Full Length Research Paper
Rotatability of Second-Order Response Surface Designs
G. O. Agadaga
Article Number - 65E1CAF929B2E | Vol. 5(2), pp. 13-31, March 2024 |
Received: 28 November 2023 | Accepted: 20 February 2024 | Published: 1 March 2024
Copyright © 2024 Author(s) retain the copyright of this article.
This article is published under the terms of the Creative Commons Attribution License 4.0.
Abstract
The concept of rotatability has been
proposed in the literature as a desirable property of a good experimental
design, with the intention of imposing stability on the scaled prediction
variance of a design. Thus this allows the prediction variance to remain
unchanged under any rotation of the coordinate axes. This study focuses on the
rotatability of the Box-Behnken designs and the Circumscribed Central Composite
Design in four and five variables. The statistical software MiniTab 16 was used
to generate the design and analyses were carried out using RStudio software.
The rotatability property of the Circumscribed Central Composite design was
found to satisfy the two rotatability conditions for second-order designs
perfectly. However, the Box-Behnken designs were either perfectly rotatable or
near rotatable. Keywords: Rotatability, Response surface, Scaled
prediction variance, Design, Properties.
References
Aanchal, N. A., Kanika, D. G. & Arun, G.
(2016). Response Surface Methodology for Optimization of Microbial Cellulase Production. Romanian Biotechnological Letters. 21(5),
11832-11841, https://rombio.unibuc.ro/wp-content/uploads/2022/05/21-5-4.pdf
Bhatra Charyulu, N. Ch., Saheb Shaik, A, &
Jayasree, G. (2022). New Series for Construction of Second Order Rotatable
Designs. European Journal of Mathematics
and Statistics, 3(2), 17-20, http://dx.doi.org/10.24018/ejmath.2022.3.2.46
Bhattacharya, S. (2020). Fabrication of Poly
(Sarcosine), Poly (Ethylene glycol), and Poly (Lactic-co-glycolic acid)
Polymeric Nanoparticles for Cancer Drug Delivery. Journal of Drug Delivery Science and Technology, 61, 102194;
https://doi.org/10.1016/j.jddst.2020.102194
Bhattacharya, S. (2021). Central Composite Design
for Response Surface Methodology and Its Application in Pharmacy. In Book: Response Surface Methodology in Engineering
Science. IntechOpen, https://doi.org/10.5772/intechopen.95835
Box, G. E. P. & Behnken, D. (1960). Some New
Three Level Designs for the Study of Quantitative Variables, Technometrics, 2, 455-475,
https://doi.org/10.2307/1266454
Box, G. E. P. & Hunter, J. S. (1957).
Multi-Factor Experimental Designs for Exploring Response Surfaces. The Annals of Mathematical Statistics, 28,
195-241, https://doi.org/10.1214/aoms/1177707047
Box, G. E. P., & Wilson, K. B. (1951). On the
Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society. Series B (Methodological), 13(1),
1-45. http://www.jstor.org/stable/2983966
Charankumar, G., Raju, P., Dasore, A., & Appa
Rao, B. V. (2020). An Empirical Study On Modified Second Order Response Surface
And Taguchi Designs For Optimizing The Process Parameters. Journal of Mathematical and Computational Science, 10(6),
3065-3073, https://doi.org/10.28919/jmcs/5026
Chigbu, P. E., Ukaegbu, E. C. & Nwanya, J. C.
(2009). On Comparing The Prediction Variances Of Some Central Composite Designs
In Spherical Regions: A Review, STATISTICA,
anno LXIX, n. 4: 285-298, http://dx.doi.org/10.6092/issn.1973-2201/3560
Chiranjeevi, P. & Victorbabu, B. Re (2021).
