Evaluation of Second-Order Response Designs with Errors in Factor Levels in the Cuboidal Region

Document Type : Original Article

Authors
Ifeoma Ogochukwu Ude 1
Abimibola Victoria Oladugba 1
Amarachi Favour Didiugwu 1
Lilian Ifeoma Nnanna 1, 2
1 Department of Statistics, University of Nigeria, Nsukka, Enugu State, Nigeria.
2 Department of Statistics, University of Regina, Regina, Canada.
10.22034/jirss.2025.2049136.1090
Abstract
Errors in factor levels often occur in response surface modeling. A design in which these errors have minimal effect is the desired design. This study evaluates the prediction capability of second-order Orthogonal Array Composite Design (OACD) and Orthogonal Uniform Composite Design (OUCD) with and without errors in factor levels for 3 ≤ k ≤ 5 factors using 2, 3, and 5 center points in the cuboidal region. Design optimality criteria (in terms of G- and IV-optimality values) and quantile dispersion plots are used to examine the prediction capability of these designs. The results show that OUCD is the preferred design in terms of G-optimality, while IV-optimality and quantile plots indicate that OACD is the preferred design in both the presence and absence of errors in factor levels.
Keywords
Response surface designs
Errors in factor levels
Predictive variance
Design optimality criteria
Quantile dispersion plot
Subjects
Ardakani MK. The impacts of errors in factor levels on robust parameter design optimization. Quality and Reliability Engineering International. 2016;32(5):1929– 1944. https://doi.org/10.1002/qre.1923. 
Ardakani MK, Das D, Wulff SS, Robinson TJ. Estimation in second-order models with errors in the factor levels. Communications in Statistics – Theory and Methods. 2011;40(9):1573–1590. https://doi.org/10.1080/03610921003689490.
Box GEP. The effects of errors in the factor levels and experimental design. Technometrics. 1963;5(2):247–262. https://doi.org/10.1080/00401706.1963.10490172.
Didiugwu AF, Oladugba AV, Ude OI. Effects of error in factor levels on orthogonal composite designs for second-order models. Communications in Statistics – Theory and Methods. 2025;54(5):1473–1491. https://doi.org/10.1080/03610926.2024. 2342943.
Donev AN. Dealing with errors in variables in response surface exploration. Communications in Statistics – Theory and Methods. 2000;29(9–10):2065–2077. https://doi.org/10.1080/03610920008832561.
Donev AN. Design of experiments in the presence of errors in factor levels. Journal of Statistical Planning and Inference. 2004;126(2):569–585. https://doi.org/10.1016/ S0378-3758(03)00161-7.
Draper NR, Beggs WJ. Errors in the factor levels and experimental design. The Annals of Mathematical Statistics. 1971;41(1):46–58. https://doi.org/10.1214/aoms/ 1177693509.
Fang J, He Z, He S, Wang G. The robustness of response surface designs with errors in factor levels. Communications in Statistics – Theory and Methods. 2019;https://doi.org/10.1080/03610926.2019.1576883.
Fang J, He Z, Song L. Evaluation of response surface designs in presence of errors in factor levels. Communications in Statistics – Theory and Methods. 2015;44(18):3769–3781. https://doi.org/10.1080/03610926.2013.858166.
Fedorov VV. Regression problems with controllable variables subject to error. Biometrika. 1974;61(1):49–56. https://doi.org/10.1093/biomet/61.1.49.
He Z, Fang J. Comparative study of response surface designs with errors-in-variables model. Transactions of Tianjin University. 2011;17(2):146–150. https://doi.org/10.1007/s12209-011-1605-5.
Khuri AI, Kim HJ, Um Y. Quantile plots of the prediction variance for response surface designs. Computational Statistics & Data Analysis. 1996;22:395–407. https://doi.org/10.1016/0167-9473(95)00046-1.
Loken E, Gelman A. Measurement error and the replication crisis. Science. 2017;355(6325):584–585. https://doi.org/10.1126/science.aal3618. 
Myers RH, Montgomery DC, Anderson-Cook CM. Process and Product Optimization Using Designed Experiments. 4 ed. Hoboken, New Jersey: John Wiley and Sons; 2016.
Vuchkov IN, Boyadjieva LN. The robustness of experimental designs against errors in the factor levels. Journal of Statistical Computation and Simulation. 1983;17(1):31–41. https://doi.org/10.1080/00949658308810607.
Winker P, Lin KJ. Robust uniform design with errors in the design variables. Statistica Sinica. 2011;21(3):1379–1396. https://doi.org/https://www.jstor.org/stable/ 24309566.
Wu F, Wang J, Ma Y. Robust parameter design based on response surface model considering measurement error. Scientia Iranica. 2021;28:2323–2332. https://doi.org/10.24200/sci.2020.54054.3062.
Xu H, Jaynes J, Ding X. Combining two-level and three-level orthogonal arrays for factor screening and response surface exploration. Statistica Sinica. 2014;24(1):269–289. https://doi.org/https://www.jstor.org/stable/26432543.
Zhang XR, Liu MQ, Zhou YD. Orthogonal uniform composite designs. Journal of Statistical Planning and Inference. 2020;206:100–110. https://doi.org/10.1016/j. jspi.2019.08.007.
Zhou YD, Xu H. Composite designs based on orthogonal arrays and definitive screening designs. Journal of the American Statistical Association. 2016;112(520):1675–1683. https://doi.org/10.1080/01621459.2016.1224718. 
Journal of the Iranian Statistical Society
Volume 24, Issue 1
June 2025
Pages 157-169

  • Receive Date 28 December 2024
  • Accept Date 12 October 2025