The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs

Kabue Timothy Gichuki; Koske Joseph; Mutiso John

doi:doi:10.11648/j.ajam.20200803.12

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The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs

Kabue Timothy Gichuki, Koske Joseph, Mutiso John

Published in American Journal of Applied Mathematics (Volume 8, Issue 3)

Received: 5 March 2020 Accepted: 10 April 2020 Published: 14 May 2020

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Abstract

In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.

Published in	American Journal of Applied Mathematics (Volume 8, Issue 3)
DOI	10.11648/j.ajam.20200803.12
Page(s)	83-88
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Optimal Designs, Rotatable Design, Balanced Incomplete Block Design and Optimal Values

References

[1]	Montgomery D.C. (2005). Design and Analysis of Experiments. In (Fifth Edit) New York. John Wily & Sons.
[2]	Myers R.H. and Montgomery, D. C. (1995). Response Surace Methodology:Process and Product Optimization Using Designed Experiments. (Fifth Edit). John Wiley $ sons.
[3]	Kiefer, J. (1959). Optimum experimental designs. Journal of the Royal Statistical Society, B21, 272–319. https://doi.org/10.1007/s00184-011-0348-6
[4]	Kiefer, J., & Wolfowitz, J. (1959). Optimum designs in regression problems. Annals of Mathematical Statistics, 30, 271–294.
[5]	Eze, F. C., & O, L. N. (2018). Alphabetic Optimality Criteria for 2K Central Composite Design Academic Journal of Applied Mathematical Sciences Alphabetic Optimality Criteria for 2 K Central Composite Design. September.
[6]	El-Gendy, N. S., Ali, B. A., Abu Amr, S. S., Aziz, H. A., & Mohamed, A. S. (2016). Application of D-optimal design and RSM to optimize the transesterification of waste cooking oil using natural and chemical heterogeneous catalyst. Energy Sources, Part A: Recovery, Utilization and Environmental Effects, 38 (13), 1852–1866. https://doi.org/10.1080/15567036.2014.967417
[7]	Iwundu, M. P. (2017). The Effects of Addition of n c Center Points on the Optimality of Box-Benhken and Box-Wilson Second-Order Designs. International Journal of Probability and Statistics, 6 (2), 20–32. https://doi.org/10.5923/j.ijps.20170602.02
[8]	Das, M., & Narasimham, V. (1962). Construction of rotatable desingns through balanced incomplete block desings. Annals of Mathematical Statistics, 33 (4).
[9]	Khuri, A. I. (2017). Response Surface Methodology and Its Applications In Agricultural and Food Sciences. Biometrics & Biostatistics International Journal, 5 (5). https://doi.org/10.15406/bbij.2017.05.00141.
[10]	Box, G. E. P., & Behnken, D. W. (1960). Some New Three Level Designs for the Study of Quantitative Variables. Technometrics, 2 (4), 455–475. https://doi.org/10.1080/00401706.1960.10489912.
[11]	Box, G., & Hunter, J. (1957). Multi-Factor Experimental Designs for Exploring Response Surfaces Author(s): G . E . P . Box and J . S . Hunter MUJLTI-FACTOR RESPONSE SURFACES ’. The Annals of Mathematical Statistics, 28 (1), 195–241.
[12]	Pukelsheim, F., & Rosenberger, J. L. (1993). Experimental designs for model discrimination. Journal of the American Statistical Association, 88 (422), 642–649. https://doi.org/10.1080/01621459.1993.10476317
[13]	Pazman, A. (1991). A Classification of N O N L I N E A R R E G R E S S I O N Models and. 8.
[14]	Wald, A. (1943). On the Efficient Design of Statistical Investigations. The Annals of Mathematical Statistics, 14 (2), 134–140.
[15]	Atkinson, A. and Donev, A. (1992). "Optimum experimental designs. Oxford University Press.
[16]	Ehrenfeld, E. (1955). “On the efficiency of experimental design.” Annals of Mathematical Statistics, 26, 247–255.
[17]	Rady, E. A., Abd El-Monsef, M. M. E., & Seyam, M. M. (2009). Relationships among Several Optimality Criteria. Interstat Journals, 247, 1–11. http://interstat.statjournals.net/YEAR/2009/articles/0906001.pdf
[18]	Rissanen, J. (1983). Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve, and extend access to The Annals of Statistics. ® www.jstor.org. The Annals of Statistics, 11 (2), 416–431.
[19]	Dette, H., & Haines, L. M. (1994). E-optimal designs for linear and nonlinear models with two parameters. Biometrika, 81 (4), 739–754. https://doi.org/10.1093/biomet/81.4.739
[20]	Pukelsheim, Fr. (2006). Optimal Design of Experiments. Society for industrial and applied mathematics.
[21]	Smith, K. (1918). On the standard deviations of adjusted and interpolated values of an observed polynomial function and its constant and the guidance they give towards a proper choice of the distribution of observations. Biometrika, 12, 1–85.
[22]	Chernoff, H. (1953). Locally optimal designs for estimating parameters. Annals of Statistics, 24, 586–602.
[23]	Yang, M. (2008). A-optimal designs for generalized linear models with two parameters. Journal of Statistical Planning and Inference, 138 (3), 624–641. https://doi.org/10.1016/j.jspi.2006.12.007
[24]	Atkinson, A. C., & Fedorov, V. V. (1975). The design of experiments for discriminating between two rival models. Biometrika, 62 (1), 57–70. https://doi.org/10.1093/biomet/62.1.57
[25]	Pukelsheim, F., & Studden, W. J. . E. (2007). Optimal Designs for Polynomial Regression. The Annals of Statistics, 21 (1), 402–415. https://doi.org/10.1214/aos/1176349033.