Construction of Second Order Slope Rotatable Designs Using Supplementary
Difference Sets. Thailand Statistician,
19(2), 261-269, https://ph02.tci-thaijo.org/index.php/thaistat/article/view/243847/165405
Czyrski, A. & Jarz?bski, H. (2020). Response
Surface Methodology as a Useful Tool for Evaluation of the Recovery of the
Fluoroquinolones from Plasma—The Study on Applicability of Box-Behnken Design,
Central Composite Design and Doehlert Design. Processes, 8(4), 473; https://doi.org/10.3390/pr8040473
Deepthi. T., Saheb Shaik, A. & Bhatra Charyulu,
N. CH. (2021). Reduction of Dimensionality Using Bayesian Approach for Second
Order Response Surface Design Model. Advances
and Applications in Mathematical Sciences, 20(9), 1937-1946,
https://www.mililink.com/upload/article/121938614aams_vol_209_july_2021_a25_p1937-1946_t._deepthi_and_n._ch._charyulu.pdf
Edmondson, R. N. (1991). Agricultural Response
Surface Experiments Based on Four-Level Factorial Designs. Biometrics, 47(4), 1435–1448. https://doi.org/10.2307/2532397
Hajji, s., Turki, T., Hajji, M. & Mzoughi, N.
(2018). Application of Response Surface Methodology for Optimization of Cadmium
Ion Removal from an Aqueous Solution by Eggshell Powder. Chemical Research in Chinese Universities 34(2), 302-310,
http://dx.doi.org/10.1007/s40242-018-7163-9
Joshy, C.G. & Balakrishna, N. (2021).
Orthogonally Blocked Second Order Response Surface Designs under Auto-
Correlated Errors. Journal of the Indian
Society of Agricultural Statistics 75(2), 169-174,
Jyostna, P., Sulochana, B. & Victor Babu, B. R.
E. (2021). Measure of Modified Rotatability for Second Order Response Surface
Designs. Journal of Mathematical and
Computational Science, 11(1), 494-519, https://doi.org/10.28919/jmcs/5121
Khuri, A. I. & Cornell, J. A. (1987). Response Surfaces (Marcel Dekker, New
York).
Khuri, A. I. (2017). Response Surface Methodology
and Its Applications In Agricultural and Food Sciences. Biometrics & Biostatistics International Journal, 5(5), 155-163.
http://dx.doi.org/10.15406/bbij.2017.05.00141.
Koech, J. K, Mutiso, M. J., & Koskei, J. K.
(2017). Response Surface Methodology Approach to the Optimization of Potato (Solanum tuberosum) Tuber Yield Using
Second-Order Rotatable Design. Journal of
Biometrics & Biostatistics, 8(3), 1000351.
http://dx.doi.org/10.4172/2155-6180.1000351
Lou, H., Li, W., Li, C. & Wang, X. (2013).
Systematic investigation on parameters of solution blown micro/nanofibers using
response surface methodology based on Box-Behnken design. Journal of Applied Polymer Science, 130(2), 1383-1391,
https://doi.org/10.1002/app.39317
Maity, S. (2023). 2 - Applications of selected
response surface design of experiments and advanced control charts in textile
engineering. Eds: R. Chattopadhyay, Sujit Kumar Sinha, Madan Lal Regar, In Book: The Textile Institute Book Series, Textile Calculation, Woodhead Publishing,
Pages 13-55, https://doi.org/10.1016/B978-0-323-99041-7.00015-1
Mead, R. & Pike, D. J. (1975). A Review of
Response Surface Methodology from a Biometric Viewpoint. Biometrics 31, 803–851.
Montgomery, D.C. (2005). Design and Analysis Experiments, 6th ed., John Wiley and Sons Inc,
N.J.
Morshedi, A. & Akbarian, M (2014). Application
of Response Surface Methodology: Design of Experiments And Optimization: A Mini
Review. Indian Journal of Fundamental and
Applied Life Sciences, 4(S4),
2434-2439, https://www.cibtech.org/sp.ed/jls/2014/04/JLS-280-S4-282-AKBARIAN-APPLICATION-REVIEW.pdf
Myers, R. H. & Montgomery, D.C. (2002). Response Surface Methodology: Process and
Product Optimization Using Designed Experiments, 2nd ed., Wiley, New York.
