemental materials for this article are available online.

Suggested Citation

  • Peter Goos & Bradley Jones & Utami Syafitri, 2016. "I-Optimal Design of Mixture Experiments," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(514), pages 899-911, April.
  • Handle: RePEc:taf:jnlasa:v:111:y:2016:i:514:p:899-911
    DOI: 10.1080/01621459.2015.1136632
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    References listed on IDEAS

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    1. Liu, Shuangzhe & Neudecker, Heinz, 1995. "A V-optimal design for Scheffé's polynomial model," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 253-258, May.
    2. Shuangzhe Liu & Heinz Neudecker, 1997. "Experiments with mixtures: Optimal allocations for becker’s models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 53-66, January.
    3. D. M. Titterington, 1978. "Estimation of Correlation Coefficients by Ellipsoidal Trimming," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(3), pages 227-234, November.
    4. Sambo, Francesco & Borrotti, Matteo & Mylona, Kalliopi, 2014. "A coordinate-exchange two-phase local search algorithm for the D- and I-optimal designs of split-plot experiments," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 1193-1207.
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    Cited by:

    1. Haosheng Jiang & Chongqi Zhang, 2022. "Construction of Full Order-of-Addition Generalization Simplex-Centroid Designs by the Directed Graph Approach," Mathematics, MDPI, vol. 10(3), pages 1-13, January.
    2. Zijlstra, Toon & Goos, Peter & Verhetsel, Ann, 2019. "A mixture-amount stated preference study on the mobility budget," Transportation Research Part A: Policy and Practice, Elsevier, vol. 126(C), pages 230-246.
    3. Rios, Nicholas & Winker, Peter & Lin, Dennis K.J., 2022. "TA algorithms for D-optimal OofA Mixture designs," Computational Statistics & Data Analysis, Elsevier, vol. 168(C).
    4. Belmiro P. M. Duarte, 2023. "Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming," Mathematics, MDPI, vol. 11(4), pages 1-17, February.
    5. Wanida Limmun & Boonorm Chomtee & John J. Borkowski, 2023. "Generating Robust Optimal Mixture Designs Due to Missing Observation Using a Multi-Objective Genetic Algorithm," Mathematics, MDPI, vol. 11(16), pages 1-33, August.
    6. Lenka Filová & Radoslav Harman, 2020. "Ascent with quadratic assistance for the construction of exact experimental designs," Computational Statistics, Springer, vol. 35(2), pages 775-801, June.
    7. Carlos de la Calle-Arroyo & Miguel A. González-Fernández & Licesio J. Rodríguez-Aragón, 2023. "Optimal Designs for Antoine’s Equation: Compound Criteria and Multi-Objective Designs via Genetic Algorithms," Mathematics, MDPI, vol. 11(3), pages 1-16, January.
    8. Hao, Honghua & Zhu, Xiaoyuan & Zhang, Xinfeng & Zhang, Chongqi, 2021. "R-optimal design of the second-order Scheffé mixture model," Statistics & Probability Letters, Elsevier, vol. 173(C).

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