Statistical Applications for Environmental Analysis and Risk Assessment

Chapter 18Nonlinear Regression

18.1 Introduction and Overview

Although the methods of linear regression are well documented, easily applied, and relatively straightforward to interpret, real-world data are often in a form that is unsuitable for linear regression. A commonly used remedy for nonlinear data is data transformation, involving alteration of the data values by applying a nonlinear transformation such as the logarithm (i.e., the data values are replaced by their logarithms), which may be effective in rendering the data suitable for linear regression. Data transformation is described in Sections 13.2 and 13.4. An obvious disadvantage of this approach is that it is not always possible to find the appropriate data transformation and even if found, interpretation of the subsequent regression results is no longer as straightforward, due to the alteration of the data structure resulting from the transformation.

Another disadvantage of data transformation is that it may result in biased predictions of the response (i.e., Y) variable. This so-called transformation bias occurs when the transformation involves the Y variable, in which case a reverse transformation (i.e., back-transformation) is needed after computing the regression, in order to obtain Y in the usual arithmetic scale. It turns out that what is recovered by back-transformation is usually the median value of Y rather than the mean value as would ordinarily be expected, and the median is in general not equal to ...