The bootstrap approach to the statistical significance of parameters in RSM model

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 only in the simple form of low-degree polynomials. The typical procedure assumes the normal distribution of the noise and uses the least square method (LSQ) to identify parameters of the model with a priori assumed structure. The terms of model are repeatedly eliminated in the specific backward stepwise regression, while three indicators (the least significance of parameters, the significance of the lack of fit and the conformity of residuals with the normal distribution) are simultaneously observed to make decision to stop or to continue elimination procedure. Practically, 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 parameters statistical significance. The bootstrap approach with simulation-based identification of parameters confidence intervals (CI) appears to be better solution than theoretically proved but only asymptotically equal t-distribution based evaluations. The case study presented in this paper utilizes data obtained during investigation on compression vertebral fractures prediction based on computer tomography (CT) and microtomography (μCT) images. The significant difference in a resolution between these two class of images leaded to different prediction models. The small sample size (23 compressed and scanned vertebraes) and the high dimensionality of detected properties imposed the necessity of an alternative approach to the analysis, other than classic one derived with a requirement of the normality. Jacek Pietraszek, Leszek Wojnar