Selection of Second Order Models’ Design Using D-, A-, E-, T- Optimality Criteria

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Wangui Patrick Mwangi
Ayubu Anapapa
Argwings Otieno


There are numerous designs for fitting second order models that can be used in conjunction with the response surface methodology (RSM) technique in optimization processes, be it in agriculture, industries and so on. Some of the designs include the equiradial, Notz, San Cristobal, Koshal, Hoke, Central Composite and Factorial designs. However, RSM can only be applied in conjunction with a single design at a time. This research aimed at choosing a design out of the most widely employed designs for fitting 2nd order models involving 3 factors for optimization of French beans in conjunction with the RSM technique. The most commonly used designs for second order models were first identified as Box-Behnken designs, Hoke D2 and Hoke D6 designs, 3k factorial designs, CCD face centred, CCD rotatable and CCD spherical. Design matrices for these 7 designs were formed and augmented with 5 centre points (chosen through lottery methods), and information and optimal design matrices were formed. Then, for each design, the analysis of D-, A- E-, T- optimality (D-Determinant, A-Average Variance, E-Eigen Value and T-Trace) was carried out according to Pukelsheim’s definitions. The results were ranked for each criterion and the ranks corresponding to each design were averaged. The design chosen was Hoke D2 with the least average- 1.75. The Hoke D2 was found to be optimal in minimizing the variance of prediction and the most economical design among the seven. The findings are in agreement with other researchers and scientist that a design may be optimal in one criterion but fails in another criterion. Further, Hoke designs are in the class of the economical designs. It is recommended that more optimality criteria be applied and a wide range of designs be involved to see whether the results would still agree with these findings.

Optimal, D-, A- E-, T- optimality, analysis, Hoke D2.

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How to Cite
Mwangi, W. P., Anapapa, A., & Otieno, A. (2019). Selection of Second Order Models’ Design Using D-, A-, E-, T- Optimality Criteria. Asian Journal of Probability and Statistics, 5(2), 1-15.
Original Research Article


Agricultural Alternatives. Snap Bean Production. PennState- College of Agricultural Sciences, Agrucultural Research and Cooperative Extension; 2002.
(Accessed on 26th April, 2019)

OECD. Common bean (Phaseolus vulgaris). In Safety Assessment of Transgenic Organisms in the Environment, Volume 6: OECD Consensus Documents, OECD Publishing, Paris. 2016;187-198.
(Accessed in April 26, 2019)

Okello JJ, Narrod C, Roy D. Food safety requirements in African green bean exports and their impact on small farmers. International Food Policy Research Institute, Washington DC, USA. IFPRI Discussion Paper 00737. 2007;1-6.

Johnson RT, Montgomery DC. Choice of second-order response surface designs for logistic and Poisson regression models. Int. J. Experimental Design and Process Optimisation. 2009;1(1).

Myers RH, Montgomery DC. Response surface methodology: Process and product optimization using designed experiments. New York: John Wiley & Sons; 1995.

Wondimu W, Tana T. Yield response of common bean (Phaseolus vulgaris L.) varieties to combined application of nitrogen and phosphorus fertilizers at Mechara, Eastern Ethiopia. J Plant Biol Soil Health- Avens Publishing Group. 2017;4(2):7.

Kiptoo GJ, Arunga EE, Kimno SK. Evaluation of French bean (Phaseolus vulgaris L.) varieties for resistance to anthracnose. Journal of Experimental Agriculture International. 2018;27(4):1-7.

Nyasani JO, Meyhöfer R, Subramaniana S, Poehling HM. Effect of intercrops on thrips species composition and population abundance on French beans in Kenya. Entomologia Experimentalis et Applicata. 2012;142:236–246.
DOI: 10.1111/j.1570-7458.2011.01217.x

Gunawan A, Chuin LH. Second order-response surface model for the automated parameter tuning problem. 2014 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM): 9-12 December 2014, Selangor. 2014;652- 656. Research Collection School of Information Systems.

Ferreira SLC, Bruns RE, Ferreira HS, Matos GD, David JM, Brandao GC, Santos WNL. Box-Behnken design: An alternative for the optimization of analytical methods. Analytica Chimica Acta. 2007;597:179–186.

Roquemore KG. Hybrid designs for quadratic response surfaces. Technometrics. 1976;18(4):419-423.
(Retrieved from JSTOR Databases, 9th October, 2018)

Neifar M, Kamoun A, Jaouani A, Ghorbel RE, Chaabouni SE. Application of Asymetrical and Hoke designs for optimization of laccase production by the White-Rot fungus fomes fomentarius in solid state fermentation. Enzyme Research. 2011;12. Article ID: 368525, (SAGE Hindawi Access to Research).
DOI: 10.4061/2011/368525

Penn State Eberly College of Science. STA 503- Design of experiments. Department of Statistics Online Programs; 2018.
(Accessed in October, 2018)

Das A. An introduction to optimality criteria and some results on optimal block design. Design Workshop Lecture Notes ISI, Kolkata. 2002;1-21.

Oyejola BA, Nwanya JC. Selecting the right central composite design. International Journal of Statistics and Applications. 2015;5(1):21-30.

Jacob E, Boon JE. Generating exact D-optimal designs for polynomial models. SpringSim '07. 2007;2.

Xiang Feng W. Optimal designs for segmented polynomial regression models and web-based implementation of optimal design software. The Graduate School at Stony Brook University; 2007.

Pukelsheim F. Optimal design of experiments. John Wiley & Sons, Inc., New York. 1993;Chapter 6:135.

Myers HR, Montgomery DC, Cook CMA. Response surface methodology- process and product optimization using designed experiments. John Wiley & Sons, Inc., Hoboken, New Jersey. 3rd Ed; 2009.

Frank IE, Todeschint R. The data analysis handbook. Data Handling in Science and Technology. 1994;14.