Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design

Eugene C. Ukaegbu; Polycarp E. Chigbu

doi:doi:10.11648/j.ajtas.20170604.16

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Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design

Eugene C. Ukaegbu, Polycarp E. Chigbu

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 4)

Received: 18 October 2016 Accepted: 3 February 2017 Published: 10 July 2017

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Abstract

Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors, is the k^th design measure, is the design’s information matrix, is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.

Published in	American Journal of Theoretical and Applied Statistics (Volume 6, Issue 4)
DOI	10.11648/j.ajtas.20170604.16
Page(s)	205-208
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), 2017. Published by Science Publishing Group

Keywords

Design Efficiency, Determinant, Information Matrix, Partial Replication, Response Surface Methodology, Trace

References

[1]	Li, J., Liang, L., Borror, C. M., Anderson-Cook, C. M. and Montgomery, D. C. (2009). Graphical Summaries to Compare Prediction Variance Performance for Variations of the Central Composite Design for 6 to 10 Factors, Quality Technology and Quantitative Management, Vol. 6 (4), 433 – 449.
[2]	Box, G. E. P and Wilson, K. B. (1951), On the Experimental Attainment of Optimum Conditions, Journal of the Royal Statistical Society, Series B, Vol. 13, 1 – 45.
[3]	Chigbu, P. E. and Ukaegbu, E. C. (2017), Recent Developments on Partial Replications of Response Surface Central Composite Designs: A Review, Journal of Statistical Application and Probability, Vol. 6 (1), 1 – 14.
[4]	Ukaegbu, E. C. and Chigbu, P. E. (2015), Graphical Evaluation of the Prediction Capabilities of Partially Replicated Orthogonal Central Composite Designs, Quality and Reliability Engineering International, Vol. 31, 707 – 717.
[5]	Kao, J.M-H. and Stufken, J. (2015), Optimal Design for Event-Related fMRI Studies: in Handbook of Design and Analysis of Experiments (Dean, Morris, Stufkten and Bingham) as Editors, CRC Press, Taylor and Francis Group, Boca Raton, London and New York.
[6]	Ukaegbu, E. C. and Chigbu, P. E. (2017). Evaluation of Orthogonally Blocked Central Composite Designs with Partial Replications, Sankhya B, Vol. 79(1), 112-141. DOI: 10.1007/s13571-016-0120-z.
[7]	Borkowski, J. J. (2003), A Comparison of Prediction Variance Criteria for Response Surface Designs, Journal of Quality Technology, Vol. 35 (1), 70–77.
[8]	Borkowski, J. J and Valeroso, E. S. (2001), Comparison of Design Optimality Criteria of Reduced Models for Response Surface Designs in the Hypercube, Technometrics, Vol. 43 (4), pp. 468–477.
[9]	Ukaegbu, E. C. and Chigbu, P. E. (2015). Characterization of Prediction Variance Properties of Rotatable Central Composite Designs for 3 to 10 Factors, International Journal of Computational and Theoretical Statistics, Vol. 2 (2), pp. 87–97.
[10]	Wong, W. K. (1994), Comparing Robust Properties of A, D, E and G-optimal Designs, Computational Statistics and Data Analysis, Vol. 18, pp. 441-448.
[11]	Searle, S. R. (1982), Matrix Algebra Useful for Statistics, John Wiley and Sons, New York.
[12]	Atkinson, A.C. and Donev, A.N. (1992), Optimum Experimental Designs, Oxford University Press, New York.

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Eugene C. Ukaegbu, Polycarp E. Chigbu. (2017). Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design. American Journal of Theoretical and Applied Statistics, 6(4), 205-208. https://doi.org/10.11648/j.ajtas.20170604.16

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ACS Style

Eugene C. Ukaegbu; Polycarp E. Chigbu. Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design. Am. J. Theor. Appl. Stat. 2017, 6(4), 205-208. doi: 10.11648/j.ajtas.20170604.16

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AMA Style

Eugene C. Ukaegbu, Polycarp E. Chigbu. Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design. Am J Theor Appl Stat. 2017;6(4):205-208. doi: 10.11648/j.ajtas.20170604.16

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@article{10.11648/j.ajtas.20170604.16,
  author = {Eugene C. Ukaegbu and Polycarp E. Chigbu},
  title = {Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {4},
  pages = {205-208},
  doi = {10.11648/j.ajtas.20170604.16},
  url = {https://doi.org/10.11648/j.ajtas.20170604.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170604.16},
  abstract = {Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors,  is the kth design measure, is the design’s information matrix,  is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.},
 year = {2017}
}

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TY  - JOUR
T1  - Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design
AU  - Eugene C. Ukaegbu
AU  - Polycarp E. Chigbu
Y1  - 2017/07/10
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.20170604.16
DO  - 10.11648/j.ajtas.20170604.16
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 205
EP  - 208
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170604.16
AB  - Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors,  is the kth design measure, is the design’s information matrix,  is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.
VL  - 6
IS  - 4
ER  -

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Author Information

Eugene C. Ukaegbu

Department of Statistics, University of Nigeria, Nsukka, Nigeria
Polycarp E. Chigbu

Department of Statistics, University of Nigeria, Nsukka, Nigeria

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Plain Text BibTeX RIS

APA Style

Eugene C. Ukaegbu, Polycarp E. Chigbu. (2017). Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design. American Journal of Theoretical and Applied Statistics, 6(4), 205-208. https://doi.org/10.11648/j.ajtas.20170604.16

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ACS Style

Eugene C. Ukaegbu; Polycarp E. Chigbu. Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design. Am. J. Theor. Appl. Stat. 2017, 6(4), 205-208. doi: 10.11648/j.ajtas.20170604.16

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AMA Style

Eugene C. Ukaegbu, Polycarp E. Chigbu. Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design. Am J Theor Appl Stat. 2017;6(4):205-208. doi: 10.11648/j.ajtas.20170604.16

Copy | Download

@article{10.11648/j.ajtas.20170604.16,
  author = {Eugene C. Ukaegbu and Polycarp E. Chigbu},
  title = {Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {6},
  number = {4},
  pages = {205-208},
  doi = {10.11648/j.ajtas.20170604.16},
  url = {https://doi.org/10.11648/j.ajtas.20170604.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170604.16},
  abstract = {Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors,  is the kth design measure, is the design’s information matrix,  is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design
AU  - Eugene C. Ukaegbu
AU  - Polycarp E. Chigbu
Y1  - 2017/07/10
PY  - 2017
N1  - https://doi.org/10.11648/j.ajtas.20170604.16
DO  - 10.11648/j.ajtas.20170604.16
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 205
EP  - 208
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20170604.16
AB  - Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors,  is the kth design measure, is the design’s information matrix,  is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.
VL  - 6
IS  - 4
ER  -

Copy | Download