On the use of nonparametric regression model in Response Surface Methodology

Abstract

The modeling phase of response surface methodology (RSM) involves the use of regression models to estimate the functional relationship between the response and the explanatory variables using data obtained from a suitable experimental design. In RSM, the Ordinary Least Squares (OLS) is traditionally used to model the data via user-specified low-order polynomials. The OLS model is found to perform poorly if the constant variance (homoscedasticity) assumption is violated. Additionally, the speci-fied polynomials are usually found inadequate for the data. The problems resulting from model inadequacy include biased estimates of the mean response function. Recently, nonparametric regression model, such as the Local Linear Regression (LLR), has been proposed to address the model inadequacy issue associated with the use of the OLS model. The LLR model is flexible, hence, can capture local trend and structure in the data that are misspecified by an inadequate OLS model. The successful application of the LLR model has been limited to studies with three unique features, namely: a single explanatory variable, fairly large sample sizes and space-filling designs. Therefore, the LLR model is scantily used in RSM which general underpinning include economy of data points (small sample size), typically sparse data, and oftentimes,
more than one explanatory variables.In this paper, we propose a new nonparametric regression models that incorporate the smoothing of residuals to provide a second opportunity of fitting part of the data that is not captured by the LLR model. Using an example from RSM literature, it is observed that the goodness-of-fits of the proposed model are considerably better when compared with those of the OLS and the LLR models.