Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Response surface methodology considering Poisson and Weibull regression models: a case study


This paper presents new statistical modeling under a Bayesian approach to analyze response surfaces in industrial experiments where the responses are given by both count data and lifetime data. As a motivation and application of the proposed methodology, we analyzed a data set related to manufacturing of 304 stainless steel components of medical tools in a metal industry in Ribeirão Preto, São Paulo State, Brazil. To manufacture batches of these components of medical tools, a cutting tool of the manufacture machine is used until there is failure of this tool that should be replaced by a new one. A first goal of this industrial sector is to identify possible factors that affect the number of manufactured components of medical tools until cutting tool failure and the manufacturing time of each unit. Multivariate linear regression models for discrete and lifetime data under a Bayesian approach were used for this purpose. The final goal of this study was to determine the optimal levels of the factors which maximize the number of manufactured components of medical tools in each batch and minimize the manufacturing time using response surface methodology.

This is a preview of subscription content, log in to check access.


  1. 1.

    Achcar JA, Brookmeyer RS, Hunter WG (1985) An application of bayesian analysis to medical follow-up data. Stat Med 4:509–520. doi:10.1002/sim.4780040411

  2. 2.

    Achcar JA, Piratelli CL, Souza RM (2013) Modeling quality control data using Weibull distributions in the presence of a change point. Int J Adv Manuf Technol 66:1611–1621. doi:10.1007/s00170-012-4444-1

  3. 3.

    Albert JH, Chib S (1993) Bayesian analysis of binary and polychotonious response data. J Am Statist Assoc 88:669–679

  4. 4.

    Anderson MJ (2004) Whitcomb PJ, Design solutions from concept through manufacture: Response surface methods for process optimization. Desktop Engineering, Accessed 19 June 2014

  5. 5.

    Bernardo JM, Smith AFM (1995) Bayesian theory. Wiley, Hoboken

  6. 6.

    Bertrand JWM, Fransoo JC (2002) Operations management research methodologies using quantitative modeling. Int J Oper Pord Man 22(2):241–261. doi:10.1108/01443570210414338

  7. 7.

    Box GEP, Wilson KB (1951) On the experimental attainment of optimum conditions. J R Stat Soc, Ser B, Methodol 13:1–45

  8. 8.

    Box GEP, Hunter WG, Hunter JS (1978) Statistics for experimenters: an introduction to design. Wiley, New York

  9. 9.

    Cameron AC, Trivedi PK (1998) Regression analysis of count data. Cambridge University Press, Cambridge

  10. 10.

    Chib S, Greenberg E (1995) Understanding the metropolis hastings algorithm. Am Stat 49:327–335. doi:10.1080/00031305.1995.10476177

  11. 11.

    Crouchley R, Davies RB (1999) A comparison of population average and random effects models for the analysis of longitudinal count data with baseline information. J R Stat Soc, Ser A 162:331–347

  12. 12.

    Draper NR, Smith H (1981) Applied regression analysis. Wiley series in probability and mathematical statistics. Wiley, New York

  13. 13.

    Dunson DB (2000) Bayesian latent variable models for clustered mixed outcomes. J R Stat Soc, Ser B 62(2):355–366. doi:10.1111/1467-9868.00236

  14. 14.

    Dunson DB (2003) Dynamic latent trait models for multidimensional longitudinal data. J Am Statist Assoc 98:555–563. doi:10.1198/016214503000000387

  15. 15.

    Gelfand AE, Smith AFM (1990) Sampling based approaches to calculating marginal densities. J Am Statist Assoc 85:398–409. doi:10.1080/01621459.1990.10476213

  16. 16.

    Gelman A, Carlin JB, Stern HS, Rubin DB (1995) Bayesian data analysis. Chapman and Hall, London

  17. 17.

    Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov Chain Monte Carlo in practice. Chapman and Hall, London

  18. 18.

    Henderson R, Shimakura S (2003) A serially correlated gamma frailty model for longitudinal count data. Biometrika 90:355–366. doi:10.1093/biomet/90.2.355

  19. 19.

    Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions. Wiley, Hoboken

  20. 20.

    Khuri AI, Cornell JA (1987) Response surfaces: designs and analyses. Marcel Dekker, New York

  21. 21.

    Lawless JF (1982) Statistical models and methods for lifetime data. Wiley, Hoboken

  22. 22.

    Montgomery DC (2009) Design and analysis of experiments. Wiley, New York

  23. 23.

    Montgomery DC, Runger GC (2010) Applied Statistics and Probability for Engineers. Wiley, Hoboken

  24. 24.

    Myers RH (1971) Response surface methodology. Allyn and Bacon, Boston

  25. 25.

    Myers RH, Khuri AI, Carter WHJ (1989) Response surface methodology: 1966–1988. Technometrics 31(2):137–153. doi:10.2307/1268813

  26. 26.

    Myers RH, Montgomery DC (1995) Response surface methodology: process improvement with steepest ascent, the analysis of response surfaces, experimental designs for fitting response surfaces. Hoboken:183–351

  27. 27.

    Nelson W (2004) Applied life data analysis. Wiley, Hoboken

  28. 28.

    Oehlert GW (2000) Design and analysis of experiments: response surface design. W.H. Freeman and Company, New York

  29. 29.

    Robert CP, Casella G (2004) Monte Carlo statistical methods. Springer, New York

  30. 30.

    Rodrigues AR, Coelho RT (2007) Influence of the tool edge geometry on specific cutting energy at high-speed cutting. J Braz Soc Mech Sci & Eng 29(3):279–283. doi:10.1590/S1678-58782007000300007

  31. 31.

    Ronald C (1997) Log-linear models and logistic regression. Springer texts in statistics. Springer, New York

  32. 32.

    Seber GAF, Lee AJ (2003) Linear regression analysis. Wiley series in probability and mathematical statistics. Wiley, New York

  33. 33.

    Shi Y, Weimer PJ (1992) Response surface analysis of the effects of pH and Dilution rate on ruminococcus flavefaciens FD-1 in cellulose-fed continues Culture. Am Soc Microbiol 58(8):2583–2591. doi:10.2307/1268813

  34. 34.

    Spiegelhalter DJ, Thomas A, Best NG, Lund D (2003) Winbugs user manual. MRC Biostatistics Unit, Cambridge

  35. 35.

    Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18(3):293–297

  36. 36.

    Wu CF, Hamada M (2000) Experiments: planning, analysis, and parameter design optimization. Wiley Interscience, New York

Download references

Author information

Correspondence to Roberto Molina de Souza.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Achcar, J.A., Faria, R.F. & de Souza, R. Response surface methodology considering Poisson and Weibull regression models: a case study. Int J Adv Manuf Technol 77, 1867–1879 (2015).

Download citation


  • Poisson regression model
  • Weibull regression model
  • Response surface analysis
  • Bayesian methods