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.


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.




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,

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,

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;

Bhattacharya, S. (2021). Central Composite Design for Response Surface Methodology and Its Application in Pharmacy. In Book: Response Surface Methodology in Engineering Science. IntechOpen,

Box, G. E. P. & Behnken, D. (1960). Some New Three Level Designs for the Study of Quantitative Variables, Technometrics, 2, 455-475,

Box, G. E. P. & Hunter, J. S. (1957). Multi-Factor Experimental Designs for Exploring Response Surfaces. The Annals of Mathematical Statistics, 28, 195-241,

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.

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,

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,

Chiranjeevi, P. & Victorbabu, B. Re (2021). Construction of Second Order Slope Rotatable Designs Using Supplementary Difference Sets. Thailand Statistician, 19(2), 261-269,

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;

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,

Edmondson, R. N. (1991). Agricultural Response Surface Experiments Based on Four-Level Factorial Designs. Biometrics, 47(4), 1435–1448.

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,

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,

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.

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.

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,

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,

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,

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,

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,

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.

Oyejola, B. A. & Nwanya, J. C. (2015). Selecting the Right Central Composite Design. International Journal of Statistics and Applications, 5(1), 21-30,

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,

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.

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,

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,

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;

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,

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),

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,

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.





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.


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Abbreviation: J. Res. Sci. Technol.
ISSN: 2971-7728 (Online)
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G. O. Agadaga