IDEAS home Printed from https://ideas.repec.org/a/taf/jnlasa/v111y2016i514p899-911.html
   My bibliography  Save this article

I-Optimal Design of Mixture Experiments

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/01621459.2015.1136632
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/01621459.2015.1136632?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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. 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.
    6. 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).
    7. 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).
    8. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Karim Abou-Moustafa & Frank P. Ferrie, 2018. "Local generalized quadratic distance metrics: application to the k-nearest neighbors classifier," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(2), pages 341-363, June.
    2. Mandal, N.K. & Pal, Manisha & Aggarwal, M.L., 2012. "Pseudo-Bayesian A-optimal designs for estimating the point of maximum in component-amount Darroch–Waller mixture model," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1088-1094.
    3. SYAFITRI, Utami & SARTONO, Bagus & GOOS, Peter, 2015. "D- and I-optimal design of mixture experiments in the presence of ingredient availability constraints," Working Papers 2015003, University of Antwerp, Faculty of Business and Economics.
    4. Nolan, D., 1999. "On min-max majority and deepest points," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 325-333, July.
    5. Nripes Mandal & Manisha Pal & Bikas Sinha & Premadhis Das, 2015. "Optimum mixture designs in a restricted region," Statistical Papers, Springer, vol. 56(1), pages 105-119, February.
    6. Yu, Yaming, 2010. "Strict monotonicity and convergence rate of Titterington's algorithm for computing D-optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 54(6), pages 1419-1425, June.
    7. Dolia, A.N. & Harris, C.J. & Shawe-Taylor, J.S. & Titterington, D.M., 2007. "Kernel ellipsoidal trimming," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 309-324, September.
    8. Haoyu Wang & Chongqi Zhang, 2022. "The mixture design threshold accepting algorithm for generating $$\varvec{D}$$ D -optimal designs of the mixture models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 345-371, April.
    9. Hilgers, Ralf-Dieter, 2000. "D-optimal design for Becker's minimum polynomial," Statistics & Probability Letters, Elsevier, vol. 49(2), pages 175-179, August.
    10. Palhazi Cuervo, Daniel & Goos, Peter & Sörensen, Kenneth, 2017. "An algorithmic framework for generating optimal two-stratum experimental designs," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 224-249.
    11. Zhang, Chongqi & Wong, Weng Kee, 2013. "Optimal designs for mixture models with amount constraints," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 196-202.
    12. GOOS, Peter & JONES, Bradley & SYAFITRI, Utami, 2013. "I-optimal mixture designs," Working Papers 2013033, University of Antwerp, Faculty of Business and Economics.
    13. Dette, Holger & Pepelyshev, Andrey & Zhigljavsky, Anatoly, 2008. "Improving updating rules in multiplicative algorithms for computing D-optimal designs," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 312-320, December.
    14. Zhang, Chongqi & Peng, Heng, 2012. "D-optimal designs for quadratic mixture canonical polynomials with spline," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1095-1101.
    15. Rosa, Samuel & Harman, Radoslav, 2022. "Computing minimum-volume enclosing ellipsoids for large datasets," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).
    16. Bernard, Carole & Vanduffel, Steven, 2015. "A new approach to assessing model risk in high dimensions," Journal of Banking & Finance, Elsevier, vol. 58(C), pages 166-178.
    17. Beirlant, J. & Mason, D. M. & Vynckier, C., 1999. "Goodness-of-fit analysis for multivariate normality based on generalized quantiles," Computational Statistics & Data Analysis, Elsevier, vol. 30(2), pages 119-142, April.
    18. 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.
    19. Martin-Martin, R. & Torsney, B. & Lopez-Fidalgo, J., 2007. "Construction of marginally and conditionally restricted designs using multiplicative algorithms," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5547-5561, August.
    20. Harman, Radoslav & Pronzato, Luc, 2007. "Improvements on removing nonoptimal support points in D-optimum design algorithms," Statistics & Probability Letters, Elsevier, vol. 77(1), pages 90-94, January.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:jnlasa:v:111:y:2016:i:514:p:899-911. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/UASA20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.