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APA Style

Kabue Timothy Gichuki, Koske Joseph, Mutiso John. (2020). The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs. American Journal of Applied Mathematics, 8(3), 83-88. https://doi.org/10.11648/j.ajam.20200803.12

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ACS Style

Kabue Timothy Gichuki; Koske Joseph; Mutiso John. The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs. Am. J. Appl. Math. 2020, 8(3), 83-88. doi: 10.11648/j.ajam.20200803.12

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AMA Style

Kabue Timothy Gichuki, Koske Joseph, Mutiso John. The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs. Am J Appl Math. 2020;8(3):83-88. doi: 10.11648/j.ajam.20200803.12

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@article{10.11648/j.ajam.20200803.12,
  author = {Kabue Timothy Gichuki and Koske Joseph and Mutiso John},
  title = {The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {83-88},
  doi = {10.11648/j.ajam.20200803.12},
  url = {https://doi.org/10.11648/j.ajam.20200803.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200803.12},
  abstract = {In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.},
 year = {2020}
}

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TY  - JOUR
T1  - The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs
AU  - Kabue Timothy Gichuki
AU  - Koske Joseph
AU  - Mutiso John
Y1  - 2020/05/14
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200803.12
DO  - 10.11648/j.ajam.20200803.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 83
EP  - 88
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200803.12
AB  - In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.
VL  - 8
IS  - 3
ER  -

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Author Information

Kabue Timothy Gichuki

Department of Physical and Mathematical Sciences, School of pure and applied Sciences, Mount Kenya University, Thika, Kenya
Koske Joseph

Department of Statistics and Computer Science, School of Biological Sciences, Moi University Eldoret, Kenya
Mutiso John

Department of Statistics and Computer Science, School of Biological Sciences, Moi University Eldoret, Kenya

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Plain Text BibTeX RIS

APA Style

Kabue Timothy Gichuki, Koske Joseph, Mutiso John. (2020). The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs. American Journal of Applied Mathematics, 8(3), 83-88. https://doi.org/10.11648/j.ajam.20200803.12

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ACS Style

Kabue Timothy Gichuki; Koske Joseph; Mutiso John. The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs. Am. J. Appl. Math. 2020, 8(3), 83-88. doi: 10.11648/j.ajam.20200803.12

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AMA Style

Kabue Timothy Gichuki, Koske Joseph, Mutiso John. The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs. Am J Appl Math. 2020;8(3):83-88. doi: 10.11648/j.ajam.20200803.12

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@article{10.11648/j.ajam.20200803.12,
  author = {Kabue Timothy Gichuki and Koske Joseph and Mutiso John},
  title = {The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs},
  journal = {American Journal of Applied Mathematics},
  volume = {8},
  number = {3},
  pages = {83-88},
  doi = {10.11648/j.ajam.20200803.12},
  url = {https://doi.org/10.11648/j.ajam.20200803.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20200803.12},
  abstract = {In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.},
 year = {2020}
}

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TY  - JOUR
T1  - The D-, A-, E- and T-optimal Values of a Second Order Rotatable Design in Four Dimension Constructed Using Balanced Incomplete Block Designs
AU  - Kabue Timothy Gichuki
AU  - Koske Joseph
AU  - Mutiso John
Y1  - 2020/05/14
PY  - 2020
N1  - https://doi.org/10.11648/j.ajam.20200803.12
DO  - 10.11648/j.ajam.20200803.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 83
EP  - 88
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20200803.12
AB  - In response surface methodology, optimal designs are experimental designs generated based on a particular optimality criterion and are optimal only for a specific statistical model. Optimality criterion are single number criteria sometimes called alphabetical optimality criteria where each one intends to capture an aspect of the ‘goodness’ of a design. Most studies on optimization of process variables have concentrated on Central Composite Designs (CCD) yet second order rotatable deigns with any number of factors with reasonably small number of points constructed using properties of balanced incomplete block designs exist. A class of experimental designs that are optimal with respect to some statistical criterion are said to be Optimal designs. These designs allow parameter estimation with increased precision using fewer experimental runs, without bias and with minimum variance thus reducing time and costs of experimentation as opposed to non-optimal designs. A measure of relative efficiency of one design over another according to an optimality criterion aids in discriminating between the two designs for the “best” design. The D-, E-, A- and T-Optimal values of the general second order rotatable design in four dimensions constructed using balanced incomplete block designs when the number of replications (r) are less than three the number of times (λ) pairs of treatments occur together in the design were found which may be used to determine the relative efficiency of the general design to the D-, E-, A- and T-Optimal designs.
VL  - 8
IS  - 3
ER  -

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