Evaluation of the Efficiency of Third-Order Response Surface Designs in the Spherical Region

Articles in Press

Document Type : Original Article

Authors
Ifeoma Ogochukwu Ude 1
Abimibola Victoria Oladugba 1
Oluwagbenga Tobi Babatunde 2
1 Department of Statistics, University of Nigeria, Nsukka
2 Department of Statistics, University of Nigeria, Nsukka; School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia.
10.22034/jirss.2026.2080191.1168
Abstract
Third-order response surface designs (RSDs) are essential when second-order models fail to capture higher-order curvature or complex prediction structures. Although several third-order designs have been proposed, comparative evaluations across multiple efficiency criteria remain limited. This study evaluates the efficiencies of five third-order RSDs: augmented Box-Behnken designs (ABBDs), augmented fractional Box-Behnken designs (AFBBDs), new augmented Box-Behnken designs (NABBDs), orthogonal array composite designs (OACD4), and orthogonal uniform composite designs (OUCD4) in the spherical region. These designs were assessed using experimental cost efficiencies (run-size and degree-of-freedom), parameter estimation efficiencies (D- and A-efficiency), and prediction accuracy efficiencies (G-, I-, T-, and per-observation G-efficiency) for 3 ≤ k ≤ 6 factors using 0 ≤ n₀ ≤ 5 center points. The results show that increasing the number of factors and center points raises run size, D-efficiency, and A-efficiency but reduces degree-of-freedom, G-, I-, and per-observation G-efficiencies. OACD4 and OUCD4 were the most economical; NABBDs and OUCD4 showed the highest D- and A-efficiency values, respectively; and OACD4 showed the best prediction performance based on G- and per-observation G-efficiencies. Overall, OACD4 was the most efficient design across most criteria.
Keywords
Third-order designs
Box-Behnken designs
Orthogonal composite designs
Prediction variance
Design efficiency
Subjects
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Journal of the Iranian Statistical Society

Articles in Press, Accepted Manuscript
Available Online from 20 August 2026

  • Receive Date 07 December 2025
  • Revise Date 29 June 2026
  • Accept Date 16 July 2026