The Response Surface Methodology revisited – comparison of analytical and non-parametric approaches

Przemysław Osocha 1  and Jordan Podgórski 1
  • 1 Cracow University of Technology, Faculty of Mechanical Engineering,, Kraków, Poland

Abstract

Since G.E.P. Box introduced central composite designs in early fifties of 20th century, the classic design of experiments (DoE) utilizes response surface models (RSM), however usually limited to the simple form of low-degree polynomials. In the case of small size datasets, the conformity with the normal distribution has very weak reliability and it leads to very uncertain assessment of a parameter statistical significance. The bootstrap approach appears to be better solution than - theoretically proved but only asymptotically equal - t distribution based evaluation. The authors presents the comparison of the RSM model evaluated by a classic method and bootstrap approach.

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  • Anderson, T.W., Darling, D.A., 1952. Asymptotic theory of certain "goodness- of-fit" criteria based on stochastic processes. Annals of Mathematical Statistics 23, 193-212.

  • Box, G.E.P., Wilson, K.B., 1951. On the Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society B 13, 1-45.

  • Efron, B., 1979. Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics 7, 1-26.

  • Gentle, J.E., Hardle, W.K., 2012. Handbook of Computational Statistics. Springer-Verlag, Berlin-Heidelberg.

  • Kempthorne, O., Hinkelmann, K., 2007. Design and analysis of experiments. Vol.1. Introduction to experimental design. John Wiley & Sons, Hoboken.

  • Kiefer, J., Wolfowitz, J., 1952. Stochastic Estimation of the Maximum of a Regression Function. The Annals of Mathematical Statistics 23, 462-466.

  • Kolmogorov, A., 1933. Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Istituto Italiano degli Attuari 4, 83-91.

  • Osocha, P., Ulewicz, R., Szataniak, P., Pietraszek, M., Kołomycki, M., Dwornicka, R., 2015. The empirical assessment of the convergence rate for the bootstrap estimation in design of experiment approach. Solid State Phenomena 235, 16-23.

  • Pietraszek, J., Wojnar, L., 2016. The bootstrap approach to the statistical significance of parameters in RSM model. ECCOMAS 2016 European Congress on Computational Methods in Applied Sciences and Engineering, Hersonissos, Crete, vol.1, 2003-2009.

  • Quenouille, M.H., 1949. Problems in Plane Sampling. The Annals of Mathematical Statistics 20, 355-375.

  • Robbins, H., Monro, S., 1951. A Stochastic Approximation Method. The Annals of Mathematical Statistics 22, 400-407.

  • Shao, J., Tu, D., 1995. The Jackknife and Bootstrap. Springer, New York.

  • Shapiro, S.S., Wilk, M.B., 1965. An Analysis of Variance Test for Normality. Biometrika 52, 591-611.

  • Smirnov, N., 1948. Table for estimating the goodness of fit of empirical distributions. Annals of Mathematical Statistics 19, 279-281.

  • Tukey, J.W., 1958. Bias and confidence in not quite large samples (abstract). The Annals of Mathematical Statistics 29, 614-614.

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