Sequential design augmentation with model misspecification

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Virginia Tech

In Response Surface Methodology (RSM) one attempts to model some variable of interest, usually as a known function of design variables. Subsequent analysis often indicates a need to move to a new region of interest. Many times the design is augmented by adding points sequentially to this new region of interest. Current methods of sequential design augmentation are used under the assumption of either correctly specified models or misspecification in the user’s model that can be quantified, such as using a first order model when a second order model is correct. However, under model misspecification the sequential placement of points in the new region of interest using usual augmentation techniques may not be optimal, especially if the misspecification in the model is not due to polynomial terms.

A new methodology, based on a modified kernel regression procedure called HATLINK, is presented that incorporates model misspecification into the sequential augmentation of points in the new region. HATLINK is a combination of parametric and nonparametric regressions and is designed to perform best when the user has specified a reasonable approximate model. Parametric regression supplies a basic fit, while nonparametric regression allows adjustments to compensate for some misspecification in the parametric model. The mixing parameter is determined adaptively through cross-validation. The augmentation is performed by a new technique called BIIV, the bias-influenced integrated prediction variance. BIIV attempts to select points that both minimizes the integrated prediction variance and the location where the current fit is the worst. Thus, BIIV incorporates an estimate of the bias due to misspecification of the parametric model into the augmentation procedure. It is shown that the designs generated by sequential design augmentation using HATLINK and BIIV are superior to designs from other methods.