Construction of twenty-six points specific optimum second order rotatable designs in three dimensions with a practical example

  • Authors

    • Tum Isaac Kipkosgei Statistics Lecturer Moi University
    2020-02-18
    https://doi.org/10.14419/ijasp.v8i1.30122
  • Response Surface Methodology, Second Order Design, Optimality, Coded levels, Natural levels.

  • Abstract

    This quadratic response surface methodology focuses on finding the levels of some (coded) predictor variables x = (x1u, x2u, x3u)' that optimize the expected value of a response variable yu from natural levels. The experiment starts from some best guess or “control†combination of the predictor variables (usually coded to x = 0 for this case x1u=30, x2u=25 and x3u =40) and experiment is performed varying them in a region around this center point.

    We go further to construct a specific optimum second order rotatable design of three factors in twenty-six points. The achievement of this is done with estimation of the free parameters using calculus in an existing second order rotatable design of twenty-six points. Such a design permits a response surface to be fitted easily and provides spherical information contours besides the realizations of optimum combination of ingredients in Agriculture, horticulture and allied sciences which results in economic use of scarce resources in relevant production processes. The expected second order rotatable design model in three dimensions is available where the responses would then facilitate the estimation of the linear and quadratic coefficients. An example involving Phosphate (x1u), Nitrogen (x2u) and Potassium (x3u) is used to represent the three factors in the coded level and converted into natural levels.

     

     

  • References

    1. [1] Box, G.E.P. (1952). Multifactor designs of first order. Journal of Biometrika 39:49-57https://doi.org/10.1093/biomet/39.1-2.49.

      [2] Bose, R. C. and Draper, N.R. (1959). Second order rotatable designs in three dimensions. Ann. Math. Stat., Vol. 30, pp. 1097-1112.https://doi.org/10.1214/aoms/1177706093.

      [3] Box, G.E.P. and Hunter, J.S. (1957). Multifactor experimental designs for exploring response surfaces. Ann.Math.Statist. 28,195-241https://doi.org/10.1214/aoms/1177707047.

      [4] Box G.E.P, Wilson K.B. (1951). On the Experimental Attainment of Optimum Conditions.Journal of the Royal Statistical Society B, 13, 1–45.https://doi.org/10.1111/j.2517-6161.1951.tb00067.x.

      [5] Draper N.R. (1960). Second order rotatable designs in four or more dimension. Ann. Math. Stat., Vol. 31, pp. 23-33.https://doi.org/10.1214/aoms/1177705984.

      [6] Draper, N.R. and Beggs, W.J. (1971). Errors in the factor levels and experimental design. Journal Annals of Mathematical Statistics 41:46-48. https://doi.org/10.1214/aoms/1177693493.

      [7] Kiefer, J.C. (1960). Optimum experimental designs V, with applications to systematic and rotatable designs. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA 1960, Volume 1 (Ed. J. Neyman). University of California, Berkeley, CA, 381-405.

      [8] Kosgei, M.K., Koske, J.K., Too, R.K., and Mutiso, J.M. (2006). On optimality of a second order rotatable design in three dimensions. The east Africa journal of statistics 2:123-128. https://doi.org/10.4314/eajosta.v1i2.39162.

      [9] Koske J.K., Mutiso J.M. Kosgei M.K. (2008). A specific optimum second order rotatable designs of twenty-four points with a practical example. Vol. 1 issues no. 1, East African Journal of pure & applied science.

      [10] Mutiso, J. M. (1998). Second and Third Order Specific and Sequential Rotatable Designs in K Dimensions. Unpublished D.Phil. Thesis, Moi University.

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  • Received date: 2019-11-09

    Accepted date: 2019-12-25

    Published date: 2020-02-18