Naseri, T., Beikia, V., Mousavi, S. V. &
Farnaudc, S (2023). A Comprehensive Review Of Bioleaching Optimization By
Statistical Approaches: Recycling Mechanisms, Factors Affecting, Challenges,
And Sustainability. Royal Society of
Chemistry Advances, 13, 23570-23589, https://doi.org/10.1039/D3RA03498D
Njoku, C. N. & Otisi, S. K. (2023). Application
of Central Composite Design with Design Expert v13 in Process Optimization. In
book: Response Surface Methodology -
Research Advances and Applications. IntechOpen, http://dx.doi.org/10.5772/intechopen.109704
Otieno-Roche, E., Koske, J. & Mutiso, J.
(2017). Construction of SECOND-ORDER Rotatable Simplex Designs. American Journal of Theoretical and Applied
Statistics. 6(6), 297-302. http://dx.doi.org/10.11648/j.ajtas.20170606.16
Oyejola, B. A. & Nwanya, J. C. (2015).
Selecting the Right Central Composite Design. International Journal of Statistics and Applications, 5(1), 21-30,
http://dx.doi.org/10.5923/j.statistics.20150501.04
Peasura, P. (2015). Application of Response Surface
Methodology for Modeling of Postweld Heat Treatment Process in a Pressure
Vessel Steel ASTM A516 Grade 70. The Scientific World Journal Volume 2015, Article ID 318475, 8 pages,
http://dx.doi.org/10.1155/2015/318475.
Rohmatussolihat, Lisdiyanti, P., Sari, M. N. &
Sukara, E. (2021). Response surface methodology for optimization of medium
components for extracellular protease production by Enterococcus faecalis InaCC B745. IOP Conferecnce Series: Earth and Environmental Science, 762,
012078. https://doi.org/10.1088/1755-1315/762/1/012078
Sulochana, B. & Victorbabu, B. R. E. (2021a).
Measure of slope rotatability for second order response surface designs under
intra-class correlation error structure using symmetrical unequal block
arrangements with two unequal block sizes. International
Journal of Statistics and Applied Mathematics, 6(1), 188-200,
https://www.mathsjournal.com/pdf/2021/vol6issue1/PartC/6-1-5-922.pdf
Sulochana, B. & Victorbabu, B. R. E. (2021b).
Measure of Slope Rotatability for Second Order Response Surface Designs Under
Intra-Class Correlated Structure Of Errors Using Central Composite Designs. Journal of Mathematical and Computational
Science, 11(1), 735-768, https://doi.org/10.28919/jmcs/5083
Usman, A. I., Aziz, A. A. & Sodipo, B.K.
(2019). Application of Central Composite Design for Optimization of
Biosynthesized Gold Nanoparticles via Sonochemical Method. SN Applied Sciences 1, 403;
https://doi.org/10.1007/s42452-019-0429-0
Veli, S., Özbay, ?., Özbay, B., Arslan, A., &
Çebi, E. (2018). Optimization of Process Variables for Treatment of Food
Industry Effluents by Electrocoagulation.
Global NEST Journal, 20(3), 551-557, https://doi.org/10.30955/gnj.002640
Yakubu, Y., Chukwu, A. U., Adebayo, B. T. &
Nwanzo, A. G. (2014). Effects of Missing Observations on Predictive Capability
of Central Composite Designs. International Journal
on Computational Sciences & Applications (IJCSA), 4(6),
https://doi.org/10.5121/ijcsa.2014.4601.
Yakubu, Y. & Chukwu, A. U. (2018). Split-Plot
Central Composite Designs Robust to a Pair of Missing Observations. Journal of Applied Sciences and
Environmental Management, 22(9), 1409–1415,
https://dx.doi.org/10.4314/jasem.v22i9.08
Zarhan, A. (2002). On the Efficiency of Designs for
Linear Models in Non-Regular Regions and the Use of Standard Designs for
Generalized Linear Models. Unpublished PhD Dissertation, Virginia Polytechnic
Institute and State University.
Authors
G. O. Agadaga
Department of Statistics, Federal Polytechnic Orogun, Nigeria. Email:
[email protected]
How to Cite this Article
Agadaga, G. O.
(2024). Rotatability of Second-Order Response Surface Designs. Journal of Research in Science and
Technology, 5(2), 13-31.
Full-Text (PDF)
View / DownloadBack to Articles
Back to Journals
ISSN: 2971-7728 (Online)
DOI:
Start Year: 2020
Published Articles: